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The document is a pre-publication version of 'Introduction to Schemes' by Geir Ellingsrud and John Christian Ottem, set to be published by Cambridge University Press. It covers various topics related to algebraic geometry, including algebraic sets, schemes, sheaves, and morphisms, organized into multiple parts with exercises included. The material is intended for personal use only and is not for redistribution or resale.

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Introduction to Schemes

Geir Ellingsrud and John Christian Ottem

April 15, 2025

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Acknowledgements 1
Preface 2
Notation 3

1 Algebraic sets 4
1.1 Algebraic sets 4
1.2 Hilbert’s Nullstellensatz 6
1.3 The structure of algebraic sets 8
1.4 Dimension 10
1.5 Regular and rational functions 11
1.6 Morphisms of affine varieties 14
1.7 Conclusion 17
1.8 Exercises 18

Part I Schemes 25

2 The spectrum of a ring 27

2.1 The spectrum of a ring 27
2.2 Affine spaces 32
2.3 Distinguished open sets 33
2.4 Irreducibility and connectedness 35
2.5 Maps between prime spectra 36
2.6 Fibers 39
2.7 Exercises 43

3 Sheaves 48
3.1 Sheaves and presheaves 48
3.2 Stalks 53
3.3 The pushforward of a sheaf 54
3.4 Sheaves defined on a basis 55
3.5 Exercises 58

4 Schemes 61
4.1 The structure sheaf on the spectrum of a ring 62
4.2 The sheaf associated to an A-module 65
4.3 Locally ringed spaces 66

iii

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4.4 Schemes 69
4.5 Morphisms into an affine scheme 70
4.6 Open embeddings and open subschemes 74
4.7 Closed embeddings and closed subschemes 75
4.8 Relative schemes 77
4.9 R-valued points 78
4.10 Affine varieties as schemes 81
4.11 Gluing two schemes together 83
4.12 Gluing sheaves 84
4.13 Gluing schemes 87
4.14 Exercises 91

5 Projective space 94
5.1 Projective space 94
5.2 The Proj construction 103
5.3 Functoriality 108
5.4 Exercises 115

6 More examples 120

6.1 Toric varieties 120
6.2 The blow-up of the affine plane 126
6.3 Line bundles on P1 129
6.4 Double covers 133
6.5 Exercises 135

Part II Basics of scheme theory 139

7 Properties of schemes 141

7.1 Reduced schemes 141
7.2 Integral schemes 142
7.3 Function fields 143
7.4 Dominant morphisms 145
7.5 Noetherian schemes 146
7.6 The dimension of a scheme 150
7.7 Exercises 153

8 Fiber products 158

8.1 Introduction 158
8.2 Fiber products of schemes 160
8.3 Examples 164
8.4 Base change 165
8.5 Scheme-theoretic fibers 168
8.6 The Segre embedding 169
8.7 Exercises 172

9 Morphisms of schemes 175

9.1 Properties of affine open subsets 175

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9.2 Morphisms of finite type 176

9.3 Affine morphisms 178
9.4 Closed embeddings 179
9.5 Finite morphisms 180
9.6 Separated schemes 185
9.7 Properties of separated schemes 188
9.8 Exercises 191
9.9 Exercises 194

10 Schemes of finite type over a field 199

10.1 The general definition of a variety 199
10.2 Schemes of finite type over a field 200
10.3 Dimension theory for schemes of finite type over a field 201
10.4 Products of varieties 202
10.5 A structure result for morphisms 203
10.6 The dimensions of the fibers of a morphism 203
10.7 Applications to intersections 205
10.8 Rational maps 207
10.9 Exercises 211

11 Local properties of schemes 217

11.1 Tangent spaces 217
11.2 Nonsingular schemes 222
11.3 The Jacobian criterion and smoothness 224
11.4 Normal schemes 228
11.5 Properties of normal schemes 233
11.6 Exercises 237

12 The functor of points 240

12.1 The functor of points 240
12.2 The fiber product in terms of the functor of points 245

Part III Quasi-coherent sheaves 247

13 More on sheaves 249

13.1 Kernels and images 249
13.2 Injective and surjective maps of sheaves 251
13.3 Exact sequences 253
13.4 The sheaf associated to a presheaf 254
13.5 Cokernels and quotients 257
13.6 The inverse image sheaf 257
13.7 Exercises 262

14 Quasi-coherent sheaves 265

14.1 Sheaves of modules 265
14.2 The tilde of a module 266
14.3 Quasi-coherent sheaves 268

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14.4 Kernels, Images and Cokernels 272

14.5 Direct sums and products 273
14.6 Tensor products 274
14.7 Finiteness conditions for OX -modules 276
14.8 The Hom-sheaf 277
14.9 Pushforwards 278
14.10 Pullbacks 280
14.11 Closed subschemes and closed embeddings 286
14.12 Exercises 288

15 Locally free sheaves 297

15.1 Basic definitions 297
15.2 Examples 298
15.3 Locally free sheaves and stalks 300
15.4 Operations on locally free sheaves 302
15.5 Locally free sheaves on P1 304
15.6 Zero sets of sections 305
15.7 Exercises 306

16 Invertible sheaves and projective space 311

16.1 Invertible sheaves and the Picard group 311
16.2 The graded tilde-functor 312
16.3 Serre’s twisting sheaf Op1q 316
16.4 The associated graded module 317
16.5 Closed subschemes of Proj R 322
16.6 Sheaves on projective space 324
16.7 Globally generated sheaves 326
16.8 Morphisms to projective space 327
16.9 Exercises 331

17 Divisors 333
17.1 Weil divisors 333
17.2 Cartier divisors 336
17.3 The class group 338
17.4 The sheaf associated to a Weil divisor 342
17.5 The divisor associated to a section of an invertible sheaf 346
17.6 Linear systems 350
17.7 Pullbacks of divisors 350
17.8 The class group of an open set 351
17.9 Quadrics 353
17.10 The 3-dimensional quadratic cone 356
17.11 Exercises 358

18 First steps in sheaf cohomology 361

18.1 Some homological algebra 362
18.2 Cech cohomology 363
18.3 Examples 366
18.4 Cech cohomology on schemes 370

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18.5 Cohomology of sheaves on affine schemes 370

18.6 Independence of the cover 372
18.7 Pushforwards and cohomology 374
18.8 Cohomology and dimension 374
18.9 Cohomology of sheaves on projective space 375
18.10 Cohomology of sheaves on projective schemes 379
18.11 Example: Plane curves 381
18.12 Example: The twisted cubic 383
18.13 Example: Hyperelliptic curves 383
18.14 Example: Complete intersections 384
18.15 Example: Bezout’s theorem 385
18.16 Example: Interpolation problems 386
18.17 Example: Non-split locally free sheaves 387
18.18 Example: The rational quartic curve 388
18.19 Exercises 389

19 Proper and projective morphisms 392

19.1 Proper morphisms 392
19.2 Projective morphisms 394
19.3 Blow-ups 397

20 Differentials 401
20.1 Derivations 401
20.2 Kähler differentials 402
20.3 Properties of differentials 404
20.4 The sheaf of differentials 406
20.5 Examples 407
20.6 The Euler sequence and differentials of Pnk 410
20.7 Nonsingularity and smoothness 411
20.8 The Tangent sheaf 413
20.9 The sheaf of p-forms 413
20.10 Application: irrationality of hypersurfaces 415
20.11 Exercises 417

Part IV Curves 421

21 Curves 423
21.1 Morphisms between projective curves 424
21.2 Extensions of rational maps 425
21.3 Sheaves on curves 428
21.4 Divisors on curves 429
21.5 The genus of a curve 435
21.6 Hyperelliptic curves 435
21.7 Exercises 439

22 The Riemann–Roch theorem 440

22.1 The Riemann–Roch formula 440

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22.2 Proof of Serre duality 444

22.3 The dualizing sheaf 445
22.4 The dualizing sheaf is the canonical sheaf 447
22.5 Exercises 449

23 Applications of the Riemann–Roch theorem 452

23.1 A criterion for being a closed embedding 452
23.2 Ample invertible sheaves and Serre’s theorems 453
23.3 Base-point freeness and very ampleness on curves 454
23.4 Curves of genus 0 456
23.5 Curves of genus 1 458
23.6 Hyperelliptic curves 460
23.7 Curves of genus 2 462
23.8 Curves of genus 3 462
23.9 Curves of genus 4 464
23.10 Exercises 466

Appendix A Some results from Commutative Algebra 469

Appendix B Further constructions and examples 498

Bibliography 513

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Thanks to Georges Elencwajg, Frank Gounelas, Johannes Nicaise, Dan Petersen, Kristian
Ranestad, Stefan Schreieder and Jørgen Vold Rennemo for comments and suggestions. Also,
thanks to Edvard Aksnes, Shamil Asgarli, Valentine Blanpain, Anne Brugård, Anakin Dey,
Søren Gammelgaard, Elias Giraud-Audine, Samit Ghosh, Kai Komori, Timo Kränzle, Alek-
sander Leraand, Simen Westbye Moe, Torger Olsson, Nikolai Thode Opdan, Erik Paemurru,
Gabriel Ribeiro, Arne Olav Vik, Magnus Vodrup, Xiangzhuo Zeng and Qi Zhu for numerous
corrections to the text. A special thanks goes to Leandros Emmanuel de Jonge for his careful
reading and helpful comments on the manuscript.

Any comments, suggestions, or corrections are welcome:

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edit?usp=sharing

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N “ t1, 2, 3, . . . u is the set of positive integers.

Ă means ‘is subset of’, i.e., the same thing as Ď.

If X and Y are two sets, then X ´ Y denotes the set of elements in X which are not in Y .

All rings are commutative with 1.

Ring maps are required to send 1 to 1.

The zero ring is not an integral domain (and hence not a field).

For a ring A, we write Ap and Af for the localizations in the multiplicative sets S “ A ´ p
and S “ t1, f, f 2 , . . . u respectively. Thus, Zppq “ t ab | p ∤ bu and Zp “ Zr p1 s.

The fraction field of an integral domain A is denoted by KpAq.

A ‘map’ is a morphism in the relevant category, e.g., a ‘map of rings’ is ring homomorphism.

We will occasionally write A “ B if there is a canonical isomorphism A » B . So for

instance, Z bZ Z “ Z.

If C is a category, we denote its opposite category by C op . This category has the same objects
as C , but for any pair of objects, the morphisms from X to Y in C op correspond to the
morphisms from Y to X in C .

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Algebraic sets

In this chapter, we discuss algebraic sets and affine varieties, which will serve as the main
motivating examples in the theory of schemes. We will confine ourself to the basic definitions,
examine a few key examples, and establish notation for later chapters. As we progress through
the book, we will develop the theory of varieties in greater depth.

1.1 Algebraic sets

Algebraic sets and affine varieties are defined over a fixed ground field k . In this chapter, we
will assume that k is algebraically closed. It is useful to keep some specific fields in mind, e.g.
the field of complex numbers C, the field of algebraic numbers Q, s or perhaps the algebraic
closure of a finite field, Fp .
s
We will denote the set k n by An pkq, and refer to it as the affine n-space. The change in
notation is meant to underline that there is more to An pkq than just the set of its elements; it
will soon be equipped with a topology, and ultimately, it will be a scheme.

Definition 1.1. For a subset S of krx1 , . . . , xn s, we define its zero set as

ZpSq “ t p P An pkq | f ppq “ 0 for all f P S u.
An algebraic set is a subset of An pkq of this form.
řr
If f1 , . . . , fr P S , then every polynomial of the form i“1 bi fi , with the bi ’s being
polynomials, also vanishes at the points of ZpSq. This means that the zero set of the ideal a
generated by the elements of S is the same as ZpSq; that is, ZpSq “ Zpaq. We will therefore
almost exclusively work with ideals and tacitly replace a set of polynomials by the ideal they
generate. Hilbert’s basis theorem (Theorem A.21) tells us that any ideal in krx1 , . . . , xn s is
finitely generated, so an algebraic subset is always described as the set of common zeros of
finitely many polynomials.
The more constraints imposed, the smaller the solution set will be, so if a and b are two
ideals with a Ă b, one has Zpbq Ă Zpaq. The assignment a ÞÑ Zpaq therefore defines
an inclusion-reversing map from the partially ordered set of ideals in krx1 , . . . , xn s to the
partially ordered set of subsets of An pkq.
Under this correspondence, different ideals can define the same algebraic set. For instance,
the ideals pxq and px2 q in krxs, both have the origin in A1 pkq as their zero set. To deal with

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Proposition 1.2. Let a and b be two ideals and tai uiPI a collection of ideals in the
polynomial ring krx1 , . . . , xn s. Then:
(i) If ařĂ b, then Zp
Ş bq Ă Zpaq
(ii) Zp iPI ai q “ iPI Zpai q
(iii) Zpabq “ Zp ?a X bq “ Zpaq Y Zpbq
(iv) Zpaq “ Zp aq.
Ş
Proof We have already proved (i) and (iv). For item (ii): p P iPI Zpai q if andŤ only if
f ppq “
ř 0 for all f P a i and all i P I . This
Ť is equivalent to f ppq “ 0 for all f P iPI ai .
Since iPI ai is the ideal generated by iPI ai , so (ii) follows.
For (iii), we have ab Ă a X b and a X b is contained in both a and b. This shows that
Zpaq Y Zpbq Ă Zpa X bq Ă Zpabq. Conversely, if p R Zpaq Y Zpbq, then there exist f P a
and g P b so that f ppq ‰ 0 and gppq ‰ 0. But then pf gqppq ‰ 0, and hence p R Zpabq.
The identities (ii) and (iii) tell us that finite unions and arbitrary intersections of algebraic
sets are again algebraic. Furthermore, as An pkq “ Zp0q and H “ Zp1q, the algebraic sets
constitute the closed sets of a topology on the affine space An pkq. It is called the Zariski
topology.
If X Ă An pkq is any subset, we get an induced Zariski topology on X by declaring that
the closed sets of X are of the form X X W , where W is a closed set in An pkq.

Examples
Example 1.3. The affine space A1 pkq is called the affine line. Every proper nonzero ideal in
krts is principal, generated by some polynomial f . Since k is algebraically closed, f factors
as a product pt ´ a1 q ¨ ¨ ¨ pt ´ ar q, and its zero set Zpf q is exactly the finite set of roots
ta1 , ..., ar u. Conversely, any finite set of points arises this way. Hence, the closed subsets of
A1 pkq are precisely the finite sets and the whole space.
This means that the Zariski topology on A1 pCq behaves very differently from the usual
topology on C. For instance, A1 pCq is not Hausdorff - any two non-empty open sets must
intersect. △
Example 1.4 (Plane conics). Plane conics are classical examples of algebraic sets. They
are defined by a single quadratic equation in A2 pkq. Three familiar examples include (i) the
circle x2 ` y 2 ´ 1 “ 0; (ii) the parabola y ´ x2 “ 0; and (iii) the hyperbola xy ´ 1 “ 0.
Note also the conic x2 ` y 2 ` 1 “ 0, which has no solutions when say k “ R. However,
over C, it becomes equivalent to xy ´ 1 “ 0, by the change of variables u “ x ` iy ,

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v “ ´x ` iy . In fact, when k is algebraically closed of characteristic ‰ 2, any conic can

be transformed via a linear change of coordinates to one of the types x2 ` y 2 ` 1 “ 0,
y ` x2 “ 0, x2 “ 0, xy “ 0 and x2 ` 1 “ 0. △
Example 1.5. A more interesting example is the so-called Clebsch cubic surface; the alge-
braic set in A3 pCq defined by the equation
x3 ` y 3 ` z 3 ` 1 “ px ` y ` z ` 1q3 .
The real points of the surface, i.e. the points in A3 pRq satisfying the equation, is depicted
below. This surface contains 27 lines, all defined over the real numbers. △

Figure 1.1 The Clebsch cubic surface

1.2 Hilbert’s Nullstellensatz

n
For a subset V Ă A pkq, we can consider the set of all polynomials which vanish at every
point of V :
IpV q “ t f P krx1 , . . . , xn s | f ppq “ 0 for all p P V u. (1.1)
Note that IpV q is naturally a radical ideal of krx1 , . . . , xn s. Moreover, if W Ă V , then
IpV q Ă IpW q.
We thereby have two inclusion-reversing mappings betwen the sets of radical ideals a
and the sets of closed sets of An pkq: a ÞÑ Zpaq and X ÞÑ IpXq. The precise relationship
between them is explained by Hilbert’s Nullstellensatz:

Theorem 1.6 (Hilbert’s Nullstellensatz). Let k be an algebraically closed field. Then

the map
␣ (
radical ideals a Ă krx1 , . . . , xn s ÝÝÑ tclosed sets X Ă An pkqu (1.2)
defined by a ÞÑ Zpaq is a bijection, with inverse defined by X ÞÑ IpXq.

To prove ?the theorem, we have to show that ZpIpXqq “ X for an algebraic set X and
IpZpaqq “ a for any ideal a. Three of the four required inclusions are straightforward, and
hold for any field k :
? ?
(i) a Ă IpZpaqq: If f P a, then f r P a for some positive integer r. Therefore,
f r vanishes on Zpaq and hence f also vanishes on Zpaq. Hence f P IpZpaqq.
(ii) X Ă ZpIpXqq: If p P X , then by definition of IpXq, f ppq “ 0 for all
f P IpXq. Therefore, p P ZpIpXqq.

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(iii) ZpIpXqq Ă? X : Write X “ Zpaq for an ideal a. Then by (i), we have

IpZpaqq Ą a Ą a. Taking zero loci, ZpIpZpaqqq Ă Zpaq, or equivalently,
ZpIpXqq Ă X .
Thus
? the if k is algebraically closed, then the inclusion
real content of the Nullstellsatz is that ?
a Ă IpZpaqq is an equality. This means that a is exactly the set of polynomial functions
which vanish on the closed set Zpaq.
We will deduce Theorem 1.6 from the following algebraic result, which is a consequence
of Noether’s Normalization Lemma. (See Theorem A.37 for a proof)

Theorem 1.7. Let k be a field and let K be a finitely generated k -algebra. If K is also a
field, then K is a finite field extension of k .

In particular, if k is algebraically closed, then there are no non-trivial algebraic field

extensions, and so the only possibility is that K “ k .
As a first consequence, we have the following result, which tells us that Zpaq is non-empty
whenever a is a proper ideal:

Theorem 1.8 (Weak Nullstellensatz). Let k be an algebraically closed field.

(i) Every maximal ideal in krx1 , . . . , xn s is of the form
m “ px1 ´ a1 , . . . , xn ´ an q
for some point pa1 , . . . , an q in An pkq.
(ii) For an ideal a, the zero set Zpaq is empty if and only if a “ p1q.

Proof (i): By Theorem 1.7, the field krx1 , . . . , xn s{m is a finite extension of k , hence equal
to k because k is assumed to be algebraically closed. For each i “ 1, . . . , n, let ai P k denote
the image of xi under the quotient map π : krx1 , . . . , xn s Ñ krx1 , . . . , xn s{m “ k . Then
clearly all the polynomials xi ´ ai belong to the kernel m of π . Since px1 ´ a1 , . . . , xn ´ an q
is already a maximal ideal, it must be equal to m.
(ii): If a “ p1q, then clearly Zpaq “ H. Conversely, if a is a proper ideal, there is a
maximal ideal m in krx1 , . . . , xn s containing it. By (i), m has the form px1 ´a1 , . . . , xn ´an q,
and consequently pa1 , . . . , an q P Zpaq, and so Zpaq is not empty.
Proof of Hilbert’s Nullstellensatz To prove the theorem, it is sufficient to prove that for any
ideal a Ă krx1 , . . . , xn s, we have
?
IpZpaqq Ă a.
Let f P IpZpaqq and consider the k -algebra obtained by inverting f and modding out by a:
A “ krx1 , . . . , xn , ts{pa ` ptf ´ 1qq.
If A ‰ 0, then there is a maximal ideal m Ă A. By the Weak Nullstellensatz, m corresponds
to a point pa1 , . . . , an , bq P An`1 pkq such that pa1 , . . . , an q P Zpaq and b ¨ f pa1 , . . . , an q ´
1 “ 0. In particular, f pa1 , . . . , an q ‰ 0, which contradicts the hypothesis f P IpZpaqq.
We deduce that A “ 0. But note that A is isomorphic to the localization krx1 , . . . , xn s{a
2
in the multiplcative set S “ t1, ?f, f , . . . u. Therefore, if A “ 0, then there is some n P N,
n
so that f P a, and hence f P a.

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Example 1.9. It is important to note that Hilbert’s Nullstellensatz only holds when the ground
field is algebraically closed. For instance, consider the ideal a “ px2 ` y 2 ` 1q in Rrx, ys,
corresponding to the conic with no real points from Example 1.4. To?vanish at the empty set
is an empty condition, so IpZpaqq “ p1q, and hence the inclusion a Ă IpZpaqq is strict.
△
Example 1.10. If k “ C, there is a quick way to prove Theorem 1.7 as follows. Suppose
that K “ Crx1 , . . . , xn s{m is a finitely generated C-algebra which is also a field. If there
1
exists an element t P K which is not algebraic over C, the fractions t´a P K for a P C form
an uncountable set of elements in K which are linearly independent over C. This is however
not possible, because K has a countable basis as a C-vector space, as it is generated by the
monomials xe11 ¨ ¨ ¨ xenn .
See also Exercise 1.8.37 for an alternative proof of Theorem 1.7. △
For an algebraic set X Ă An pkq, we define the coordinate ring of X as the quotient
ApXq “ krx1 , . . . , xn s{IpXq.
where IpXq is the ideal of polynomials that vanish on X .
The elements of ApXq can be viewed as polynomial functions on X : any polynomial
in krx1 , . . . , xn s defines a function X Ñ k by restriction, and two polynomials f and g
define the same function on X precisely when their difference f ´ g vanishes on X , that is,
f ´ g P IpXq.

1.3 The structure of algebraic sets

A consequence of the Hilbert’s Nullstellensatz, the geometric properties of algebraic sets can
be translated into algebraic properties of the ideals of the polynomial ring krx1 , . . . , xn s. To
formulate these geometric properties, we recall a few notions from point set topology.
A topological space X is defined to be irreducible if it is nonempty and cannot be written
as the union of two proper closed subsets; that is, if X “ Z Y Z 1 with Z and Z 1 closed, then
either Z “ X or Z 1 “ X .

Proposition 1.11. An algebraic set X Ă An pkq is irreducible if and only if the ideal
IpXq is prime, i.e., ApXq is an integral domain.

Proof Suppose X “ X1 YX2 with X1 and X2 proper closed subsets. Then IpX1 q Ą IpXq
and IpX2 q Ą IpXq are strict inclusions, so we may pick fi P IpXi q ´ IpXq for i “ 1, 2.
Then f1 f2 P IpXq, and hence IpXq is not prime.
Conversely, suppose IpXq is not prime, so there exist f1 , f2 so that fi R IpXq for i “ 1, 2,
but f1 f2 P IpXq. Then X1 “ X X Zpf1 q and X2 “ X X Zpf2 q are closed proper subsets
and X1 Y X2 “ X X Zpf1 f2 q “ X , so X is not irreducible.
The following is a preliminary definition of an affine variety. We will revisit the definition
in Chapter ?? after we have introduced schemes.

Definition 1.12. An affine variety is an irreducible algebraic set in An pkq.

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Example 1.13. The affine space An pkq is irreducible, as krx1 , . . . , xn s is an integral domain.
△
Example 1.14. The closed set Zpxyzq Ă A3 pkq is not irreducible, as it can be written as a
union of three planes ZpxqYZpyqYZpzq. Likewise, Zpxz, yzq “ Zpx, yqYZpzq Ă A3 pkq
is not irreducible, being the union of a plane and a line. △
We say that a topological space X is Noetherian if all descending chains of closed subsets
X0 Ą X1 Ą X2 Ą ¨ ¨ ¨
in X eventually stabilize, that is, there is an integer N such that Xi “ Xi`1 for all i ě N .

Proposition 1.15.
(i) Every algebraic set X is a Noetherian topologial space.
(ii) Every Noetherian topological space has a unique decomposition
X “ X1 Y ¨ ¨ ¨ Y Xr (1.3)
where Xi Ă X is a closed irreducible subset and Xi Ć Xj for all i ‰ j .

Proof (i): If X0 Ą X1 Ą X2 Ą ¨ ¨ ¨ is a chain of closed subsets, then IpX0 q Ą IpX1 q Ą

IpX2 q Ą ¨ ¨ ¨ is a chain of ideals in krx0 , . . . , xn s, which must stabilize because the polyno-
mial ring is Noetherian. Since Xi “ ZpIpXi qq, we see that the chain of closed subsets also
stabilizes.
(ii): Suppose for a contradiction that X does not admit such a decomposition. Then X
cannot be irreducible, so we may write X “ X1 Y X11 where X1 , X11 are proper closed
subsets. As X “ X1 Y X11 , the statement in (ii) must therefore fail for one of X1 or X21 .
Suppose that it fails for X1 . Then again X1 cannot be irreducible, so we write X1 “ X2 YX21 .
Continuing in this way, we obtain an infinite chain of closed subsets X Ą X1 Ą X2 Ą . . . ,
where each inclusion is strict, contradicting (i). Removing any redundant subsets (where
Xi Ă Xj ) gives the desired decomposition (1.3)
As to uniqueness, suppose X “ X1 Y ¨ ¨ ¨ Y Xr and Y “ Y1 Y ¨ ¨ ¨ Y Ys are two such
decompositions. For each i, we have
s
ď
Xi “ pXi X Yj q.
j“1

Since Xi is irreducible, Xi “ Xi X Yj for some j , meaning Xi Ă Yj . Likewise, Yj Ă Xk

for some k . Thus, Xi Ă Xk , and by the minimality condition (Xi Ć Xℓ unless i “ ℓ), we
must have i “ k and hence Xi “ Yj . Similarly, every Yj must equal some Xi . Therefore,
the two decompositions are the same up to reordering.
The irreducible sets appearing in the decomposition (1.3) are called the irreducible compo-
nents of X . They are exactly the closed irreducible subsets which are maximal with respect
to inclusion.
Example 1.16. Consider the algebraic set X “ ZpIq in A3 pkq, where I is the ideal
I “ pxz ´ y 2 , x2 ´ yq.

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Let us find the irreducible components of X . Let p “ pa, b, cq P X be a point. Then the
second equation implies that b “ a2 . Plugging this into the first equation, we get ac ´ a4 “ 0,
which implies that either a “ 0 or c “ a3 . Hence p lies in one of the irreducible subsets
X1 “ Zpx, yq or X2 “ Zpy ´ x2 , z ´ x3 q. Conversely, a point in X1 clearly lies in X , and
if p “ pa, b, cq P X2 , it holds that b “ a2 and c “ a3 so that ac ´ b2 “ a4 ´ a4 “ 0, and p
lies in X . Hence we find that
X “ Zpx, yq Y Zpy ´ x2 , z ´ x3 q.
In geometric terms, X is the union of the z -axis and the ‘twisted cubic curve’ (see Example
1.33). △
Example 1.17. Consider the algebraic set ZpIq Ă A2 pkq, where I is the ideal
I “ py ´ x2 , x2 ` py ´ 1q2 ´ 1q.
Over the real numbers, we recognise the points of ZpIq as the intersection points of the
parabola y “ x2 and the circle of radius 1 with centre in p0, 1q. To find these intersection
points, one can either argue as in the previous example, or directly write the ideal as an
intersection:
I “ py ´ x2 , x2 ` px2 ´ 1q2 ´ 1q
“ py ´ x2 , x2 px ´ 1qpx ` 1qq
“ py ´ x2 , x2 q X py ´ x2 , x ´ 1q X py ´ x2 , x ` 1q
“ py, x2 q X py ´ 1, x ´ 1q X py ´ 1, x ` 1q. (1.4)
This shows that ZpIq consists of the three points p0, 0q, p1, 1q, and p´1, 1q.
The decomposition of ideals (1.4) contains more refined information than just the compo-
nents of the zero set. In this example, the ideal py, x2 q reflects the fact that the two curves
intersect at the origin p0, 0q with ‘multiplicity 2’, unlike the two other intersection points,
which have multiplicity 1. Geometrically, this corresponds to the fact that the parabola and
the circle have a common tangent at p0, 0q. △

1.4 Dimension
One of the advantages of the correspondence an algebraic set X and its coordinate ring ApXq,
is the ability to define and study invariants of X . As a first example of this, let us define
the dimension of an algebraic set. For the coordinate ring ApXq, there is a well-established
notion of the Krull dimension, which is defined as the supremum of the lengths of chains of
prime ideals. Using the correspondence between prime ideals and irreducible closed subsets
as a guide, we make the following similar definition, which works for any topological space.

Definition 1.18 (Dimension). Let X be a topological space. The dimension of X is the

supremum of the lengths n of chains
Z0 Ă Z1 Ă ¨ ¨ ¨ Ă Zn
of distinct irreducible closed subsets of X .

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This definition is formulated for any topological space, but it mostly makes sense for spaces
with ‘Zariski-like’ topologies such as the ones found in algebraic geometry. For instance, Rn
with the standard topology would be zero-dimensional according to the above definition, as
the only irreducible closed subsets are the singleton sets.
The following proposition follows almost immediately:

Proposition 1.19. Let X Ă An pkq be an algebraic set. Then

dim X “ dim ApXq.

Proof If Z0 Ă Z1 Ă ¨ ¨ ¨ Ă Zr is a chain of distinct irreducible subsets in X , then

IpZ0 q Ą IpZ1 q Ą ¨ ¨ ¨ Ą IpZr q is a chain of distinct prime ideals in krx1 , . . . , xn s
containing IpXq. These chains correspond exactly to the chains of distinct prime ideals in
the ring ApXq “ krx1 , . . . , xn s{IpXq. As the Krull dimension is defined as the suprema of
these lengths, we get the claim.

Example 1.20. The affine n-space An pkq has dimension n. This follows because the Krull
dimension of the polynomial ring krx1 , . . . , xn s is equal to n. While this may seem intuitive,
it is in fact not so easy to prove. What is clear, is that dim Ank pkq ě n, because the following
chain of linear spaces

Zpx1 , . . . , xn q Ă Zpx2 , . . . , xn q Ă ¨ ¨ ¨ Ă Zpxn q

has length n (and each linear space is irreducible). However, the opposite inequality dim An pkq ď
n requires some effort (see Proposition A.31). △
Example 1.21 (Hypersurfaces). Let f P krx1 , . . . , xn s be a non-constant polynomial, and
let X “ Zpf q Ă An pkq be the zero set of f . Then ApXq “ krx1 , . . . , xn s{pf q which has
Krull dimension n ´ 1. This is a consequence of Krull’s Principal Ideal Theorem (Theorem
A.29). It follows that dimpXq “ n ´ 1. △
Example 1.22. The algebraic sets in A2 pkq can be classified according to their dimension:
Dimension 0: X “ tp1 , . . . , pr u is a finite set of points.
Dimension 1: X “ Zpf q, where f P krx, ys is a non-constant polynomial.
Dimension 2: X “ A2 pkq.
In contrast, the classification of closed subsets of A3 pkq is much more complex, as there is
no simple description of the ideals in krx, y, zs defining closed subsets of dimension 1. △

We will study the notion of dimension in more detail in Chapter ??.

1.5 Regular and rational functions

The coordinate ring ApXq of an affine variety X is an integral domain, so it has a fraction
field, which we will denote by KpXq. The field KpXq is called the function field or field of
rational functions on X , and its elements are called rational functions.
A rational function is by definition an equivalence class of fractions a{b where a, b P ApXq
and b ‰ 0, where two fractions a{b and c{d represent the same element if ad “ bc in ApXq.

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Figure 1.2 A Kummer surface

If ApXq is a UFD, a rational function f has a unique minimal representation as a fraction

a{b where a, b P ApXq. In general however, there may be different expressions.
While rational functions are formally defined as fractions of polynomials, we may interpret
them as functions over open sets. A rational function f P KpXq is said to be regular at a
point p P X if it can be expressed as a fraction f “ a{b with a, b P ApXq and bppq ‰ 0. If
this holds, we define the value f ppq of f at p to be appq{bppq P k . One checks that this is
independent of the choice of representatives a{b. If f is regular at every point in some open
set U , then f defines a function
f : U ÝÝÑ k. (1.5)
For instance, an element a{b P KpXq will define a function Dpbq Ñ k , although it may be
regular on a larger open set as well, as illustrated below:
Example 1.23. Consider the affine variety X Ă A4 pkq defined by the equation xw ´yz “ 0.
In the function field KpXq, we have an equality x{y “ z{w, and the corresponding rational
function is therefore regular over the open set Dpyq Y Dpwq, which is strictly larger than
both Dpyq and Dpwq (for instance, it contains both p0, 1, 0, 0q and p0, 0, 0, 1q). △
Rational functions are uniquely determined by their values over open sets: if two rational
functions f, g P KpXq are regular on some open set V and f ppq “ gppq for all p P V , then
f “ g in KpXq. To see this, write f “ a{b and g “ c{d in KpXq. Shrinking V if necessary,
we may assume that appq{bppq “ cppq{dppq for every p P V . Then the polynomial function
ad ´ bc vanishes on an open set V and hence on all of X . But then ad ´ bc “ 0 in ApXq
as well, and hence a{b “ c{d in KpXq.
If p P X is a point, the set of rational functions which are regular at p forms a subring of
KpXq called the local ring of X at p:
OX,p “ t f P KpXq | f is regular at p u. (1.6)
The ring OX,p is indeed a local ring with maximal ideal
mp “ t f P OX,p | f ppq “ 0 u.
In fact, if p “ pa1 , . . . , an q P X , and we write mp “ px1 ´ a1 , . . . , xn ´ an q for the
maximal ideal in ApXq, then
OX,p “ ApXqmp .

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To see this, note that as ApXq is an integral domain, the localization ApXqmp is naturally
a subring of KpXq. Then an element f P KpXq lies in ApXqmp if and only if f can be
written as a{b with b R mp , which is equivalent to f being regular at p.
If U Ă X is an open set, we set
OX pU q “ t f P KpXq | f is regular at every point in U u (1.7)
č
“ OX,p .
pPU

Note that each OX pU q is a sub k -algebra of KpXq. Moreover, if V Ă U , then any f P

KpXq which is regular on U is also regular at V , so there is an inclusion OX pU q Ă OX pV q.
Example 1.24. Let U “ A1 pkq ´ t0u be the complement of the origin in A1 . Then
KpA1 pkqq “ kpxq, and a rational function f is regular in U if and only if it is of the form
f “ apxq{xn for some polynomial apxq. This means that OA1 pkq pU q “ krx, x´1 s. △
This example generalizes as follows:

Proposition 1.25. Let X be an affine variety.

(i) If a rational function g P KpXq is regular at every point of X , then g is a
polynomial function. In other words,
OX pXq “ ApXq.
(ii) More generally, if f P ApXq and Dpf q “ t p P X | f ppq ‰ 0 u, then
OX pDpf qq “ ApXqf .

Proof (i) follows from (ii) by taking f “ 1.

(ii): Clearly ApXqf Ă OX pDpf qq. Conversely, given g P KpXq, define the ideal
ag “ t b P ApXq | bg P ApXq u. This ideal has the property that g is regular at p if and
only if p R Zpag q. To see this, note that p R Zpag q if and only if some b P af does not
vanish at p, which in turn is equivalent to f being of the form g “ a{b for some b with
bppq ‰ 0. Therefore, if g is regular on all of Dpf q, it follows that Zpag q Ă Zpf q, and the
Nullstellensatz implies that f n P ag for some n ą 0. But then f n g P ApXq, which shows
that g is an element of ApXqf .
The above proposition is a sort of ‘local-to-global principle’: being regular is a local
condition, which has to be verified near every point, but the conclusion is that a rational
function which is regular at every point can be represented globally by a polynomial function.
Example 1.26. Consider the complement of the origin in A2 pkq,
U “ A2 pkq ´ tp0, 0qu
We claim that every rational function f P OA2 pkq pU q is the restriction of a polynomial
function, that is, OA2 pkq pU q “ krx, ys. To see this, identify the function field of A2 pkq, with
kpx, yq, so that f can be expressed as a quotient of two polnomials without common factors
f “ a{b. As krx, ys is a unique factorization domain, this expression is unique. This means
that f cannot be not regular at the points in Zpbq.

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If b is a constant, then f is a polynomial, so the conclusion follows. If b is not constant,

then Zpbq will contain infinitely many points (e.g., by Example 1.21). In particular, f will
fail to be regular in other points other than the origin tp0, 0qu, contradicting f P OX pU q. △

1.6 Morphisms of affine varieties

Let X and Y be two affine varieties. For a continuous map f : X Ñ Y and a regular function
g P OY pV q, the pullback of g by f is defined as the composition

f 7 pgq “ g ˝ f,

which is a regular function on the open set f ´1 pV q.

Definition 1.27. A continuous map f : X Ñ Y is called a morphism if it pulls regular

functions back to regular functions, that is, if f 7 pgq P OX pf ´1 V q for every open set
V Ă Y and every g P OY pV q.

If f : X Ñ Y and g : Y Ñ Z are two morphisms, the composition g ˝ f : X Ñ Z is a

morphism. Indeed, this map is continuous, and it pulls back regular functions because

pg ˝ f q7 phq “ f 7 pg 7 phqq.

A morphism of affine varieties f : X Ñ Y is said to be an isomorphism if it has an inverse

map that is also a morphism.
Starting with n regular functions f1 , . . . , fn P OX pXq, then we can define a morphism
because

f : X ÝÝÑ An pkq
p ÞÑ pf1 ppq, . . . , fn ppqq. (1.8)

To verify that f is continuous, it suffices to check that the preimage of any closed set in
An pkq is closed in X . Indeed, if W “ Zpg1 , . . . , gr q Ă An pkq, then

f ´1 pW q “ t p P X | gi pf1 ppq, . . . , fn ppqq “ 0 for all i “ 1, . . . , r u

“ Zpf 7 pg1 q, . . . , f 7 pgr qq.

Finally, we check that f pulls back regular functions to regular functions. Let g P
kpy1 , . . . , yn q be a rational function on An pkq and assume g is regular on an open set
V Ă An pkq. Let p P f ´1 pV q and q “ f ppq. Then locally around q , we may write g “ a{b
where a, b are polynomials in y1 , . . . , yn . Then in a neighborhood of p P V , we have

apf1 pxq, . . . , fn pxqq

f 7 pgqpxq “ ,
bpf1 pxq, . . . , fn pxqq
which, after expanding, can be written as a quotient of polynomials where the denominator
does not vanish at p. Therefore f 7 pgq P OX pf ´1 V q and f is a morphism.

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Proposition 1.28. Let X be an affine variety. Then any morphism f : X Ñ An pkq is of

the form
f : X ÝÝÑ An pkq; f ppq “ pf1 ppq, . . . , fn ppqq (1.9)
where fi P ApXq for i “ 1, . . . , n.

Proof Let f : X Ñ An pkq be a morphism. The coordinate functions y1 , . . . , yn are regular

functions on An pkq, so their pullbacks fi7 pyi q are regular on X . The morphism defined
by them, X Ñ An pkq, coincides with f , as both are defined by the same set of regular
functions.

Theorem 1.29 (The main theorem of affine varieties). Let X and Y be affine varieties.
Then there is a one-to-one correspondence between morphisms f : X Ñ Y and maps of
k -algebras ϕ : ApY q Ñ ApXq.

Proof The correspondence is defined as follows: if f : X Ñ Y is a morphism, then the

pullback f 7 : OY pY q “ ApY q Ñ OX pXq “ ApXq is a map of k -algebras.
To define the inverse, let ϕ : ApY q Ñ ApXq be a map of k -algebras. Suppose that
Y Ă An pkq and let y1 , . . . , yn be the coordinate functions on An pkq. Now set fi “ ϕpyi q
for i “ 1, . . . , n. The f1 , . . . , fn define a morphism

f : X ÝÝÑ An pkq.

For any polynomial h P kry1 , . . . , yn s, we have

f 7 phqpxq “ hpf pxqq “ hpf1 pxq, . . . , fn pxqq “ ϕphpy1 , . . . , yn qqpxq (1.10)

where the last equality holds because both sides of the equation are maps of k -algebras
in h and take the same values f1 , . . . , fn on the generators y1 , . . . , yn . This shows that
f 7 phq “ ϕphq “ 0 for every h P IpY q, as h is zero in ApY q. Therefore, the image of f is
contained in Y “ ZpIpY qq.
Finally, (1.10) shows that f 7 “ ϕ, so the correspondence is one-to-one.

If X Ă Am pkq and Y Ă An pkq are affine varieties, the theorem implies that any morphism
f : X Ñ Y is the restriction of a morphism of the form

Am pkq ÝÝÑ An pkq; p ÞÑ ph1 ppq, . . . , hp ppqq (1.11)

where h1 , . . . , hn P krx1 , . . . , xm s are polynomials.

We could have taken (1.11) as the definition of morphisms of affine varieties. However, as
we will see, the above definition generalizes well to general varieties.

Corollary 1.30. Two affine varieties X and Y are isomorphic if and only if the k -algebras
ApXq and ApY q are isomorphic.

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y 2 “ x3

Figure 1.3 The cuspidal cubic curve

Examples
Example 1.31. Any linear map f : An pkq m
řÑ A pkq is a morphism. Indeed, the components
hi of f are linear polynomials hi pxq “ j aij xj . △

Example 1.32 (The cuspidal cubic curve). Consider the algebraic set X “ Zpy 2 ´ x3 q in
A2 pkq. The affine coordinate ring of X is equal to

ApXq “ krx, ys{py 2 ´ x3 q.

This is an integral domain because the polynomial y 2 ´ x3 is irreducible.
Consider the morphism f : A1 pkq Ñ A2 pkq defined by t ÞÑ pt2 , t3 q. The image of f is
contained in X Ă A2 pkq, and, in fact, f is a bijection between A1 pkq and X . Indeed, observe
that f ptq “ p0, 0q only for t “ 0. Moreover, for points in X ´p0, 0q, the assignment t “ y{x
defines the inverse. However, f is not an isomorphism. As f 7 pxq “ t2 and f 7 pyq “ t3 , the
induced ring map
f 7 : krx, ys{py 2 ´ x3 q ÝÝÑ krts
has image krt2 , t3 s and so is not surjective.
This means that in order to check that a morphism is an isomorphism, it is not enough to
check that it is injective and surjective. Also the pullback map f 7 has to be an isomorphism.
△
Example 1.33. Consider the morphism

f : A1 pkq ÝÝÑ A3 pkq

t ÞÝÑ pt, t2 , t3 q.
If x, y, z are coordinates on A3 pkq, the image of f is the irreducible algebraic set X “
Zpy ´ x2 , z ´ x3 q. X is called the twisted cubic curve. The word ‘twisted’ refers to the fact
that X is not contained in a linear plane in A3 pkq.
The corresponding map of k -algebras is defined by

ϕ : krx, y, zs ÝÝÑ krts

mapping x ÞÑ t, y ÞÑ t2 , and z ÞÑ t3 . It is not hard to see that the kernel of ϕ is exactly
the ideal a “ py ´ x2 , z ´ x3 q. Clearly, a Ă Ker ϕ. Conversely, if P P krx, y, zs is any

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y “ x2

z “ x3

Figure 1.4 The twisted cubic curve

polynomial, then using y “ x2 and z “ x3 to remove powers of y and z , we can write

P px, y, zq “ P px, x2 , x3 q mod a.
2 3
If P P Ker ϕ, then P px, x , x q is the zero polynomial, and so P P a.
As ϕ is clearly surjective, we see that
ApXq “ krx, y, zs{a » krts.
This also implies that f induces an isomorphism X » A1 pkq. △
Example 1.34. Let X “ Zpxy ´ 1q Ă A2 pkq and consider the first projection
f : X ÝÝÑ A1 pkq; px, yq ÞÑ x
The map f is a bijection onto the open set U “ A1 pkq ´ t0u. The inverse is given by
gpxq “ px, x´1 q (note that x´1 is indeed a regular function on A1 pkq ´ t0u). On the
algebraic side, this reflects the fact that OA1 pkq pU q “ krx, x´1 s » krx, ys{pxy ´ 1q. △
Example 1.35. Consider the morphism f : A2 pkq Ñ A2 pkq given by px, yq ÞÑ px, xyq.
Then the image of f is equal to Dpxq Y tp0, 0qu. This shows that the image of an algebraic
set need not even be an open set in an algebraic set. △
Example 1.36 (Products). If X Ă Am pkq and Y Ă An pkq are algebraic sets, we can define
the product X ˆ Y Ă Am`n pkq as the subset
X ˆ Y “ t pa1 , . . . , an , b1 , . . . , bn q | pa1 , . . . , an q P X, pb1 , . . . , bn q P Y u.
This is indeed an algebraic set: if X “ Zpf1 , . . . , fr q and Y “ Zpg1 , . . . , gs q, then
X ˆ Y “ Zpf1 , . . . , fr , g1 , . . . , fs q. Moreover, the projection maps π1 : Am`n pkq Ñ
Am pkq and π2 : Am`n pkq Ñ An pkq are morphisms. △

1.7 Conclusion
The main theme of this chapter is that the geometric properties of an affine variety X Ă
An pkq is reflected in the algebraic properties of the coordinate ring
ApXq “ krx1 , . . . , xn s{IpXq.

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For instance, by the Nullstellensatz the mappings V ÞÑ IpV q and a ÞÑ Zpaq give mutually
inverse inclusion-reversing bijections between the objects in the following table:

A LGEBRA G EOMETRY
radical ideals of ApXq closed subsets of X
prime ideals of ApXq closed irreducible subsets of X
maximal ideals of ApXq points of X

Since IpXq is a radical ideal, the k -algebra ApX ) is always a reduced ring, that is, there
are no zerodivisors. Moreover, ApXq is an integral domain if and only if X is irreducible
(Proposition 1.11).
Conversely, every reduced finitely generated k -algebra A is isomorphic to ApXq for some
algebraic set X . Indeed, choose a presentation A “ krx1 , . . . , xn s{a for some ideal a; then
X “ Zpaq is an algebraic set in Am pkq with ApXq “ A. Likewise, any finitely generated
k -algebra which is also an integral domain is of the form ApXq for some affine variety X .
In category theory terms, the affine varieties form a category, AffVar, where the objects are
affine varieties and the arrows betwen them are morphisms of affine varieties. By Theorem
1.29, if X and Y are two affine varieties, the assignment f ÞÑ f 7 defines a bijection
HomAffVar pX, Y q ÝÝÑ HomAlg{k pApY q, ApXqq. (1.12)
In sum, this means that the functor X ÞÑ ApXq defines an equivalence of categories between
the category of affine varieties and the opposite category of finitely generated k -algebras
without zero divisors (that is, the category where the objects are k -algebras, but where all
arrows are formally reversed).

1.8 Exercises
Exercise 1.8.1. Show that X “ t px, yq | y ´ cospxq “ 0 u is not an algebraic set in A2C .
What is the closure of X in the Zariski topology?
Exercise 1.8.2. Show that the algebraic set Zpy 2 ´ x3 ´ 1q Ă A2 pkq is irreducible.
Exercise 1.8.3. Let f : A1 pkq Ñ A1 pkq be an isomorphism. Show that f is given by a linear
polynomial ax ` b, where a, b P k and a ‰ 0.
Exercise 1.8.4. Show that the product X ˆ Y of two affine varieties X and Y satisfies the
following undersal property: For any affine variety V , with two morphisms f1 : V Ñ X
and f2 : V Ñ Y , there is a unique morphism f : V Ñ X ˆ Y so that f1 “ π1 ˝ f and
f2 “ π2 ˝ f .
Exercise 1.8.5. Consider the algebraic set in A3 pkq given by Y “ Zpx3 ` y 3 ` z 3 ´ 3xyzq.
Decide whether Y is irreducible and find its dimension.
Exercise 1.8.6. Let X “ Zpy 2 ´ x3 ` 1, z ´ x2 q Ă A3 pkq. Show that X is isomorphic to
the cubic curve C “ Zpv 2 ´ u3 ` 1q Ă A2k .
Exercise 1.8.7. a) Show that the polynomial y 2 ´ x3 ´ x is irreducible.

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b) Describe the Zariski topology on Zpy 2 ´ x3 ´ xq Ă A2 pkq.

Exercise 1.8.8. Compute a primary decomposition of the ideal I “ px2 y, y 3 , xz, yzq. Use
this to find the irreducible components of ZpIq Ă A3 pkq.
Exercise 1.8.9. For each of the following ideals a find a decomposition of Zpaq into irre-
ducible components.
a) px3 , x2 y, xy 3 q.
b) pyz, xz, y 3 , x2 yq.
c) px2 ´ y, xz ´ y 2 , x3 ´ xzq.
Exercise 1.8.10. Find the irreducible compoents of the algebraic sets below.
a) Zpx2 ´ y 2 q Ă A2 pkq.
b) Zpx ´ yz, xz ´ y 2 q Ă A3C .
c) Zpx2 ` y 2 , x2 ´ z 2 ´ 1q Ă A3C .
Exercise 1.8.11. Compute a primary decomposition for the following ideals and describe
their corresponding closed subsets.
a) I “ px2 y 2 , x2 z, y 2 zq in krx, y, zs.
b) I “ px2 y, y 2 xq in krx, ys.
c) I “ px3 y, y 4 xq in krx, ys.
d) I “ px, y, x ´ yzq in krx, y, zs.
e) I “ px2 ` py ´ 1q2 ´ 1, y ´ x2 q in krx, ys.
Exercise 1.8.12. Show that the Zariski topology on A2 pkq is not the product topology on
A2 pkq “ A1 pkq ˆ A1 pkq.
Exercise 1.8.13. Let C “ Zpf q, D “ Zpgq be two 1-dimensional affine algebraic sets in
A2 pkq with C irreducible. Show that either C X D is a finite set, or C Ă D.
Exercise 1.8.14. Consider the algebraic set in A2 pkq given by
X “ Zpx2 ´ y 3 , x2 ` y 2 ´ 1q
a) Show that X consists of 6 points
b) If π : A2 pkq Ñ A1 pkq denotes the projection onto the x-axis, find the polyno-
mial describing the closed set Y “ πpXq in A1 pkq.
Exercise 1.8.15 (The diagonal). Let X be an affine variety and consider the map
∆: X Ñ X ˆ X
x ÞÑ px, xq
a) Show that ∆ is a morphism of affine varieties.
b) Let X “ An pkq, and let x1 , . . . , xn , y1 , . . . , yn be coordinates on An pkq ˆ
An pkq. Show that the ideal defining the image of ∆ is given by
I “ px1 ´ y1 , . . . , xn ´ yn q
c) In general, for an affine variety X , show that the image of ∆ is closed in X ˆX ,
and that ∆ gives an isomorphism X Ñ ∆pXq. H INT: Use the two projections
X ˆ X Ñ X.

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Exercise 1.8.16. Prove that the coordinate ring of the affine curve X “ Zpy ´ x5 q is
isomorphic to krts.
Exercise 1.8.17. Identify Anm pkq with the space of m ˆ n-matrices over the field k . For
a given integer r, show that the set of matrices of rank less than r is an algebraic set. Is it
irreducible?
Exercise 1.8.18. Let us continue the previous exercise with m “ n.
a) Show that the set of symmetric matrices, i.e. matrices such that AT “ A, is an
2
algebraic set in An pkq.
2
b) Show that the set GLn pkq of invertible matrices is Zariski open in An pkq.
c) Show that the set SLn pkq of matrices with determinant one is an algebraic set
2
in An pkq.
d) Show that the set X of matrices A such that Ar “ 0 for a given r P N, form
2
an algebraic set in the affine space An pkq. Compute the ideal IpXq for n “ 2
and r “ 2.
Exercise 1.8.19. Let S Ă Anˆn pCq denote the set of diagonalizable matrices. Determine
the Zariski closure of S .
Exercise 1.8.20. With notation as in Example 1.23:
a) Verify that xw ´ yz is an irreducible polynomial.
b) Show that the rational function x{y is not regular in any open set containing
the locus where y “ w “ 0.
c) Show that there is no single distinguished open set so that Dpbq contains all the
points where x{y is regular.
Exercise 1.8.21. Let x0 , . . . , xn be coordintes on the affine pn ` 1q-space An`1 pkq and let
f “ f px1 , . . . , xn q be a polynomial in x1 , . . . , xn .
a) Show that the algebraic set X “ Zpx0 ´ f q is isomorphic to An pkq.
b) For which f ’s is the algebraic set X “ Zpx20 ´ f q irreducible?
c) Find a bijection between the open set An pkq ´ Zpf q in An pkq and the algebraic
set Zpx0 f ´ 1q in An`1 pkq.
Exercise 1.8.22.
a) Let ϕ : A Ñ B be a ring map. Show that ϕ´1 p is a prime ideal if p Ă B is
prime.
b) Assume further that A and B are finitely generated k -algebras, and k is al-
gebraically closed. Show that ϕ´1 m is a maximal ideal if m Ă B is one.
H INT: Use the Nullstellensatz to see that A{ϕ´1 m “ k .
Exercise 1.8.23. Let A be a ring and let a be an ideal
? Ş
a) Show that a “ pĄa p, where the intersection is taken over all the prime
?
ideals containing a. H INT: If f R a the ideal aAf is a proper ideal in the
localization Af , hence contained in a maximal ideal.
Ş if A “ krx1 , . . . , xn s over an algebraically
b) Using the Nullstellensatz,?show that
closed field k , we have a “ mĄa m where the intersection is over all the
maximal ideals containing a.

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Exercise 1.8.24. Let X “ Zpf q and Y “ Zpgq be two algebraic sets in A2 pkq with X
irreducible. Show that either X X Y is a finite set, or X Ă Y .
Exercise 1.8.25. Show that the image of the map
ϕ : A1 pkq ÝÝÑ A3 pkq
t ÞÝÑ pt2 , t3 , t6 q
is given by V px3 ´ y 2 , z ´ x3 q. Show that ϕ is bijective. Is ϕ an isomorphism of affine
varieties?
Exercise 1.8.26. Show that the image of the map
ϕ : A1 pkq ÝÝÑ A3 pkq
t ÞÝÑ pt3 , t4 , t5 q
is given by V px4 ´ y 3 , z 3 ´ x5 , y 5 ´ z 4 q. Show that ϕ is bijective. Is ϕ an isomorphism of
affine varieties?
Exercise 1.8.27. Let X “ Zpx2 ` y 2 ´ 1q Ă A2 pkq, show that the two morphisms
A2 pkq Ñ A1 pkq defined by px, yq ÞÑ x2 and px, yq ÞÑ 1 ´ y 2 define the same morphism
when restricted to X .
Exercise 1.8.28. Consider the curve X defined by the equation y 2 “ x3 ` x. Show that
y{x and px2 ` 1q{y define the same rational function f on X . Show that f is regular on
U “ X ´ tp0, 0qu. Show that there are no polynomials a and b so that f can be represented
as a{b on all of U .
Exercise 1.8.29. Show that a subset Y Ă X is irreducible if and only if the closure Y is
irreducible. In particular, the closure of a singleton is irreducible.
Exercise 1.8.30. Let X be a topological space.
a) Show that if a subset Z Ă X is irreducible, then so is the closure Z ;
b) Show that X is irreducible if and only if every non-empty open subset is dense.
c) Show that X is irreducible if and only if every pair of nonempty open sets
U, V Ă X intersect.
d) If f : X Ñ Y is a continuous map, show that f pXq is irreducible if X is.
Exercise 1.8.31. Let X be a topological space and let Z Ă X be an irreducible component
of X . Let U be an open subset of X and assume that U X Z is nonempty. Show that Z X U
is an irreducible component of U .
Exercise 1.8.32. Let X Ă A3 pkq be the affine variety defined by the equation xw ´ yz “ 0.
a) Compute OX pU q for the open set U “ Dpxq Y Dpyq.
b) Compute the local ring OX,p where p “ p0, 0, 0q.
Exercise 1.8.33. Show that any open set in An pkq can be written as a finite union of
distinguished open sets.
Exercise 1.8.34. Show that the open set A2 ´ tp0, 0qu is not isomorphic to an affine variety.

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Exercise 1.8.35. Show that any irreducible conic in A2 pkq is isomorphic to either Zpy ´ x2 q
or Zpxy ´ 1q.
Exercise 1.8.36. Let X be an irreducible space with at least 2 points. Show that X is not
Hausdorff.
Exercise 1.8.37 (Alternative proof of Hilbert’s Nullstellensatz). Let k be a field.
a) Show that krxs contains infinitely many irreducible polynomials. H INT: Try
to imitate Ecuclid’s proof of the infinitude of prime numbers.
b) Show that kpx1 , . . . , xn q is not finitely generated as a k -algebra.
c) Suppose k Ă K Ă L are field extensions and that L is both a finitely generated
k -algebra and a k -vector space of finite dimension. Show that K is also finitely
generated over k .
d) Prove Theorem 1.7.
Exercise 1.8.38 (Varieties defined by matrices). Identify the space of n ˆ n matrices with
2
entries in k with affine n2 -space An pkq, where the coordinates xij correspond to the matrix
entries for 1 ď i, j ď n.
(i) Show that the general linear group GLn pkq, consisting of the invertible n ˆ n
2
matrices, is an open subset of An pkq.
(ii) Show that the special linear group SLn pkq, consisting of the n ˆ n matrices
2
with determinant 1, is a closed subset of An pkq. Is SLn pkq irreducible?
(iii) The orthogonal group Opnq (respectively, the special orthogonal group SOpnq)
consists of the n ˆ n matrices A such that AT A “ In (respectively, AT A “ In
and detpAq “ 1), where In is the n ˆ n identity matrix. Show that Opnq and
2
SOpnq are closed subsets of An pkq. Are they irreducible?
(iv) Show that the locus of matrice
Exercise 1.8.39 (Nilpotent matrices). As in Exercise 1.8.38 identify the space of 2 ˆ 2-
matrices with A4 pkq via the map p ac db q ÞÑ pa, b, c, dq. Show that the two ideals in kra, b, c, ds
a “ pa2 ` bc, ab ` bd, ac ` cd, bc ` d2 q and b “ pad ´ bc, a ` dq
?
satisfy a ‰ b but I “ J . Furthermore, show that
Zpaq “ Zpbq “ tM | M is nilpotentu Ă A4 pkq.
.
Exercise 1.8.40. Show that if X is a Hausdorff space, then the only irreducible subsets are
the one-point sets txu.
Exercise 1.8.41. Show that a topological space is Noetherian if and only if every non-empty
collection of closed subsets has a minimal element. Equivalently, every collection of open
subsets has a maximal element.
Exercise 1.8.42 (The Frobenius morphism). Let k be an algebraically closed field of positive
characteristic p.
a) Show that the mapϕ : krts Ñ krts defined by t ÞÑ tp , is a map of k -algebras.

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b) Describe the corresponding morphism F : A1 pkq Ñ A1 pkq. Show that F is

bijective, but not an isomorphism.
Exercise 1.8.43. Let U Ă X be an open subset of an affine variety X , and let h P OX pU q.
Show that the zero set Zphq “ t p P U | hppq “ 0 u is closed in U .
Exercise 1.8.44. Let X and Y be affine varieties and let U Ă X be an open set.
a) Show that the morphisms f : U Ñ An pkq are exactly the morphisms of the
form
f : U ÝÝÑ An pkq; f ppq “ pf1 ppq, . . . , fn ppqq
where fi P OX pU q for i “ 1, . . . , n.
b) Show that there is a one-to-one correspondence between morphisms f : U Ñ Y
and maps of k -algebras ϕ : ApY q Ñ OX pU q.

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Schemes

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The spectrum of a ring

In this chapter we make the first step towards the notion of a scheme, by defining the spectrum
of a ring. The spectrum of a ring A, denoted by Spec A, is a topological space with a topology
whose closed sets are defined by the ideals of A, reminiscent of the Zariski topology on affine
varieties.
To motivate the definition, let us assume for a moment that A “ ApXq is the coordinate
ring of an affine variety X Ă An pkq. By Hilbert’s Nullstellensatz, there is a one-to-one
correspondence between the points p “ pa1 , . . . , an q of X and the maximal ideals mp “
px1 ´ a1 , . . . , xn ´ an q in A. Therefore, we can identify X with the set of the maximal
ideals of A, with no loss of information. Note that a point p P X lies in Zpaq precisely
when a Ă mx . Therefore, under this identification, the closed sets Zpaq now take the form
t m | a Ă m u of maximal ideals containing a. This shows that the ring A not only determines
the underlying set X , but also its topology. Furthermore, morphisms between affine varieties
correspond to ring maps between their coordnate rings.
The rings appearing A in this setting are rather special. They are integral domains and
finitely generated k -algebras. There is also the assumption that k is algebraically closed,
which is essential in order to have the above correspondence between points and maximal
ideals.
There is a natural way of generalizing this to all rings, which involves including all prime
ideals, rather than just the maximal ideals. Given a ring A, the spectrum Spec A of A is
simply the set of prime ideals of A. This set is then equipped with a topology, called the
Zariski topology, whose closed sets are the sets of the form V paq “ t p P Spec A | a Ă p u
where a is any ideal in A.
The idea of replacing maximal ideals by prime ideals is central in scheme theory. From a
functorial perspective, this is a natural choice, because inverse images of prime ideals under
ring maps are prime ideals, and hence a ring map A Ñ B induces a map Spec B Ñ Spec A.
If X and Y are affine varieties, the induced map ApY q Ñ ApXq, in fact, pulls maximal
ideals back to maximal ideals, but this is generally not true for arbitrary ring maps (a simple
example is the inclusion Z Ñ Q).

2.1 The spectrum of a ring

Let A be a ring. As always, A is commutative with 1.

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Definition 2.1. The spectrum of A, denoted Spec A, is the set of all prime ideals in A:
Spec A “ t p | p Ă A is a prime ideal u.

The set Spec A has a Zariski topology similar to the Zariski topology as defined in Chapter
1. The definitions are very similar; the closed sets are those of the form
V paq “ t p P Spec A | p Ą a u. (2.1)
where a Ă A is an ideal. Note that a Ă b implies V paq Ą V pbq.
To draw a closer analogy to the zero sets Zpaq from Chapter 1, we can think of the sets
(2.1) as the ‘vanishing set’ of a set of functions on Spec A. To make this precise, we introduce
the concept of a residue field.

Definition 2.2. For a point p P A, we define the residue field κppq to be the field of
fractions of the integral domain A{p.

The field κppq is canonically isomorphic to Ap {pAp , where Ap is the localization of A at

p, which is a local ring with maximal ideal pAp .
Given f P A, we denote by f ppq P κppq the image of f modulo p. In this way, we can
view f as a function on Spec A. Note however, that unlike the case of the coordinate ring
of an affine variety, the values of f do not lie in a fixed field k , but rather in many different
fields κppq, each associated with a different prime ideal p.
With this notation, it makes sense to talk about the vanishing set of f , that is,
V pf q “ t p P Spec A | f ppq “ 0 in κppq u.
Note that f ppq “ 0 if and only if f P p. Therefore, this set is exactly the vanshing set
associated to the principal ideal a “ pf q as defined in (2.1). More generally, if a Ă A is an
ideal, the set (2.1) can be written as
V paq “ t p P Spec A | f ppq “ 0 for all f P a u. (2.2)
The advantage of expressing V paq in this way comes from the fact that all the identities
for the Zpaq remain valid for V paq, and the proofs are essentially the same.
The next lemma is analogous to Proposition 1.2. It tells us that the vanishing sets indeed
form the closed sets of a topology on Spec A. We call this the Zariski topology.

Lemma 2.3. The vanishing sets in Spec A satisfy the following properties.
(i) (Arbitrary intersections): For any collection of ideals tai uiPI in A,
č `ÿ ˘
V pai q “ V ai . (2.3)
iPI iPI

(ii) (Finite unions): For two ideals a and b,

V paq Y V pbq “ V pa X bq “ V pa ¨ bq. (2.4)
Additionally, V pAq “ H and V p0q “ Spec A.

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The next lemma is about inclusions between the closed sets of Spec A. We recognize them
as analogues of some of the statements about algebraic sets in Proposition 1.2.

Lemma 2.4. For two ideals a and b in A, the following hold:

?
(i) V paq “ V p aq ? ?
(ii) V paq Ă V pb) if and only if b Ă a
(iii) V paq “ H if and only if a “ A a
(iv) V paq “ Spec A if and only if a Ă p0q.

Proof We recall the following identity for the radical of an ideal (see page 474):

? č
a“ p. (2.5)
aĂp

?
From? this, we see that a and a are contained in the same prime ideals. Therefore V paq “
V p aq. Hence we get (i).
(ii): If V paq Ă V pbq, then (2.5) implies that
? č č ?
b“ pĂ p“ a.
pPV pbq pPV paq

? ? ?
Conversely,
? if b Ă a, then any prime ideal which contains a automatically contains
b, so V paq Ă V pbq. This proves (ii).
? Statement (iii) follows from Lemma 2.3 because V paq “ V p1q “ H if and only if
a “ p1q, which happens
a if and only if a “ p1q. Similarly, (iv) holds because V paq “ V p0q
if and only if a Ă p0q.

As in the definition (1.1), we define the ideal of a subset S Ă Spec A as

č
IpSq “ t f P A | f pxq “ 0 for all x P S u “ p. (2.6)
pPS

Note that IpSq is a radical ideal.

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Proposition 2.5. For any subset S Ă Spec A, we have

V pIpSqq “ S. (2.7)
In particular, the closure tpu of the one-point set tpu is equal to V ppq.
Ş
Proof First off all, S Ă V pIpSqq, because if p P S then clearly p Ą pPS p “ IpSq and
so p P V pIpSqq. Conversely, if V pbq is any closed subset containing S , then this means that
for any p P S , we have p P V pbq and hence p Ą b. Taking the intersection over all p’s in
S , we see that IpSq Ą b. By Lemma 2.4, we get that V pIpSqq Ă V pbq. This shows that
V pIpSqq is the smallest closed set which contains S , and so (2.7) follows.
As a consequence, we get the following analogue of the Nullstellensatz:

Corollary 2.6. Let A be a ring. Then the map a ÞÑ V paq defines a bijection
␣ (
radical ideals a Ă A ÝÝÑ tclosed sets W Ă Spec Au, (2.8)
with inverse defined by W ÞÑ IpW q.
Ş ?
Proof If a is any ideal, then IpV paqq “ pPV paq p “ a by the formula (2.5). Hence
IpV p´qq is the identity map on radical ideals a. Conversely, if W is a closed set, then
V pIpW qq “ W “ W by Lemma 2.4, so also V pIp´qq is the identity.

Corollary 2.7. A point in p P Spec A is closed if and only if p is a maximal ideal.

Proof V ppq “ tpu if and only if p is the only prime ideal which contains p, which precisely
means that p is maximal.
Corollary 2.7 shows that the Zariski topology on Spec A behaves very differently from
not only the Euclidean topology on manifolds, but also the usual Zariski topology on affine
varieties. Indeed, the spectrum of a general ring typically contains many non-closed points
corresponding to the prime ideals which are not maximal.

Definition 2.8 (Generic points). A point x in a closed subset Z of a topological space X

is called a generic point for Z if txu “ Z .

In our context, the point p P Spec A is the generic point of the closed set V ppq. The point
p is in fact the only generic point of V ppq, because if V ppq “ V pqq, then Lemma 2.4 implies
that both p Ă q and q Ă p.

First examples
Example 2.9 (Fields). If K is a field, the prime spectrum Spec K has only one point, which
corresponds to the only prime ideal in K , the zero ideal. △
Example 2.10 (The integers). In the ring of integers Z, there are two types of prime ideals:
the zero ideal p0q and the maximal ideals ppq, one for each prime number p.

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As Z is a principal ideal domain, any ideal is of the form pnq for some integer n. It follows
that the closed subsets are of the form V pnq “ V pp1 q Y ¨ ¨ ¨ Y V ppr q where the pi are the
prime factors of n. In other words, the closed sets are either finite sets of closed points or the
whole space. Dually, the non-empty open sets are the complements of finite sets of closed
points. This means that Spec Z is not Hausdorff, as every nonempty open set contains p0q.

p0q p2q p3q p5q p7q p11q p13q p17q ...

Figure 2.1 Spec Z

The residue field at a closed point ppq is equal to κppq “ Z{p “ Fp , whereas the residue
field at p0q is equal to Q. Each element f of the ring Z gives rise to a function on Spec Z
with values in the various residue fields. For instance, the integer f “ 17 takes the values
f pp0qq “ 17, f pp2qq “ 1̄, f pp3qq “ 2̄, f pp5qq “ 2̄, f pp7qq “ 3̄, . . . , in the fields Q, F2 ,
F3 , F5 , F7 , . . . , respectively, where the bar indicates the class modulo the relevant prime.△
Example 2.11 (A polynomial ring). Consider the spectrum of the polynomial ring in one
variable, Crts. Since Crts is a principal ideal domain, every ideal is of the form pf ptqq for
some polynomial f ptq. If f ptq is not a constant, we may factor it into a product of terms of
the form t ´ ai , so that
V pf ptqq “ V pt ´ a1 q Y ¨ ¨ ¨ Y V pt ´ an q
Therefore, the closed sets are either the empty set; the whole space; or a finite set of closed
points. As in the previous example, Spec Crts is not Hausdorff.
At the prime p “ pt ´ aq, the residue field κppq is equal to Crts{pt ´ aq » C. Under
this isomorphism, a polynomial f P Crts maps to its value f paq P C. Hence the ‘value’
f ppq P κppq is identified with the usual evaluation of the polynomial at t “ a. △
Example 2.12. If A is an integral domain, the zero ideal p0q is prime, and as V p0q “ Spec A,
it is the generic point of all of Spec A. This explains the ‘fat’ points in the pictures in
Examples 2.10 and 2.14 - the closures of these points are the whole space. △
Example 2.13. The ring A “ Crts{pt2 q is not a field, but has only one prime ideal, namely
the ideal ptq. Note that the ideal p0q is not prime as t2 “ 0, but t R p0q.
The ring A “ Crts{ptpt ´ 1qq has a spectrum which consists of two points. By the Chinese
Remainder Theorem,
A » Crts{t ˆ Crts{pt ´ 1q » C ˆ C,
which has exactly two prime ideals, namely 0 ˆ C and C ˆ 0. △
Example 2.14 (Discrete valuation rings). Consider a discrete valuation ring A, such as
krtsptq or Zppq . (See Appendix A for background on discrete valuation rings). The ring A
has exactly two prime ideals, the maximal ideal m and the zero ideal p0q. Therefore, Spec A
consists of just two points: Spec A “ tx, ηu with x corresponding to the maximal ideal m
and η to p0q. The closed sets are H, txu and tx, ηu. Therefore tηu “ Spec A ´ txu is open,
meaning that η is an open point!

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The open sets are H, Spec A and tηu. Again Spec A is not Hausdorff, as there are no
open sets that separate x and η . △

x
η

Figure 2.2 The spectrum of a DVR

2.2 Affine spaces

The most important examples of prime spectra are the affine spaces.

Definition 2.15. For each non-negative integer n, we define the affine n-space as
An “ Spec Zrt1 , . . . , tn s.
More generally, for a ring R, we define the affine n-space over R by
AnR “ Spec Rrt1 , . . . , tn s.

When R “ k is a field, each n-tuple pa1 , . . . , an q P k n determines a maximal ideal

m “ pt1 ´ a1 , . . . , tn ´ an q in the polynomial ring A “ krt1 , . . . , tn s, and hence a point of
Ank “ Spec A with residue field equal to
A{m “ krt1 , . . . , tn s{pt1 ´ a1 , . . . , tn ´ an q » k.
Conversely, every maximal ideal m in A with residue field A{m isomorphic to k will be
of the form pt1 ´ a1 , . . . , tn ´ an q: the ai are uniquely determined as the images of ti via
the quotient map A Ñ A{m “ k . Hence, there is a bijection between k n and the points in
SpecpAq with residue field isomorphic to k .
For these ‘traditional points’, we will frequently switch between the n-tuple pa1 , . . . , an q P
k n and the maximal ideal m “ pt1 ´ a1 , . . . , tn ´ an q for the corresponding point in Ank .
If k is algebraically closed, then Hilbert’s Nullstellensatz tells us that every maximal ideal
is of the form m “ pt1 ´ a1 , . . . , tn ´ an q, so every closed point of Ank will be of the form
pa1 , . . . , an q. This means that An pkq, as introduced in Chapter 1, forms a subset of the affine
n-space Ank , and the Zariski topology on An pkq is simply the topology induced from Ank .
In general, the affine space Ank also contains many points which are not of the ‘traditional’
type. There can be maximal ideals with residue fields not isomorphic to k , and prime
ideals which are not maximal (the zero ideal for instance). Hence, Ank is strictly larger than
the previously defined An pkq. The differences between Ank and An pkq become even more
apparent if k is not algebraically closed.
Example 2.16 (The affine line). The prime spectrum A1k “ Spec krts is called the affine line
over k . The polynomial ring krts is a principal ideal domain, so the prime ideals are either of
the form pf ptqq where f ptq is an irreducible polynomial, or the zero ideal p0q. In the first

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case, the ideals are automatically maximal. Therefore, A1k has two types of points: the closed
points and the generic point η .
When k is algebraically closed, the maximal ideals are all of the form pt ´ aq for a P k ,
and their residue fields are isomorphic to k .
When k is not algebraically closed, there can be other closed points in Spec krts besides
the ones of the form pt ´ aq. An interesting example is when k “ R. By the Fundamental
Theorem of Algebra, a non-zero prime ideal p of Rrts is of the form p “ pf ptqq where f ptq
is either linear, that is, f ptq “ t ´ a for an a P R; or f is quadratic with two conjugate
complex non-real roots, that is, f ptq “ pt ´ aqpt ´ āq with a P C but a R R. The closed
points in Spec Rrts may therefore be identified with the set of pairs ta, āu with a P C. For
instance, the maximal ideal pt2 ` 1q corresponds to the pair ti, ´iu. The residue field at this
point is isomorphic to Rrts{pt2 ` 1q » C.
In general, if a maximal ideal m in krts is generated by the irreducible polynomial f ptq,
the residue field at the corresponding point in A1k is the extension of k obtained by adjoining
a root of f .
Affine spaces over non-algebraically closed fields can be quite complicated. Even the
affine line A1Q over Q is quite mysterious, as the monic irreducible polynomials in Qrts have
a very intricate structure. △
Example 2.17 (The affine plane). When k is algebraically closed, the prime ideals of krx, ys
come in three types: the maximal ideals, which are all of the form px ´ a, y ´ bq for a, b P k ;
the prime ideals of the form p “ pf q, where f P krx, ys is an irreducible polynomial; and
the zero ideal p0q.
Note that the point pf q belongs to the closed set V pf q. In addition to this point, V pf q
contains the maximal ideals px ´ a, y ´ bq which contain pf q, or equivalently, f pa, bq “ 0.
Hence the points of V pf q correspond to the points of the ‘plane curve’ defined by the equation
f px, yq “ 0.

px ´ a, y ´ bq pf q p0q

closed points generic points of curves generic point

2.3 Distinguished open sets

The Zariski topology of Spec A has a convenient basis consisting of particularly simple open
sets. For an element f P A, we define the distinguished open set, Dpf q to be the complement
of the closed set V pf q, that is,
Dpf q “ t p P Spec A | f R p u “ Spec A ´ V pf q.

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These open sets will turn out to be very important in computations involving Spec A.

Lemma 2.18. For a ring A and elements f, g P A, we have

(i) Dpf q X Dpgq “ Dpf gq
(ii) Dpgq Ă Dpf q if and only if g n P pf q for some n P N. In particular, one
has Dpf q “ Dpf n q for all n.

Proof (i): if p is a prime ideal, then f R p and g R p hold if and only if f g R p.

(ii): the inclusion Dpgq Ă Dpf q holdsa if and only if V pf q Ă V pgq, and by Lemma 2.4
on page 29, this is equivalent to pgq Ă pf q, i.e., g n P pf q for some n P N.

Lemma 2.19.
(i) The distinguished open sets tDpf quf PA form a basis for the topology of
Spec A
(ii) A collection tDpfi quiPI forms an open cover of Spec A if and only if the
fi ’s generate the unit ideal. Equivalently, there are finitely many indices
i1 , . . . , ir P I and a1 , . . . , ar P A such that
1 “ a1 fi1 ` ¨ ¨ ¨ ` ar fir . (2.9)

Proof (i): We need to show that every open subset U of Spec A can be written as the union
of distinguished open sets. By definition, the complement U c of U is of the form U c “ V paq
with a Ă A an ideal. If we choose a set of generators tfi uiPI for a (not necessarily a finite
set), then we have
`ÿ ˘c ` č ˘c ď
U “ V paqc “ V pfi q “ V pfi q “ Dpfi q. (2.10)
iPI iPI iPI

(ii): By the identity (2.10) withř U “ Spec A, the collection of open sets tDpfi quiPI
covers Spec
ř A if and only if V p iPI pfi qq “ H. By Lemma 2.4, this in turn happens if and
only if iPI pfi q “ p1q, or in other words, the fi generate the unit ideal. But this happens if
and only if 1 can be expressed as a combination of finitely many of the fi ’s.

In algebraic geometry, a topological space is said to be quasi-compact if every open

covering has a finite subcover. (This terminology is a bit unfortunate, as spaces with this
property are usually called ‘compact’. The distinction becomes meaningful when considering
non-Hausdorff spaces, which are common in algebraic geometry.

Corollary 2.20. Spec A is quasi-compact.

Proof Let tUi uiPI be any open cover of Spec A. Then each Ui can be covered by distin-
guished open sets Dpfij q with fij P A. Since the Dpfij q’s cover Spec A, finitely many of
them will suffice, by Lemma 2.19. As each Ui contains Dpfij q, we can cover Spec A with
finitely many Ui as well.

Example 2.21. In the affine line A1k over a field, every closed set is of the form V pf q for
some polynomial f , so every open set is a distinguished open set Dpf q.

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In A2k “ Spec kru, vs, the set U “ A2k ´ V pu, vq is open, but not of the form Dpf q. Still,
we have U “ Dpuq Y Dpvq. △
Example 2.22 (The circle). Consider the spectrum X “ Spec Rrx, ys{px2 ` y 2 ´ 1q. The
maximal ideal m “ px, y ´ 1q defines the point p0, 1q on X . Even though the ideal m is not
a principal ideal (Exercise 2.7.10), the complement X ´ tmu is a distinguished open set. In
fact, it is equal to Dpy ´ 1q because modulo the relation x2 “ 1 ´ y 2 , we have
m2 “ px2 , xpy ´ 1q, py ´ 1q2 q “ py ´ 1q.
△

2.4 Irreducibility and connectedness

Recall from Chapter 1 that a topological space X is irreducible if it cannot be written as the
union of two proper closed subsets. From Proposition 1.11, we know that the coordinate ring
of an affine variety is an integral domain, and simple examples indicate that reducibility of
Spec A is closely linked to zero divisors in A (see Example 2.28 below). In general, one has
the following:

Proposition 2.23. Let A be a ring.

(i) A closed subset Z Ă Spec A is irreducible if and only if Z is of the form
Z “ V ppq for some prime ideal p.
(ii) The space Spec A itself is irreducible if and only if A has
ajust one minimal
prime ideal; in other words, if and only if the nilradical p0q is prime.

Proof (i): By Proposition 2.5 we have V ppq “ tpu, and this is irreducible, being the closure
of a ?
singleton.
Ş For the? reverse implication, let V paq Ă Spec A be a closed subset. Recall
that a “ aĂp p. If a is not prime, there must be more than one prime involved in the
?
intersection.?We may divide them into two different groups, thereby representing a as an
intersection a “ b X b1 where b and b1 are the intersections of the primes in the two groups,
and hence are different radical ideals. From this, we get that V paq “ V pbq Y V pb1 q, and
V paq is not irreducible. `a ˘
The statement (ii) follows from (i), because Spec A “ V p0q , by Lemma 2.4.
A consequence of the proposition is that Spec A is irreducible whenever A is an integral
domain, as in that case p0q is a minimal prime ideal. However, Spec A may certainly be
irreducible for other rings as well. For example, the ring A “ Crts{pt2 q is not an integral
domain, and yet has only one prime ideal, namely the principal ideal ptq. By part (ii) tells us
that this example is typical for such rings: every zerodivisor in the ring is nilpotent. In the
spirit of the analogy with functions, there are non-zero functions which vanish everywhere:
the element t is nonzero in A, but it becomes zero in the residue field Crts{ptq » C.
Example 2.24. The spectrum Spec Crx, ys{py 2 ` x5 ` 1q is irreducible, as the polynomial
y 2 ` x5 ` 1 is irreducible, and generates a prime ideal. △
2 2
Example 2.25. The spectrum Spec a Crx, y, z, ws{px , xy, y , xw ´ yzq is irreducible. In-
deed, the nilradical is given by p0q “ px, yq, and this is a prime ideal. △

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Recall that a topological space is connected if it cannot be written as a disjoint union of

two proper open subsets. All of the examples we have seen until now, with the exception of
Example 2.13, are connected.
Example 2.26 (A disconnected spectrum). Suppose that A “ A1 ˆ A2 is the direct product
of two non-trivial rings A1 and A2 . In A we have the two orthogonal idempotents e1 “ p1, 0q
and e2 “ p0, 1q; they satisfy the relations e21 “ e1 , e1 e2 “ 0, e22 “ e2 and e1 ` e2 “ 1.
The spectrum Spec A decomposes as the disjoint union Spec A “ V pe1 q Y V pe2 q of
the two closed sets V pei q; indeed, since e1 ` e2 “ 1, it holds that V pe1 q X V pe2 q “
V pe1 , e2 q “ H. And since e1 e2 “ 0, either e1 P p or e2 P p for each prime p P Spec A, so
the V pei q’s cover Spec A. △
In fact, there is a converse to this example.

Proposition 2.27. A spectrum Spec A is disconnected if and only if A is isomorphic to

a direct product A “ A1 ˆ A2 of non-trivial rings A1 and A2 .

While it would certainly be possible to give a direct proof of this proposition at the present
stage, we will wait until the next chapter to do so. There, we will see a much more conceptual
proof using the structure sheaf (see Example 4.8 on page 64). For reduced rings however, the
argument is straightforward (see Exercise 2.7.5).
Example 2.28. Note that any irreducible space is also connected. The converse does not
hold: the spectrum X “ Spec krx, ys{pxyq is connected but not irreducible. The coordinate
functions x and y are zerodivisors in the ring krx, ys{pxyq, and their zero-sets V pxq and
V pyq show that X has two components. Since these two components intersect at the origin,
X is connected. △

2.5 Maps between prime spectra

Let A and B be two rings and let ϕ : A Ñ B be a ring map between them. The inverse
image ϕ´1 p of a prime ideal p Ă B is a prime ideal: that ab P ϕ´1 p means that ϕpabq “
ϕpaqϕpbq P p, so at least one of ϕpaq or ϕpbq has to lie in p. This means that there is an
induced map of spectra
Specpϕq : Spec B ÝÝÑ Spec A (2.11)
p ÞÑ ϕ´1 ppq.
This map is continuous in the Zariski topology, because preimages of closed sets are closed;
this follows from item (i) in the next proposition.

Proposition 2.29. ϕ : A Ñ B be a ring map and let f : Spec B Ñ Spec A be the

induced map of spectra. Then
(i) f ´1 V paq “ V pϕpaqBq for each ideal a Ă A.
(ii) f ´1 Dpgq “ Dpϕpgqq for each g P A.
(iii) f pV pbqq “ V pϕ´1 bq for each ideal b of B .

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Proof To prove (i), let a Ă A be an ideal. Then we have the following equalities:
f ´1 V paq “ t p Ă B | a Ă ϕ´1 p u “ t p Ă B | ϕpaq Ă p u “ V pϕpaqBq.
Indeed, as a Ă ϕ´1 ϕpaq, the inclusion ϕpaq Ă p holds if and only if a Ă ϕ´1 p.
For (ii), note that for each element g P A, we have
f ´1 Dpgq “ t p Ă B | g R ϕ´1 p u “ t p Ă B | ϕpgq R p u “ Dpϕpgqq.

Finally, we prove (iii): according to Corollary 2.5 on page 30, the closure f pV pbqq equals
V paq with a where a is the ideal given by
č č
a“ p“ ϕ´1 q.
pPf pV pbqq bĂq

Here the second equality holds because p P f pV pbqq implies that p “ ϕ´1 q for some q with
b Ă q. So we get that
č `č ˘ ? a
a“ ϕ´1 q “ ϕ´1 q “ ϕ´1 p bq “ ϕ´1 b.
bĂq bĂq

Hence V paq “ V pϕ´1 bq, which gives the desired identity.

The following propositions gives prototypical examples of induced maps on spectra.

Proposition 2.30. For an ideal a Ă A, the quotient map A Ñ A{a induces a homeo-
morphism
»
f : SpecpA{aAq ÝÝÑ V paq Ă Spec A

Proof If ϕ : A Ñ A{a denotes the quotient map, the map p ÞÑ ϕ´1 p gives an inclusion
preserving one-to-one correspondence between prime ideals in A{a and prime ideals in A
containing a, with inverse given by q ÞÑ q{a. This shows that f is a continuous bijection
onto the closed subset V paq. To show that f is a homeomorphism, it suffices to show that it
is closed, and this follows from the equalities
␣ (
f pV pb{aqq “ p P Spec A | b{a Ă p{a P SpecpA{aq “ V pbq.

By the proposition, if the ring map ϕ : A Ñ B is surjective, then Spec B maps homeo-
morphically onto the closed subsetaV pKer ϕq Ă Spec A.
In the special case, where a “ p0q is the nilradical of A, we have V paq “ Spec A, and
the proposition implies the following:
a
Corollary 2.31. Let A be a ring and let Ared “ A{ p0q be the reduction of A. Then
the map A Ñ Ared induces a homeomorphism
SpecpAred q » Spec A.

We next consider the maps of spectra induced by localization maps. The next result implies

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in particular that distinguished open sets Dpf q can themselves be viewed as prime spectra.
This fact will be important later on.

Proposition 2.32.
(i) For any f P A, there is a canonical homeomorphism
»
ι : SpecpAf q ÝÝÑ Dpf q Ă Spec A.
(ii) More generally, if S Ă A is a multiplicative subset, the localization map
ℓ : A Ñ S ´1 A induces a homeomorphism
»
ι : SpecpS ´1 Aq ÝÝÑ D Ă Spec A,
where D “ t p P Spec A | p X S “ H u.

Proof It suffices to prove the statement (ii). Let q Ă S ´1 A be a prime ideal. As q does not
contain 1, we see that ℓ´1 pqq does not contain any elements of S . In particular, ℓ´1 pqq P D.
Conversely, if p Ă A is a prime ideal with p X S “ H, that is p P D, then pS ´1 A is a prime
ideal in S ´1 A, and we have p “ ℓ´1 ppS ´1 Aq. From this it follows that ι is continuous,
injective, with image D. To show that ι is a homeomorphism, it is enough to show that it is
closed. This follows from the following equalities:
ιpV paqq “ t ℓ´1 pqq | q Ą a u
“ t p P Spec A | p Ą ℓ´1 paq and p X S “ H u
“ V pℓ´1 paqq X D.

For two distinguished open sets with Dpgq Ă Dpf q, we may write g r “ cf for some
c P A and r P N (statement (ii) of Lemma 2.18). Hence Dpgq “ Dpg r q “ Dpcf q. This
shows that any distinguished open set contained in Dpf q is of the form Dpaf q for some
a P A.
Algebraically, the inclusion Dpgq Ă Dpf q corresponds to the fact that f becomes invert-
ible in Ag . This means that the localization map A Ñ Ag factors uniquely as A Ñ Af Ñ Ag .
Explicitly, the map Af Ñ Ag sends a{f n to acn {g rn . This ring map identifies Dpgq with
the distinguished open set Dpg{1q in SpecpAf q, where g{1 denotes the image of g in Af .

Examples
Example 2.33 (Reduction modulo a prime p). The reduction mod p-map Z Ñ Fp induces a
map Spec Fp Ñ Spec Z. The one and only point in Spec Fp is sent to the point in Spec Z
corresponding to the maximal ideal ppq.
Likewise, the inclusion Z Ă Q induces a map Spec Q Ñ Spec Z. It sends the unique
point in Spec Q to the generic point η of Spec Z. △
Example 2.34 (The twisted cubic curve). Let k be a field. The ring map ϕ : krx, y, zs Ñ krts
given by x ÞÑ t, y ÞÑ t2 , z ÞÑ t3 correspondes to a map of prime spectra
f : A1k ÝÝÑ A3k .

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As ϕ is surjective, f is a homeomorphism onto the closed set defined by a “ Ker ϕ “

py ´ x2 , z ´ x3 q. This is the scheme version of the twisted cubic curve from Example 1.33.
△
Example 2.35. Let k be a field. The natural inclusion ϕ : krxs Ñ krx, ys{pxy ´ 1q induces
a morphism
Spec krx, ys{pxy ´ 1q ÝÝÑ A1k “ Spec krxs.
On the level of closed points, when k is algebraically closed, this maps pa, a´1 q to a. Since
krx, ys{pxy ´ 1q is an integral domain, Spec krx, ys{pxy ´ 1q has a unique generic point η ,
and this is mapped to the generic point of A1k . Note that Spec krx, ys{pxy´1q » Dpxq Ă A1k
via this morphism. In particular, the image is dense, but not closed in A1k . △

2.6 Fibers
To understand a map of spectra f : Spec B Ñ Spec A, it is often useful to understand the
fibers of f , that is, the inverse images of points. If f is induced by a ring map ϕ : A Ñ B
and y P Spec A is a point corresponding to p in A, the fiber f ´1 pyq consists of the primes q
in B such that p “ ϕ´1 pqq.
If y P Spec A is a closed point, so that p is a maximal ideal, then tyu “ tyu “ V ppq, so
Proposition 2.29 implies that f ´1 V ppq “ V ppBq. In particular, the fiber f ´1 pyq is a closed
set, homeomorphic to Spec B{pB .
If y is not closed, the fiber f ´1 pyq may or may not be closed in Spec B . While we still
have f ´1 V ppq “ V ppBq, this closed set may contain other primes than the ones mapping to
y . For instance, in the sitation when p “ p0q is prime in A, then V ppBq “ Spec B .
To describe the fiber, we consider the localization Bp “ S ´1 B in the multiplicative set
S “ ϕpA ´ pq.
The idea is that the ‘extra primes’ in V ppBq which do not map to p will localize to non-proper
ideals in Bp .
Consider the composition
SpecpBp {pBp q ÝÝÑ SpecpBp q ÝÝÑ Spec B, (2.12)
which is induced by the localization map B Ñ Bp and the quotient map Bp Ñ Bp {pBp .

Proposition 2.36. The composition (2.12) induces a homeomorphism

f ´1 ppq » SpecpBp {pBp q.
If p P Spec A is a closed point, then f ´1 ppq is homeomorphic to SpecpB{pBq.

Proof The first map is a homeomorphism onto the subset V ppBp q (Proposition 2.29),
and the second is a homeomorphism onto the set of primes q P Spec B disjoint from S
(Proposition 2.32). Therefore the composition is a homeomorphism onto the set of prime
ideals q P Spec B such that q X S “ H and qBp Ą ϕppqBp . The second equality implies
that q Ą ϕppq and hence ϕ´1 pqq Ą p. The first equality then implies that ϕ´1 pqq “ p (if

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ϕ´1 pqq Ą p were strict, then elements in ϕ´1 pqq ´ p would map to elements of q X S ).
Therefore, the image of (2.12) consists of precisely those prime ideals q in B so that ϕ´1 pqq “
p, that is, f pqq “ p.
Example 2.37. Consider the map
f : Spec Crx, y, zs{pxy ´ zq ÝÝÑ Spec Crzs,
induced by the ring map Crzs Ñ Crx, y, zs{pxy ´ zq “ B . Let us compute the fibers
f ´1 ppq over the maximal ideals p “ pz ´ aq. Note that
B{pB “ Crx, y, zs{pxy ´ z, z ´ aq » Crx, ys{pxy ´ aq.
There are two cases. If a ‰ 0, then xy ´ a is an irreducible polynomial, and so Spec B{pB
is irreducible. This is intuitive, as it corresponds to the hyperbola V pxy ´ aq in A2C . If a “ 0,
we are left with Spec Crx, ys{pxyq, which is not irreducible. It has two components, the two
coordinate axes V pxq and V pyq.
Let us also consider the fiber over the generic point η of Spec Crzs, which corresponds
to p “ p0q. In this case, the ring pB{pBqp is the localization of B with respect to the
multiplicative set S “ Crzs ´ t0u; that is, the ring
Cpzqrx, ys{pxy ´ zq.
This is again an integral domain, so the fiber f ´1 pηq is irreducible. This fiber may be regarded
as a hyperbola in the affine plane A2Cpzq over the field Cpzq. △
Example 2.38. Let k be a field and consider the map
f : Spec krx, ys{px ´ y 2 q ÝÝÑ Spec krxs
induced by the injection krxs Ñ krx, ys{px ´ y 2 q. Geometrically this corresponds to the
projection of the ‘horizontal’ parabola onto the x-axis.
If a P k , the fiber f ´1 ppq over the maximal ideal p “ px ´ aq is the spectrum of the ring
B{pB “ krx, ys{px ´ y 2 , x ´ aq » krys{py 2 ´ aq.
Let us assume that k has characteristic different from 2. Several cases can occur:
(i) If a ‰ 0 and a has a square root in k , say b2 “ a, the polynomial y 2 ´ a factors
as py ´ bqpy ` bq, and by the Chinese Remainder Theorem, the fiber becomes
the product
` ˘
Spec krys{py ´ bq ˆ krys{py ` bq ,
which is the disjoint union of two copies of Spec k . ?
(ii) If a ‰ 0, ?but does not have a square root in k , then the fiber equals Spec kp aq,
where kp aq is a quadratic field extension of k . The fiber is a single point,
but with?‘multiplicity 2’ (in the sense that the degree of the field extension
k Ă kp aq is 2).
(iii) The final case is when a “ 0. The fiber then equals Spec krys{py 2 q, which
is just a single point. But again there is a ‘multiplicity 2’, accounted for by
the presence of nilpotent elements in the ring (as vector space over k the ring
krys{py 2 q has dimension 2).

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△
Example 2.39 (The Möbius Strip). Let A “ Rrx, ys{px2 ` y 2 ´ 1q and consider the
A-algebra B “ Aru, vs{pvx ´ uyq. This induces a map
f : Spec Aru, vs{pvx ´ uyq ÝÝÑ Spec A.
Let us compute the fibers of f . As x2 ` y 2 “ 1 in A, x and y generate the unit ideal, and
so Spec A is covered by the two affine subsets Dpxq and Dpyq. If p P Dpxq, then x is
invertible in Ap , and so we compute
Bp {pBp “ pAp {pqru, vs{pv ´ x´1 uyq » κppqrus,
and the fiber is homeomorphic to A1κppq . A similar argument works when p P Dpyq. Hence
all fibers are isomorphic to affine lines. △
Example 2.40. The inclusion Rrts Ă Crts induces a map from the affine line over C to the
affine line over R:
π : Spec Crts ÝÝÑ Spec Rrts.
By Example 2.16 there are three cases to consider for the fiber π ´1 pyq of a point y P A1R .
(i) The point y corresponds to the maximal ideal pt ´ aq with a P R. Then the fiber
is given by
π ´1 pyq “ Spec pCrts{pt ´ aqq » Spec C,
and the fiber is a single closed point with residue field C.
(ii) The point y is a closed point corresponding to p “ pf ptqq where f P Rrts has
two conjugate complex roots a, ā. Then
π ´1 pyq “ Spec Crts{pf ptqq » Spec pCrts{pt ´ aq ˆ Crts{pt ´ āqq .
Hence the fiber consists of two closed points, both with residue field C.
(iii) The point y equals the generic point η . Then π ´1 pηq is the spectrum of the
localization S ´1 Crts “ Cptq where S “ Rrts´p0q. Therefore, π ´1 pηq consists
of a single point with residue field Cptq.
The Galois group G “ GalpC{Rq » Z{2 acts on the fibers in this example. More precisely,
consider the conjugation map Crts Ñ Crts on the polynomial ring Cř rts given byřconjugating
the coefficients of the polynomials; that is, sending a polynomial i ai ti to i āi ti . This
defines an automorphism
ι : Spec Crts ÝÝÑ Spec Crts.
Note that the subring Rrts Ă Crts is fixed by the conjugation map, so the following diagram
commutes:
ι
Spec Crts Spec Crts
π π

Spec Rrts

Therefore, G “ xid, ιy » Z{2 acts by automorphisms on the fibers of π , and Spec Rrts can

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be viewed as the quotient space of Spec Crts by G; i.e. the space of G-orbits. Indeed, by
Example 2.16, the closed points of Spec Rrts correspond exactly to the orbits of G and the
generic point of Spec Crts is invariant and corresponds to the generic point of Spec Rrts. △
Example 2.41 (The Gaussian integers). The inclusion Z Ă Zris induces a morphism
f : Spec Zris ÝÝÑ Spec Z.
We will study Spec Zris by studying the fibers of this map. If p P Z is a prime, the fiber over
ppq P Spec Z is given by V pppqZrisq. This is homeomorphic to the spectrum
ˆ ˙
Fp rxs
Spec pZris{pq » Spec .
x2 ` 1
The ring Fp rxs{px2 ` 1q is not an integral domain precisely when the equation x2 ` 1 “ 0
has a solution modulo p. This happens1 if and only if p “ 2 or p ” 1 mod 4. By Fermat’s
Two Square Theorem, this is equivalent to p “ a2 ` b2 being a sum of two squares, or in
other words, that p “ pa ` ibqpa ´ ibq factors in Zris. This means that there are three cases:
(i) p “ 2. Then p2qZris “ p1 ` iq2 Zris. Therefore, f ´1 pp2qq consists of the single
point corresponding to the prime ideal p1 ` iq.
(ii) p ” 3 pmod 4q. Then the ideal ppq stays prime in Zris and the fiber f ´1 pppqq
consists of the single point x corresponding to ppqZris. The residue field κpxq
is in fact isomorphic to Fp rxs{px2 ` 1q » Fp2 , the field with p2 elements.
(iii) p ” 1 pmod 4q. In this case, we may factor p “ pa ` ibqpa ´ ibq in Zris and
so the fiber consists of the two points with residue fields Fp ,
f ´1 pppqq “ V pppqZrisq “ tpa ` ibq, pa ´ ibqu.
Note that the complex conjugation map σ : Zris Ñ Zris acts on Spec Zris and the fibers of
the map f . For instance, the primes sitting over p5q are p2 ` iq and p2 ´ iq, and complex
conjugation sends one to the other. We therefore picture Spec Zris as a curve lying above
Spec Z, with σ permuting the points in each fiber (though some are fixed by σ ).
p3 ` 2iq p4 ` iq
p2 ` iq
p0q p1 ` iq p3q p7q p11q
...

p2 ´ iq
p3 ´ 2iq p4 ´ iq

...
p0q p2q p3q p5q p7q p11q p13q p17q

Figure 2.3 The spectrum Spec Zris

△
1 Here is a quick proof for p ‰ 2: The elements t1, . . . , p ´ 1u modulo p can be partitioned into subsets of the
form Cpxq “ tx, ´x, x´1 , ´x´1 u. If |Cpxq| ‰ 4, then either x “ x´1 (meaning Cpxq “ t1, ´1u); or
x “ ´x´1 (meaning Cpxq “ tx, ´xu consists of the two solutions to x2 ` 1 “ 0 mod p). This implies that
p ´ 1 “ |Cpxq| is conguent to 2 mod 4 if x2 ` 1 “ 0 has no solutions, and to 0 mod 4 otherwise.
ř

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Example 2.42 (The affine line A1Z ). Consider the affine line A1Z “ Spec Zrts and the
morphism f : Spec Zrts Ñ Spec Z induced by the inclusion Z Ă Zrts.
There are two cases for a fiber f ´1 pyq of a point y P Spec Z. If y corresponds to the closed
point ppq P Spec Z, the fiber f ´1 pyq consists of all primes p Ă Zrts such that p X Z “ ppq.
According to Proposition 2.36, this is given by
V pppqZrtsq “ SpecpZrts{pZrtsq “ A1Fp .
Likewise, if y “ η is the generic point of Spec Z, Proposition 2.36 tells us that the fiber
f ´1 pηq is the spectrum of the localization S ´1 Zrts “ Qrts, where S “ Z ´ p0q. In other
words,
f ´1 pηq “ Spec Qrts “ A1Q .
The scheme A1Z is shown in Figure 2.42. In the figure, we have depicted the two closed sets
V p6x ` 1q and V px2 ` 3q. Note that V p6x ` 1q is disjoint from the fibers above the primes
2 and 3 (why?). The closed subset V px2 ` 3q should be compared to Example 2.41). △

A1F2 A1F3 A1F5 A1F7 A1Q

´ ¯
p7, x ` 5q x2 ` 3
´ ¯
5, x2 ` 3
p3, xq
p7, x ` 2q

p2, x ` 1q p6x ` 1q

p5, x ` 1q

p7, 6x ` 1q
p0q

p2q p3q p5q p7q ... p0q

´ ¯
V p6x ` 1q V x2 ` 3

Figure 2.4 The affine line Spec Zrxs

2.7 Exercises
Exercise 2.7.1. Show that Spec A is the empty set if and only if A is the zero ring.
1
Exercise 2.7.2. Describe Spec Zr 255 s.
Exercise 2.7.3. Consider the discrete valuation ring R “ Cruspuq . Describe Spec Arts and
the map Spec Arts Ñ Spec A induced by the inclusion A Ă Arts. Which fibers are closed?
Exercise 2.7.4. Let p be a prime ideal in a ring A. Show that there is a canonical inclusion
A{p ãÑ Ap {pAp and that this yields an identification of Ap {pAp with the fraction field of
A{p.
Exercise 2.7.5 (Disconnected spectra).

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a) Show that a topological space X is disconnected if and only if it has two

orthogonal idempotents, i.e., continuous maps e1 , e2 : X Ñ t0, 1u such that
e21 “ e1 , e22 “ e2 , e1 e2 “ 0 and e1 ` e2 “ 1. H INT: Try the characteristic
functions of two open sets.
b) Let A be a reduced ring. Show that Spec A is disconnected if and only if
A “ A1 ˆ A2 for two non-trivial rings A1 and A2 . H INT: Find two ideals
a, b such that a ` b “ A and a X b “ p0q. Use this to find two non-trivial
idempotents.
Exercise 2.7.6. Let A be an Artinian ring. Show that X “ Spec A is a finite set, and the
topology is the discrete topology. H INT: Write A as a product of its localizations.
Exercise 2.7.7. With the notation of Proposition 2.32, show that the set D can be written
D “ t p | sppq ‰ 0 in κppq for all s P S u
Find an example of an S so that D is not an open set.
Exercise
a 2.7.8. Show that Dpf q “ H if and only if f is nilpotent. H INT: Use the identity
Ş
p0q “ pPSpec A p.
Exercise 2.7.9. Show that Dpf q “ Dpgq if and only if there are integers m, n such that
g m “ u ¨ f n for some unit u P A.
Exercise 2.7.10. Show that the ideal m “ px, y ´ 1q in A “ Rrx, ys{px2 ` y 2 ´ 1q is not
principal.
Exercise 2.7.11. Check that for a nested inclusion Dphq Ă Dpgq Ă Dpf q, we have
ρf h “ ρgh ˝ ρf g .
Exercise 2.7.12. Let A be a ring, let a be an ideal in A and let tfi uiPI be elements from
a. Show that the open distinguished sets Dpfi q cover Spec A ´ V paq if and only if some
power of each element f P a lies in the ideal generated by the fi ’s.
Exercise 2.7.13. Let k be a field and let A “ krt0 , t1 , . . . s be a polynomial ring in countably
many variables. Let m be the maximal ideal m “ pt0 , t1 , . . . q. Show that U “ Spec A ´ m
is not quasi-compact. Conclude that U is not the spectrum of a ring. H INT: Consider the
open cover tDpti quiě0 .
Exercise 2.7.14. Find an open set of A1Z which is not a distinguished open set.
Exercise 2.7.15. Let ϕ : A Ñ B and ψ : B Ñ C be two ring maps. Show that, with the
notation of (2.11), we have Specpψ ˝ ϕq “ Specpϕq ˝ Specpψq.
Exercise 2.7.16 (The Gaussian integers). For a complex number z “ a ` ib, we let N pzq “
a2 ` b2 denote the norm of z .
a) Show that for any two α, β P Zris, there exists µ and ρ in Zris such that
α “ µβ ` ρ where N pβq ą N pρq. H INT: Consider the lattice generated by
β, iβ and consider the square nearest α.
b) Show that Zris is a PID. H INT: If a Ă Zris is a non-zero ideal, show that a is
generated by a non-zero element in a of minimal norm.

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c) Conclude that Zris is a UFD.

Exercise 2.7.17. Show that the Zariski topology on Spec A is Hausdorff if and only if every
prime ideal p is maximal.
?
Exercise 2.7.18. ShowŞthat the closed points in Spec A form a dense set if and only if 0
equals the intersection mĂA m of all maximal ideals in A. H INT: Corollary 2.5 on page 30.
Exercise 2.7.19. Let A be an integral domain and U Ă Spec A an open non-empty subset.
Show that there is no closed point in U if and only if there is an f P A such that Af is a field.
H INT: Consider distinguished open subsets Dpf q Ă U .
Exercise 2.7.20. Let tAi uiPI be an infinite sequence of non-trivial rings, and let X be the
disjoint union of the spectra Spec Ai . Show that X is not homeomorphic to a spectrum of a
ring.
Exercise 2.7.21. Recall that a ring is local if there is only one maximal ideal.
a) Show that A is local if and only if Spec A has a unique closed point.
b) Give examples of local rings A so that Spec A consists of (i) one point; (ii) two
points; (iii) infinitely many points.
c) A map of rings ϕ : A Ñ B is said to be local if ϕpmA q Ă mB . Show that ϕ is
local if and only if the induced map f : Spec B Ñ Spec A maps the unique
closed point of Spec B to that of Spec A.
d) Give an example of a map of rings f : A Ñ B which is not local. Describe
your example in terms of the corresponding map on spectra.
Exercise 2.7.22. Show that Spec A has just onea element if and only if A is a local ring
all whose non-units are nilpotent, i.e. the radical p0q of the ring is a maximal ideal. For
Noetherian rings this is equivalent to the ring being an Artinian local ring.
Exercise 2.7.23. Perform the analysis of the fibers of the map in Example 2.38 on page 40
when the field k has characteristic 2.
Exercise 2.7.24. With reference to Example 2.41 on page 42:
a) Show that the fiber of ϕ over a prime ideal ppq is homeomorphic to
Spec Fp rxs{px2 ` 1q
and that dimFp Fp rxs{px2 ` 1q =2. H INT: Use that Zris “ Zrxs{px2 ` 1q.
b) Show that Fp rxs{px2 ` 1q is a field if and only if x2 ` 1 does not have a root
in Fp .
c) Show that Fp rxs{px2 ` 1q is a field if and only if ppqZris is a prime ideal.
Exercise 2.7.25. Consider the ring map
ϕ : Crx, ys ÝÝÑ Crx, y, zs{pxz ´ yq
which induces f : Spec Crx, y, zs{pxz ´ yq Ñ A2C . Show that the map f on the level of
closed points sends pa, ab, bq to pa, abq, and the generic point to the generic point. Show that
in this example, the image is neither open nor closed: it equals Dpxq Y V px, yq.

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Exercise 2.7.26. Let p and q be two different prime numbers and consider the morphism
ϕ : A1k Ñ A2k induced from the map krx, ys Ñ krts which is defined by the assignments
x ÞÑ tp and y ÞÑ tq . Determine all fibers of ϕ.
Exercise 2.7.27. Let A “ Crx, y, zs{pxy, xz, yzq and X “ Spec A. Consider the map
f : X Ñ A1 corresponding to the C-algebra homomorphism Crts Ñ A that sends t to
x ` y ` z . Determine all fibers of f . H INT: The set of closed points of X is the union of
the three coordinate axes in C3 , and the map sends a point to the sum of the coordinates.
Exercise 2.7.28. Describe all fibers of the following morphisms.
a) f : Spec Crx, ys{pxy ´ 1q Ñ Spec Crxs
b) f : Spec Crx, ys{px2 ´ y 2 q Ñ Spec Crxs
c) f : Spec Zrx, ys{pxy ´ nq Ñ Spec Z, where n is a non-zero integer.
d) Spec Crx, ys{px2 ` xy ` 1q Ñ Spec Crys.
Exercise 2.7.29. Describe the fibers of the map
A1k Ñ Spec krx, ys{py 2 ´ x3 ´ x2 q
induced by x ÞÑ t2 ´ 1, y ÞÑ t3 ´ t.
Exercise 2.7.30. Determine all the fibers of the morphism
? ?
Spec Zrp1 ` 5q{2s Spec Zr 5s
? ?
induced by the natural inclusion Zr 5s Ă Zrp1 ` 5q{2s.
Exercise 2.7.31. Let A “ Zrx, ys{px2 ´ y 2 ´ 5q and consider the morphism f : Spec A Ñ
Spec Z. Compute the fibers over p0q, p2q, p3q and p5q. What happens if you replace A with
the ring Zrx, ys{p3x2 ´ 3y 2 ´ 15q?
Exercise 2.7.32. Describe the following prime spectra
a) Spec Crxs{px3 ` x2 q
b) Spec Rrxs{px3 ` x2 q.
Exercise 2.7.33. Let A and B be finitely generated k -algebras, and let f : Spec B Ñ
Spec A be the map induced by a ring map A Ñ B . Show that f maps closed points to closed
points.
Exercise 2.7.34. Study the fibers of the morphisms
a) Spec Zrts{pt2 ` t ` 1q Ñ Spec Z;
b) Spec Qptqrxs{px3 ` 3x ` 1q Ñ Spec Qptq.
Exercise 2.7.35. Let A “ Rrx, ys{px2 ` y 2 ` 1q.
a) Show that for each a, b, c P R with pa, bq ‰ p0, 0q, the ideal m “ pax`by`cq
is a maximal ideal of A.
b) Show that every non-zero prime ideal is of this form. Use this to describe
Spec A. H INT: A{m is a finite extension of R. Show that 1, x, x2 are linearly
dependent, and likewise 1, y, y 2 .
Exercise 2.7.36. Show that Spec Qrxs is homeomorphic to Spec Z.

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Exercise 2.7.37. Let X “ Spec A, and let U “ SpecpAf q Ă X be a distinguished open

set. If V is a distinguished open set in U , show that V is a distinguished open set in X as
well.
Ş
Exercise 2.7.38. Let X “ Spec A show that the intersection U ĂX,U ‰H U is equal to the
set of generic points of X .
Exercise 2.7.39. Let A “ Cra, b, cs and R “ Arxs{pax2 ` bx ` cq. Study the fibers of the
morphism Spec R Ñ Spec A.
Exercise 2.7.40. Let f : Spec B Ñ Spec A be a map of spectra induced by a ring map
ϕ : A Ñ B.
a) Show that if ϕ is injective, then the image of f is dense in Spec A. a
b) More generally, show that f pSpec Bq is dense if and only if Ker ϕ Ă p0q.
c) Define ϕ : Qrxs Ñ C by sending x to π . Show that ϕ is injective. Describe the
morphism Spec C Ñ Spec A1Q .
Exercise 2.7.41. Describe the spectrum of the ring
A “ t pa, bq P Z ˆ Z | a ” b pmod 2q u.
Exercise 2.7.42. Let X be a topological space with an open cover tUi uiPI , and let tUj ujPJ
be a finite subcover where J Ă I . Show that if the sheaf sequence for F with respect to the
cover tUi ujPJ is exact, then it is also exact for the full cover tUi uiPI .
Exercise 2.7.43. Describe the following prime spectra:
a) Spec Zrxs{px2 ` 2q
b) Spec Zrxs{px2 ´ 2q
c) Spec Zrxs{px3 ` 2q
d) Spec Zrx, ys{px2 ` y, y 2 ` 1q

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Sheaves

3.1 Sheaves and presheaves

A common theme in mathematics is to study spaces by describing them in terms of their
local properties. A manifold is a space which looks locally like Euclidean space; a complex
manifold is a space which looks locally like open sets in Cn ; an algebraic variety is a space
that looks locally like the zero set of a set of polynomials. Here it is clear that point set
topology alone is not enough to fully capture the essence of these three notions. However, in
each case, the spaces come equipped with a distinguished set of functions that adequately
define them, respectively the C 8 -functions, the holomorphic functions, and the polynomials.
Sheaves provide a general framework for discussing such functions; they are objects that
satisfy basic axioms valid in each of the examples above. To explain what these axioms are,
let us consider the primary example of a sheaf: the sheaf of continuous maps on a topological
space X . By definition, X comes with a collection of ‘open sets’, and these encode what
it means for a map f : X Ñ Y to another topological space Y to be continuous: for every
open U Ă Y , the set f ´1 pU q should be open in X . For two topological spaces X and Y ,
we can define, for each open U Ă X , a set of continuous maps
CpU, Y q “ t f : U Ñ Y | f is continuous u.
Note that if V Ă U is another open set, then the restriction f |V to V of a continuous function
f is again continuous, so we obtain a map
ρU V : CpU, Y q ÝÝÑ CpV, Y q
f ÞÑ f |V .
Moreover, note that if W Ă V Ă U , we can restrict to W by first restricting to V , and so
ρU W “ ρV W ˝ ρU V . The collection of the sets CpU, Y q together with their restriction maps
ρU V constitutes the sheaf of continuous maps from X to Y .
An essential feature of continuity is that it is a local property; f is continuous if and only
if it is continuous in a neighbourhood of every point, and two continuous maps that are
equal in a neighbourhood of every point, are equal everywhere. A second property is that
continuous functions can be glued together: given an open covering tUi uiPI of an open set U ,
and continuous functions fi P CpUi , Y q that agree on the intersections Ui X Uj (formally:
fi pxq “ fj pxq for all i and j and all x P Ui X Uj ), we can patch the maps fi together to
form a continuous map f : U Ñ Y , which satisfies f |Ui “ fi for each i; simply define
f pxq “ fi pxq for any i such that x P Ui .
Essentially, a sheaf on a topological space is a structure that encodes these properties. In

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each of the examples above, there is a corresponding sheaf of C 8 -functions, respectively

holomorphic functions, and regular functions .
One may think of a sheaf as a collection of distinguished sets of functions, but they can
also be much more general mathematical objects, which in a certain sense behave as sets of
functions. The main aspect is that we want the distinguished properties to be preserved under
restrictions to open sets, that the objects are determined from their local properties, and that
‘gluing’ is allowed.

Presheaves
The concept of a sheaf may be defined for any topological space, and the theory is best
studied at this level of generality. We begin with the definition of a presheaf.

Definition 3.1 (Presheaf). Let X be a topological space. A presheaf of abelian groups F

on X consists of the following two sets of data:
(i) For each open U Ă X , an abelian group FpU q.
(ii) For each pair of nested opens V Ă U , a map of groups
ρU V : FpU q ÝÝÑ FpV q.
These are called restriction maps and must satisfy the following two conditions:
(iii) For any open U Ă X , we have ρU U “ idF pU q .
(iv) For any three open subsets W Ă V Ă U , one has ρU W “ ρV W ˝ ρU V .

We will usually write s|V for ρU V psq when s P FpU q. The elements of FpU q are usually
called sections (or sections over U ). The notation ΓpU, Fq for the group FpU q is also
common usage; here Γ is the ‘global sections’-functor (it is functorial in both U and Fq.
The notion of a presheaf is not confined to presheaves of abelian groups. One can also
consider presheaves of sets, rings, vector spaces etc. In fact, for any category C one may
define presheaves with values in C. The definition is essentially the same as for presheaves of
abelian groups, the only difference being that one requires that the FpU q are objects from C,
and of course, that restriction maps are all morphisms in C. We are certainly going to meet
sheaves with more structure beyond that of abelian groups, e.g. sheaves of rings, but they will
usually have an underlying structure of abelian group, so we start with these. We will also
encounter sheaves of sets. Most of the results we establish for sheaves of abelian groups can
be proved for sheaves of sets as well, as long as they can be formulated in terms of sets, and
the proofs are essentially the same.

Sheaves
We are now ready to give the main definition of this chapter:

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Definition 3.2 (Sheaf). A presheaf F is a sheaf if it satisfies the two conditions:

(i) (Locality axiom) Suppose U Ă X is an open set with an open covering
U “ tUi uiPI . If s, t P FpU q are sections such that
s|Ui “ t|Ui
for all i, then s “ t.
(ii) (Gluing axiom) If U and U are as in (i), and if si P FpUi q is a collection of
sections that satisfy
si |Ui XUj “ sj |Ui XUj
for all i, j P I , then there exists a section s P FpU q so that s|Ui “ si for all
i.

These two axioms mirror the properties of continuous functions mentioned in the introduction.
The Locality axiom says that sections are uniquely determined from their restrictions to
smaller open sets. The Gluing axiom says that you are allowed to patch together local sections
to a global one, provided they agree on overlaps.
A presheaf G is a subpresheaf of a presheaf F if GpU q Ă FpU q for every open U Ă X ,
and such that the restriction maps of G are the restrictons of those of F . If F and G are
sheaves, G is naturally called a subsheaf.
There is a convenient way of formulating the two sheaf axioms simultaneously. For each
open cover U “ tUi uiPI of an open set U Ă X , there is a sequence
α ś β ś
0 FpU q iPI FpUi q i,jPI FpUi X Uj q, (3.1)

where the maps α and β are defined by the two assignments αpsq “ ps|Ui qi , and βpsi q “
psi |Ui XUj ´ sj |Ui XUj qi,j . Then F is a sheaf if and only if these sequences are exact. Indeed,
exactness at FpU q means that α is injective, i.e. that s|Ui “ 0 for all i implies that s “ 0
(this is equivalent to the Locality axiom). Exactness in the middle means that Ker β “ Im α;
that is, elements si satisfying si |Ui XUj ´ sj |Ui XUj “ 0 come from an element s P FpU q
(the Gluing axiom).
This reformulation is sometimes handy when proving that a given presheaf is a sheaf.
Moreover, since FpU q “ Ker β , we can often use it to compute FpU q if the FpUi q’s and
the FpUi X Uj q’s are known.

Example 3.3 (The empty set). If F is a sheaf of abelian groups, we define FpHq “ 0. This
is essentially forced upon us by the sheaf axioms: the empty set is covered by the empty
open covering, and since the empty product equals 0, the sheaf sequence (3.1) implies that
FpHq “ 0 as well. △

Morphisms between (pre)sheaves

A morphism (or simply a map) of (pre)sheaves ϕ : F Ñ G is a collection of maps of abelian
groups ϕU : FpU q Ñ GpU q, one for each open set in X , which are required to be compatible
with the restriction maps. In other words, the following diagram commutes for each inclusion

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V Ă U of open sets:
FpU q ϕU
GpU q
ρU V ρU V (3.2)
ϕV
FpV q GpV q.

In this way, presheaves of abelian groups on X , together with their morphisms form a
category, which is denoted by PAbpXq. Likewise, the we have the category of sheaves of
abelian groups, AbpXq, which forms a subcategory.
As usual, a map ϕ between two (pre)sheaves F and G is an isomorphism if it has a
two-sided inverse, i.e. a map ψ : G Ñ F such that ϕ ˝ ψ “ idG and ψ ˝ ϕ “ idF .

Examples
Example 3.4 (Continuous functions). Take X “ Rn and let CpX, Rq be the sheaf whose
sections over an open set U is the ring of continuous real valued functions on U , and whose
restriction maps ρU V are just the good old restriction of functions. Then CpX, Rq is a sheaf
of rings (functions can be added and multiplied), and both sheaf axioms are satisfied. Indeed,
any function f : X Ñ R which restricts to zero on an open covering of X takes the value
zero at every point, so it is the zero function. Also, given continuous functions fi : Ui Ñ R
that agree on the overlaps Ui X Uj , we can form the continuous function f : U Ñ R by
setting f pxq “ fi pxq for any i such that x P Ui . △

Example 3.5 (Holomorphic functions). For a second familiar example, let X Ă C be an open
set. On X one has the sheaf AX of holomorphic functions. That is, for any open U Ă X ,
the sections AX pU q is the ring of complex differentiable functions on U . Just like in the
example above, one checks that AX forms a sheaf. In fact, AX is a subsheaf of the sheaf of
continuous functions U Ñ C. △

Example 3.6. More generally, for any two topological spaces X and Y , the presheaf
Cp´, Y q of continuous maps f : U Ñ Y forms a sheaf (they are sheaves of sets, because
we cannot in general add or multiply maps).
Furthermore, any presheaf of the form

FpU q “ t f : U Ñ Y | f satisfies p˚q u

is a subsheaf of Cp´, Y q, provided p˚q is a condition that can be checked locally. For instance,
we can take p˚q to mean ‘differentiable’, ‘smooth’, ‘analytic’, and so on. Locality holds
because it holds in Cp´, Y q. Given local sections fi P FpUi q, agreeing on the overlaps,
we can glue them to an element f P CpU, Y q, and the result lies in FpU q, because the
restriction of f to Ui is equal to fi , which satisfies p˚q. See Example 3.8 for an example
where p˚q is not a local condition. △

One of the main examples in this book will be the following:

Example 3.7 (Affine varieties). Let k be an algebraically closed field, and let X be an affine

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variety in An pkq. As in Section 1.5, we define, for an open set U Ă X , the ring of all rational
functions which are regular in U :
OX pU q “ t f P KpXq | f is regular in U u.
Note that if V Ă U , then OX pU q Ă OX pV q, so this defines a presheaf by letting the
restriction maps be the inclusions.
The sheaf axioms are also satisfied: The Locality axiom holds, because if f P KpXq
restricts to 0 in some OX pU q, then this simply means that f “ 0 in KpXq. The Gluing
axiom holds, because if fi P OX pUi q is a collection of rational functions that agree on the
overlaps Ui X Uj of an open covering, then they are all equal to the same element f in KpXq.
This rational function f must in turn be regular in all of U , because if p P U is a point, then
p lies in some Uj , and hence f “ fj can be written as a{b with bppq ‰ 0. It follows that OX
defines a sheaf of rings, which is called the sheaf of regular functions on X .
We obtain subsheaves sheaves of OX by imposing vanishing conditions on the regular
functions. For example, if p P X is a point, one has the sheaf denoted mx of holomorphic
functions vanishing at x. This is an example of an ideal sheaf: for each open U Ă X , mx pU q
is an ideal of the ring OX pU q. △
Example 3.8 (A presheaf which is not a sheaf). Let us continue the set-up in Example 3.5
to construct an example of a presheaf which is not a sheaf. Let X “ C ´ t0u, and let AX
denote the sheaf of holomorphic functions. Inside AX we find a subpresheaf given by
FpU q “ t f P AX pU q | f “ g 2 for some g P AX pU q u.
While F satisfies the Locality axiom (being a subpresheaf of a sheaf), the Gluing axiom
fails. Consider for instance, the function f pzq “ z in FpXq. For each point a P C ´ t0u,
we can find an open neighbourhood Ua and a local square root ga such that ga2 “ z . The ga
can moreover be chosen to agree on the overlaps Ua X Ua1 by consistently choosing signs.
However, there is no continuous function gpzq on C ´ t0u such that gpzq2 “ z , as any such
function would change sign when z transveses a loop around the origin. △
Example 3.9 (Constant presheaves). For any space X and any abelian group A, one has the
constant presheaf defined by ApU q “ A for any nonempty open set U (and ApHq “ 0).
This is not a sheaf in general. For instance, if X “ U1 Y U2 is a disjoint union, and
A “ Z, then any choice of integers a1 , a2 P Z will give sections of ApU1 q and ApU2 q, and
they automatically agree over the intersection, which is empty. But if a1 ‰ a2 , they cannot
be glued to an element in ApXq “ Z. In fact, the constant presheaf is a sheaf if and only if
any two non-empty open subsets of X have non-empty intersection. Algebraic varieties with
the Zariski topology are examples of such spaces.
There is a quick fix for this. We can define the following sheaf AX by letting
AX pU q “ t f : U Ñ A | f is continuous u
where we give A the discrete topology. As before, we also must put AX pHq “ 0. For a
connected open set U , we then have AX pU q “ A. More generally,
ś since f must be constant
on each connected component of U , we have AX pU q » π0 pU q A, where π0 pU q denotes
the set of connected components of U .
The new presheaf AX is called the constant sheaf on X with value A. It is a sheaf (e.g. by

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Example 3.6). That being said, the sheaf AX is not quite worthy of its name, as it is not quite
constant. △

3.2 Stalks
Suppose we are given a presheaf F of abelian groups on a topological space X . With every
point x P X there is an associated abelian group Fx called the stalk of F at x. The stalk
captures the local behavior of sections of F near x (ignoring differences that occur away
from x.) The elements of Fx are called germs of sections or just germs, near x; they are
essentially the sections of F defined in some sufficiently small neighbourhood of x.
The group Fx is formally defined as the direct limit of the groups FpU q as U runs through
the directed set of open neighbourhoods U of x (ordered by inclusion)1 :

Fx “ lim
ÝÑ FpU q.
U Qx

Concretely,
š the group Fx can be defined as follows. We begin with the disjoint union
xPU FpU q whose elements we index as pairs ps, U q where U is an open neighbourhood
of x and s is a section in FpU q. We want to identify sections that coincide near x; that is, we
declare ps, U q and ps1 , U 1 q to be equivalent, and write ps, U q „ ps1 , U 1 q, if there is an open
V Ă U X U 1 with x P V such that
s|V “ s1 |V .
This is clearly a reflexive and symmetric relation, and it is transitive as well: if ps, U q „
ps1 , U 1 q and ps1 , U 1 q „ ps2 , U 2 q, one may find open neighbourhoods V Ă U X U 1 and
V 1 Ă U 1 X U 2 of x over which s and s1 , respectively s1 and s2 , coincide. Clearly s and
s2 then coincide over the intersection V 1 X V . The relation „ is therefore an equivalence
relation.

Definition 3.10. The stalk Fx at x P X is defined as the set of equivalence classes

ž
Fx “ FpU q{ „ .
U Qx

In case F is a sheaf of abelian groups, the stalks Fx are all abelian groups. This is not
completely obvious, because sections over different open sets can not be added directly.
However, if ps, U q and ps1 , U 1 q are given, the restrictions s|V and s1 |V to any open V Ă
U X U 1 can be added, and this suffices to define an abelian group structure on the stalks.

The germ of a section

For any neighbourhood U of x P X , there is a natural map of abelian groups FpU q Ñ Fx
that sends a section s P FpU q to the equivalence class of ps, U q in Fx . This class is called
the germ of s at x, denoted sx .
1 For background on direct limits, see Appendix A

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Taking the germ is compatible with restrictions: one has sx “ ps|V qx for any other open
neighbourhood V of x contained in U , or in other words, the following diagram commutes:

FpU q Fx
ρU V (3.3)

FpV q.

When working with sheaves and stalks, it is useful to keep in mind the following principles:
‚ The germ sx of a section s vanishes if and only if s vanishes on some neighbourhood of x.
That is, there exists an open neighbourhood U of x such that s|U “ 0.
‚ All elements of the stalk Fx are germs. That is, each element of Fx is of the form sx for
some section s over some open neighbourhood of x.
‚ A sheaf F is the zero sheaf if and only if all stalks are zero, i.e. Fx “ 0 for all x P X .
A map of presheaves ϕ : F Ñ G induces for every point x P X a map between the stalks
ϕx : Fx ÝÝÑ Gx .
This map sends a pair ps, U q to the pair pϕU psq, U q, and since ϕ behaves well with respect
to restrictions, this assignment is compatible with the equivalence relations: if ps, U q and
ps1 , U 1 q are equivalent and s and s1 coincide on an open set V Ă U X U 1 , the diagram (3.2)
gives
ϕU psq|V “ ϕV ps|V q “ ϕV ps1 |V q “ ϕU 1 ps1 q|V .
One checks that pϕ ˝ ψqx “ ϕx ˝ ψx and pidF qx “ idFx , so the assignments F ÞÑ Fx and
ϕ ÞÑ ϕx define a functor from the category of sheaves to the category of abelian groups.
Example 3.11. Let X “ C, and let AX be the sheaf of holomorphic functions in X . The
stalk AX,p at a point p P X is determined as follows: Two holomorphic functions f and g
have the same germ at p if they agree on some neighbourhood of X . This in particular implies
that f and g must have the same Taylor series at p. The stalk AX,p is therefore identified
with the ring of power series that converge in a neighbourhood of p. △

3.3 The pushforward of a sheaf

Given a continuous map f : X Ñ Y , and a sheaf F on X , we define the pushforward f˚ F
on Y by defining
pf˚ FqpU q “ Fpf ´1 U q,
and the restriction maps Fpf ´1 U q Ñ Fpf ´1 V q to be those coming from F .

Definition 3.12. The sheaf f˚ F is called the pushforward or the direct image of F .

It is straightforward to verify that f˚ F is a sheaf and not merely a presheaf. Indeed, if

tUi u is an open covering of U , then tf ´1 Ui u is an open covering of f ´1 U . A set of gluing
data for f˚ F and the given covering consists of sections si P ΓpUi , f˚ Fq “ Γpf ´1 Ui , Fq

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that agree on the intersections. This means that they coincide in ΓpUi X Uj , f˚ Fq, which
equals Γpf ´1 Ui X f ´1 Uj , Fq, and they may therefore be glued together to a section in
Γpf ´1 U, Fq “ ΓpU, f˚ Fq, as F is a sheaf. The Locality axiom follows for f˚ F , because it
holds for F .
The pushfoward is also functorial in f : if g : X Ñ Y and f : Y Ñ Z are continuous
maps between topological spaces, and F is a sheaf on X , one has an equality of sheaves on
Z
pf ˝ gq˚ F “ f˚ pg˚ Fq. (3.4)
Example 3.13 (Skyscraper sheaves). Let ι : txu Ñ X be the inclusion of a closed point in
X and let A be the constant sheaf of an abelian group A on txu. The skyscraper sheaf of A
at x, denoted Apxq, is the pushforward sheaf ι˚ A. Explicitly, the sections are given by
#
A if x P U,
ApxqpU q “
0 otherwise.
△

3.4 Sheaves defined on a basis

Recall that a basis for the topology on X is a collection of open subsets B such that every
open set of X can be written as a union of members of B . In many situations, it turns out to
be convenient to define a sheaf by saying what it should be over the open sets in a specific
basis for the topology on X . This leads to the concept of a B -sheaf.
Formally, a B -presheaf F consists of:
(i) For each U P B , an abelian group F pU q.
(ii) For each pair V Ă U in B , a restriction map ρU V : F pU q ÝÝÑ F pV q.
These are required to satisfy the relations ρU U “ idF pU q and ρU W “ ρV W ˝ ρU V for each
triple W Ă V Ă U of open sets in B .
Since the intersections V X V 1 of two sets V, V 1 P B need not lie in B , we need to clarify
what we mean in the sheaf axioms. We say that a B -presheaf F is a B -sheaf if it satisfies
the additional axioms:
(iii) (Locality axiom): If tUi uiPI is an open cover of U P B with Ui P B for all i P I ,
and s P F pU q is such that s|Ui “ 0 for every i, then s “ 0.
(iv) (Gluing axiom): if tUi uiPI is an open cover of U P B with Ui P B for all i P I ,
and si P F pUi q are sections such that for any open set V P B contained in
Ui X Uj , we have si |V “ sj |V , then there exists a section s P F pU q such that
s|Ui “ si for all i P I .
Likewise, we define a morphism of B -sheaves ϕ : F Ñ G to be a collection of maps
ϕU : F pU q Ñ GpU q for U P B which are compatible with the restriction maps.
If the basis B has the property that U X V P B for every U, V P B , then a B -presheaf
F is a B -sheaf if and only if the following sequence is exact for every U P B and covering
tUi uiPI with Ui P B .
ź ź
0 ÝÝÑ F pU q ÝÝÑ F pUi q ÝÝÑ F pUi X Uj q (3.5)
iPI i,jPI

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The whole point of introducing B -sheaves is expressed in the following proposition. The
B -sheaf construction will be used when we define the structure sheaf in Chapter 4.

Proposition 3.14. Let X be a topological space and let B be a basis for the topology on
X . Then:
(i) Every B -sheaf F extends to a sheaf F on X , which is unique up to isomor-
phism.
(ii) Every morphism of B -sheaves ϕ0 : F Ñ G extends uniquely to a morphism
of sheaves ϕ : F Ñ G .
(iii) The stalk of the extended sheaf F at a point x P X is given by
Fx “ lim
ÝÑ F pU q. (3.6)
U PB, xPU

The basic idea is to define FpU q as the set of all possible gluings of sections of F over
open sets in B that cover U . More formally, a section s P FpU q is given by a set of sections
si P F pVi q for some open cover tVi uiPI of U with Vi P B , such that
si |W “ sj |W
for any W P B with W Ď Vi X Vj . If we define FpU q and GpU q as the sets of compatible
sections of F and G in this way, we can define the extended sheaf map ϕU : FpU q Ñ GpU q
by sending psi q to ϕ0 psi q.
This is indeed the best way to think of an element of FpU q, but it is important to note that
each section depends on the choice of a covering Vi . To define the group FpU q in a way that
is independent of any particular cover, we need a more formal construction. The approach
below defines FpU q by working with all possible covers of U by subsets in B at once. The
technical details of the construction is not terribly important, so the proof below may safely
be skipped on first reading. The main thing to remember is that the sheaf F exists; for all
later arguments using F , we can rely on the intuitive definition of sections as collections of
compatible local sections, as described above.

Proof Let U Ă X be any open subset. Let BU Ă B denote the open sets in B which are
contained in U and define
# +
ź ˇ
FpU q “ psV q P F pV q ˇ sV |W “ sW , @ W Ă V in B . (3.7)
ˇ
V PBU

An element of FpU q is therefore given by a set of sections sV P F pV q, such that whenever

W Ă V in BU , we have the compatibility condition sV |W “ sW .
Note that if U 1 Ă U , then BU 1 Ă BU , and so the projection maps give us restriction maps
FpU q Ñ FpU 1 q. It is not difficult to check that this makes F into a presheaf. The two sheaf
axioms for F require more work.
Locality: Suppose s “ psV q P FpU q is a collection of compatible elements and tUi uiPI
is a covering of U such that s|Ui “ 0 for every i. Let V P BU be any open set. Since B is
a basis, we can find a covering V of V consisting of open sets B P B such that for each
B P V , there is an index ipBq such that B Ă UipBq . Now s|Ui “ 0 means that sB “ 0 for

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every B Ă Ui and so sV |B “ sB “ 0 for every B P V . In particular, by the Locality axiom

for F , we must have sV “ 0. Since this happens for any V , we see that s “ 0 as well.
Gluing: Let tUi uiPI be an open cover of U and let si “ psi qV P FpUi q be a collection
of elements so that si |Ui XUj “ sj |Ui XUj for all i, j . This means that for each i and j , and
B Ă Ui X Uj in B , we have
psi qB “ psj qB . (3.8)
Fix V P BU . We would like to define an element sV P F pV q from the si and then set
s “ psV q in FpV q. As above, we choose an open cover V of V by open sets B P B such
that each B is contained in some UipBq . For each B P V , we define sB “ psipBq qB P F pBq
(by (3.8), this is independent of the choice of index ipBq). Then the elements sB glue to a
unique element sV P F pV q, because if W P B is contained in B X B 1 with B, B 1 P V ,
then by (3.8),
1 1
sB |W “ psipBq qB |W “ psipBq qW “ psipB q qW “ psipB q qB 1 |W “ sB 1 |W .
By uniqueness, the elements sV constructed in this way are moreover compatible, so it makes
sense to define s “ psV q P FpU q. It is clear that s|Ui “ si for every i. The presheaf F
therefore satisfies the Gluing axiom, and hence F is a sheaf.
Note that if U is an open set in B , there is an obvious map
ιU : F pU q ÝÝÑ FpU q.
sending t P F pU q to the collection sV “ t|V . This is clearly injective, because the composi-
tion of ιU with the projection FpU q Ñ F pU q to the ‘U -th component’ is the identity. ιU
is also surjective, because if psB q P F pU q is any element, then sB is determined by sU , as
sB “ sU |B for every B Ă U .
(iii): Let us check that the stalk of the extended sheaf F is indeed given by Fx as defined
in (3.6). By what we just showed, the natural maps ιV : F pV q Ñ FpV q are isomorphisms
for every V P B . As we may use open sets of B when computing the stalk, we obtain an
isomorphism
»
Fx “ lim
ÝÑ F pV q ÝÝÑ lim
ÝÑ FpV q “ lim
ÝÑ FpV q “ Fx .
V Qx,V PB V Qx,V PB V Qx

(ii): Saying that ϕ0 : F Ñ G is a map of B -sheaves amounts to saying that the following
diagram commutes for each pair V 1 Ă V of opens in B :
pϕ0 qV
FpV q GpV q

FpV 1 q pϕ0 qV 1
GpV 1 q.

Taking the products over all open subsets V P BU , we obtain a natural map FpU q Ñ GpU q
which extends ϕ0 . These maps are moreover compatible with the restriction maps, so we
get a map of sheaves ϕ : F Ñ G . The map ϕ must be unique, as it is determined by ϕ0 on
stalks, and two sheaf maps ϕ, ϕ1 : F Ñ G which induce the same maps on stalks are equal
(Exercise 3.5.9)

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3.5 Exercises
Exercise 3.5.1 (Differential operators). Let X “ R and let C r pX, Rq be the sheaf of
functions f : U Ñ R which are r times continuously differentiable.
a) Show that C r pX, Rq is a subsheaf of CpX, Rq.
b) Show that the differential operator D “ d{dx defines a morphism of sheaves
D : C r pX, Rq Ñ C r´1 pX, Rq.
Exercise 3.5.2. Let X be the set with two elements with the discrete topology. Find a presheaf
on X for which the Locality axiom fails. H INT: Define FpXq to be much bigger than
Fptpuq and Fptquq.
Exercise 3.5.3. In the notation of Example 3.5, the differential operator gives a map of
sheaves D : AX Ñ AX , where as previously X Ă C is an open set.
a) Show that the assignment
FpU q “ t f P AX pU q | Df “ 0 u
defines a subsheaf F of AX .
b) Show that if U is a connected open subset of X , one has FpU
ś q “ C.
c) For a not necessarily connected set U , show that FpU q “ π0 pU q C where the
product is taken over the set π0 pU q of connected components of U . Deduce
that F is the constant sheaf with value C.
Exercise 3.5.4. Let X Ă C be an open set.
a) For U Ă X open, define KpU q to be the group of meromorphic functions on U
(these are functions holomorphic on all of U except for a set of isolated points,
where they have poles). Show that KpU q is a sheaf of groups which contains
the sheaf of holomorphic functions AX as a subsheaf.
b) For points a1 , . . . , ar P X and n1 , . . . , nr natural numbers, define FpU q to
be the set of meromorphic functions in U which are holomorphic away from
the ai ’s and having a pole order bounded by ni at ai . Show that F is a sheaf of
abelian groups.
c) Are K and F sheaves of rings?
Exercise 3.5.5. Let X be a topological space and let Z Ă X be a closed subset with
inclusion ι : Z Ñ X . For a sheaf F on Z , describe the stalks of ι˚ F .
Exercise 3.5.6. Let X be a topological space and define F by setting FpXq “ 0, FpHq “ 0
and FpU q “ Z for every other open set, and with restriction maps FpU q Ñ FpV q equal to
the identity if U and V are different from H and X , and the zero map otherwise. Show that
F is a presheaf. Is F a sheaf?
Exercise 3.5.7 (The sheaf of homomorphisms). Given two presheaves F and G , we may
form a presheaf HompF, Gq by letting the sections over an open U be given by
HompF, GqpU q “ HomAbpXq pF|U , G|U q. (3.9)
If V Ă U , the restriction map sends ϕ : F|U Ñ G|U to the restriction ϕ|V : F|V Ñ G|V .
Show that HompF, Gq is a sheaf whenever G is a sheaf.

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Exercise 3.5.8. Let F be a sheaf and let s, t P FpU q be two sections. Show that s “ t if
and only if sx “ tx for every x P U .
Exercise 3.5.9. Let ϕ, ψ : F Ñ G be maps of presheaves and assume that G is a sheaf. Prove
that ϕ “ ψ if and only if ϕ and ψ induce the same maps on all stalks, i.e. ϕx “ ψx for every
x P X . H INT: Use Exercise 3.5.8.
Exercise 3.5.10. Let ϕ : F Ñ G be a map of sheaves. We say that ϕ is injective (resp.
surjective) if ϕx : Fx Ñ Gx is injective (resp. surjective) for every point x P X .
a) Show that ϕ is injective if and only if ϕU : FpU q Ñ GpU q is injective for every
open set U .
b) Give an example of a map of sheaves ϕ so that ϕx is surjective for all x P X ,
but ϕX : FpXq Ñ GpXq is not surjective. H INT: Example 3.8.
Exercise 3.5.11. Verify the formula (3.4).
Exercise 3.5.12. Denote by t˚u a one point set. Let X be a topological space and f : X Ñ
t˚u be the one and only map. Show that f˚ F “ ΓpX, Fq (where strictly speaking ΓpX, Fq
stands for the constant sheaf on t˚u with value ΓpX, Fq).
Exercise 3.5.13. Let X be a topological space and x P X a point that is not necessarily
closed. Let ι : txu Ñ X be the inclusion. Let A be the constant sheaf on txu with value the
group A. Show that the stalks of ι˚ A are
#
A if y P txu
pι˚ Aqy “
0 otherwise.
Exercise 3.5.14. Let F and G be two sheaves on a space X and assume there is an open
covering U of X and isomorphisms θU : F|U » G|U that match on intersections; that is,
θU |U XU 1 “ θU 1 |U XU 1 . Show that there is an isomorphism θ : F » G extending the θU ’s.
H INT: It is possible to do this via B -sheaves.
Exercise 3.5.15. Let U “ tUi uiPI be an open cover of a space X and let U 1 “ tUi uiPJ be a
subcover (J Ă I ). Show that the sequence (3.1) is exact for the covering U if and only if it is
exact for U 1 .
Exercise 3.5.16 (The restriction of a sheaf). Let U Ă X be an open subset of a topological
space X and let F be a sheaf on X . Define F|U by the assignment F|U pV q “ FpV q for
V Ă U.
a) Show that F|U defines a sheaf on U , and that the stalks at points in U are equal
to the corresponding stalks of F .
b) Let G be a sheaf on U . Let ι : U Ñ X be the inclusion map. Show that
ι˚ pGq|U » G .
Exercise 3.5.17. Let X “ R with the standard topology.
a) Let F be the subpresheaf of Cp´, Rq defined by
FpU q “ t f : U Ñ R | f is bounded u
Show that F is not a sheaf.

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b) Let G be the subpresheaf of Cp´, Rq defined by

GpU q “ t f : U Ñ R | f p0q “ f p1q u
Show that G is not a sheaf.
Exercise 3.5.18. Let X be aŤtopological space where every open cover of any openŤ set has a
finite subcover (i.e., if U “ iPI Ui , then there exists a finite J Ă I such that U “ iPJ Ui ).
Show that a presheaf F is a sheaf if and only if it satisfies the Locality and Gluing axioms for
all finite covers.

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Schemes

In this chapter, we introduce the main objects of this book, namely schemes. We begin by
studying affine schemes, which serve as the building blocks for schemes in general. Every
scheme admits an open cover consisting of affine schemes, and understanding the mechanics
of affine schemes is essential for understanding schemes in general.
An affine scheme, like any scheme, has two components: a topological space and a sheaf
of rings. The topological space is the spectrum of a ring, and the sheaf of rings is the structure
sheaf, which will be defined in Section 4.1.
To motivate the definition of the structure sheaf, recall the sheaf of regular functions on
an affine variety (Example 3.7). Let X Ă An pkq be an affine variety with coordinate ring
A “ ApXq. For each open set U Ă X , the ring OX pU q consists of the rational functions
that are regular at every point of U . The elements of OX pU q can be viewed either as elements
of the function field KpXq, or as functions f : U Ñ k . Moreover, each restriction map
OX pU q Ñ OX pV q can be seen as an inclusion of subrings of KpXq, or as an actual
restriction map on functions.
For a distinguished open set Dpf q, the ring of regular functions OX pDpf qq is simply
the localization Af (Proposition 1.25). This is intuitive, because the elements of the form
a{f n define regular functions on Dpf q. Moreover, if Dpgq Ă Dpf q, the restriction map
corresponds to the canonical localization map Af Ñ Ag .
In fact, the sheaf OX is completely determined by knowing that OX pDpf qq “ Af for
each f P A. This is because the distinguished open sets form a basis for the Zariski topology
on Spec A and

č
OX pU q “ OX pDpf qq. (4.1)
Dpf qĂU

If one tries to carry out the above construction for a general ring A, there are a few issues
that arise. First, there may not exist a natural field k for which we can view elements of
OX pU q as functions U Ñ k . More critically, the localization maps Af Ñ Ag may fail to
be injective (so an intersection like (4.1) does not make sense). This happens already in the
simple example when A “ krx, ys{pxyq, which corresponds to the union of x-axis and the
y -axis in the affine plane. Here the nonzero element x P A maps to 0 via the localization
map A Ñ Ay . So this behaviour is not a big surprise; it naturally appears once we allow
reducible spaces into our discussion.

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4.1 The structure sheaf on the spectrum of a ring

Motivated by the above discussion, we define the structure sheaf O using the basis B of
distinguished open sets Dpf q. For each Dpf q P B , we define
OpDpf qq “ Af . (4.2)
If Dpgq Ă Dpf q, we take the restriction map OpDpf qq ÝÝÑ OpDpgqq to be the canonical
localization map Af Ñ Ag .
There is a small subtlety here, namely that different f ’s may define the same distinguished
open set U “ Dpf q (for instance, Dpf q “ Dpf 2 q). Of course, one can get around this by
simply fixing once and for all a representative f for each U . A more canonical approach is to
work with a localization which does not involve any choices – defining OpU q “ SpU q´1 A,
where SpU q is the multiplicative set
SpU q “ t s P A | s R p for all p P U u. (4.3)
The elements of OpU q are the fractions a{s with a, s P A and sppq ‰ 0 for all p P U .
If V Ă U are two distinguished open sets, then SpU q Ă SpV q, and there is a canonical
localization map SpU q´1 A Ñ SpV q´1 A, which gives restriction maps OpU q Ñ OpV q.
The localization SpU q´1 A is canonically isomorphic to Af . To see this, note that since
f P SpU q, there is a canonical localization map Af Ñ SpU q´1 A. This map is injective: if
af ´n P Af maps to 0 in SpU q´1 A, there is an s P SpU q such that sa “ 0. By definition,
a have spa
we pq ‰ 0 for every p P Dpf q, and hence Dpf q Ă Dpsq. This is equivalent to
pf q Ă psq, and we may write f n “ cs for some c P A and n P N. Multiplying
sa “ 0 by c now shows that f n a “ 0, and so a “ 0 in Af . The map is also surjective: if
a{s P SpU q´1 A, then writing f n “ cs as before, we have a{s “ ca{f n , which is in the
image of Af Ñ SpU q´1 A.
In light of this, we will from now on simply write OpDpf qq “ Af , bearing in mind that
O is defined in terms of a canonical localization.

Proposition 4.1. O is a B -sheaf of rings.

Proof Step 1: Finite covers. Suppose that Dpf q is covered by finitely many distinguished
opens Dpf1 q, . . . , Dpfr q. We need to show that the following sequence is exact:
śr β śr
0 Af α i“1 Afi i,j“1 Afi fj (4.4)

where α psq “ ps{1, . . . , s{1q and βps1 , . . . , sr qi,j “ si {1 ´ sj {1.

We will do this by a series of reductions. First observe that since (4.4) is a sequence of
Af -modules, it is exact if and only if it is exact after being localized at every prime ideal
p P SpecpAf q “ Dpf q. After reordering the indices, we may assume that p P Dpf1 q, which
means that f1 is invertible in Ap .
Since localization commutes with finite direct products, the localized sequence takes the
form
śr β śr
0 pAp qf α i“1 pAp qfi i,j“1 pAp qfi fj . (4.5)

Here we’ve used the isomorphisms pAfi qp “ pAp qfi and pAfi fj qp “ pAp qfi fj .

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Replacing A by Ap , we therefore reduce to showing the exactness of:

α śr β śr
0 A i“1 Afi i,j“1 Afi fj . (4.6)

where f1 is invertible in A. Now α is injective because its first component is the localization
map ℓ : A Ñ Af1 “ A, which is an isomorphism because f1 is invertible. For exactness in
the middle: given pai {fini qri“1 P Ker β , we have ai {fini “ a1 {f1n1 in Af1 fi » Afi for all
i ě 2. Therefore, if we define s “ ℓ´1 pa1 {f1n1 q P A, then s satisfies αpsq “ pai {fini qri“1 .
This proves that Ker β “ Im α. Ť
Step 2. General covers. Now suppose that Dpf q “ iPI Dpfi q is an arbitrary cover
by distinguished open sets. Since Dpf q is quasi-compact, we may choose a finite subset
J “ ti1 , . . . , ir u Ă I so that tDpfj qujPJ forms a subcover.
Locality axiom: if s P Af maps to zero in Afi for every i P I , then in particular, it maps
to zero in each Afi for i P J , so by Step 1, we have s “ 0 in Af .
Gluing axiom: given compatible elements si P Afi for i P I (meaning si {1 “ sj {1 in
Afi fj for all i, j P I ). Then Step 1 provides an s P Af such that si “ s{1 P Afi for all
i P J . We claim that s in fact induces si for all i P I . For this, fix an index α P I . Applying
Step 1 to the finite covering consisting of Dpfi q, i P J Y tαu, we get an element s1 P Af
such that s1 {1 “ si in Afi for all i P J and also s{1 “ sα . As s and s1 coincide in each Afi
for i P J , we must also have s “ s1 in Af by uniqueness in Step 1, and hence s{1 “ sα
holds in Afα as well.
Using Proposition 3.14 on page 56, we may now make the following definition:

Definition 4.2. The structure sheaf OSpec A on Spec A is the unique sheaf extending the
B -sheaf O.

Using the sheaf sequence (3.1), we may compute OSpec A pU q for any open set U : cover U
by distinguished open sets Dpfi q, i P I ; then OSpec A pU q can be identified with the ring
#ˆ ˙ ź +
ai ˇ a
ˇ i aj
OSpec A pU q “ P Afi ˇ ni “ nj in Afi fj for all i, j P I . (4.7)
fini iPI
fi fj
That being said, we will basically never need to know the ring OSpec A pU q for other open
sets than distinguished open sets U “ Dpf q.

Proposition 4.3.
(i) ΓpSpec A, OSpec A q “ A.
(ii) If x P Spec A is a point corresponding to a prime ideal p, then
OSpec A,x “ Ap ,

Proof We defined OSpec A so that OSpec A pDpf qq “ Af for every every f P A. Taking f “
1, we see that the (i) holds. For (ii), note that we may compute the stalk using distinguished
open sets:
OX,x “ lim ÝÑ OSpec A pDpf qq “ lim
ÝÑ Af .
xPDpf q f Rp

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Note that any element a{s P Ap comes from some Af for f P A ´ p. Hence there is a
surjective map limf Rp Af Ñ Ap . This is also injective: an element a{f n maps to 0 in Ap
ÝÑ
precisely when there is an element s P A ´ p such that as “ 0, but then a{f n “ 0 in Ag ,
where g “ sf . This proves (ii).

Examples
Example 4.4. For a field K , the structure sheaf OSpec K is a constant sheaf with the value K
at the single point of Spec K . △
Example 4.5. The structure sheaf of Spec Z satisfies OSpec Z pDpnqq “ Zr n1 s for each
natural number n. The stalks of OSpec Z at the closed point ppq is equal to OSpec Z,p “ Zppq
and at the generic point the stalk equals OSpec Z,p0q “ Zp0q “ Q. △
Example 4.6. Let X “ Spec Crts. Then the stalk of OX at the generic point η “ p0q is
equal to OX,η “ Cptq. Each closed point p P X corresponds to a maximal ideal pt ´ aq, and
the stalk of OX at p is equal to OX,p “ Crtspt´aq . △
Example 4.7. We continue Example 2.14 about spectra of DVR’s. The spectrum X “ Spec A
has the two points x and η , and there are three open sets: H, tηu, and X . Note that η “ Dpxq
is a distinguished open in X . The structure sheaf takes the following values at these opens:
OX pHq “ 0, OX pXq “ A, OX ptηuq “ Ax “ K,
where K denotes the fraction field of A. The stalks are given by OX,x “ Apxq “ A and
OX,η “ Ap0q “ K . △
Example 4.8 (Disconnected spectra). The structure sheaf may be used to prove that a ring
A whose spectrum Spec A is not connected, decomposes as the direct product of two rings.
Suppose X “ U1 Y U2 , where U1 and U2 are open and closed subsets with U1 X U2 “ H.
The sheaf exact sequence takes the form

0 OX pXq “ A OX pU1 q ˆ OX pU2 q OX pU1 X U2 q “ 0,

and we deduce that there is an isomorphim of rings A » OX pU1 q ˆ OX pU2 q. △

Example 4.9. The structure sheaf behaves particularly well when the base ring A is an
integral domain. In this case, all the localizations Af are subrings of the field of fractions
K of A, and for Dpgq Ă Dpf q, the localization maps Af Ñ Ag are simply inclusions of
subrings. This implies that OX pU q is a subring of K for any open set U . Indeed, covering
Ş sets tDpfi quiPI , the formula (4.7) shows that OX pU q is identified
U by distinguished open
with the intersection iPI Afi inside K . Moreover, by Proposition A.5, we have
č č č č
OX pU q “ Afi “ Ap “ Ap .
iPI iPI pPDpfi q pPU

Consequently, when A is an integral domain, the intuition from affine varieties is valid: the
rings OX pU q are subrings of the fraction field of A, and the elements f P OX pU q are
precisely those for which, for each point p P U , we may write f “ a{b and bppq ‰ 0. △

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4.2 The sheaf associated to an A-module

The contruction of the structure sheaf can be extended for arbitrary A-modules. For each
A-module M , we define a presheaf M Ă on Spec A, whose sections over distinguished open
sets Dpf q are given by:
M
ĂpDpf qq “ Mf .

If Dpgq Ă Dpf q, the restriction map is given by the canonical localization map Mf Ñ Mg
(sending m{f n to man {g nr , where g r “ af ). The same proof as for the structure sheaf
(Proposition 4.1 on page 62), shows that this is actually a B -sheaf, which extends uniquely
to a sheaf on Spec A, which we will continue to denote by M Ă.
The sheaf M has the following properties:
Ă

‚ Stalks: For a point x P Spec A corresponding to a prime ideal p,

M
Ăx “ Mp .

‚ Global sections:
ΓpSpec A, M
Ăq “ M. (4.8)
‚ For any open set U covered by distinguished open sets tDpfi quiPI , there is an exact
sequence
ś β ś
0 ΓpU, M
Ăq
i Mfi i,j Mfi fj ,

where β is defined by
ˆ ˙
m1 ms mi mj
β , . . . , ns “ ´ nj (4.9)
f1n1 fs i,j fini fj
This tilde-construction is functorial in M . For any A-linear map ϕ : M Ñ N , there is an
induced map ϕ r: MĂÑN r . To define ϕr, it suffices, by Proposition 3.14, to specify it over the
distinguished open sets Dpf q. Here, we define ϕ̃Dpf q : M ĂpDpf qq Ñ N r pDpf qq to be the
n n
localization of ϕ, that is, ϕf : Mf Ñ Nf given by m{f ÞÑ ϕpmq{f . This defines a map of
B -sheaves because the following diagram commutes for each f and g with Dpgq Ă Dpf q:
ϕf
Mf Nf

ϕg
Mg Ng

Indeed, writing g r “ af as above, the composition Mf Ñ Mg Ñ Ng sends m{f n to

ϕpman q{g rn “ an ϕpmq{g rn , which is is the same as the image via Mf Ñ Nf Ñ Ng .
We clearly have ϕĆ ˝ ψ “ ϕr ˝ ψr, whenever ϕ and ψ are composable A-linear maps.
Consequently the tilde-operation defines a covariant functor from the category ModA of
A-modules to the category AbpXq of sheaves on X “ Spec A.
The sheaves MĂ are rather special sheaves, and they play an important role in algebraic
geometry. We will study them in detail in Chapter 14.

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Example 4.10. Let A be a ring and let I Ă A be an ideal. Then Ir is an ideal sheaf in OSpec A .
r q is an ideal of OX pU q. For
This means that for each U Ă Spec A, the space of sections IpU
U “ Dpf q, we have IpDpf qq “ IAf , the ideal generated by I in Af .
r △
Example 4.11. Let A “ kru, vs{pu2 ` v 2 ´ 1q and X “ Spec A. Consider the A-module
M given by the quotient
M “ pAe1 ‘ Ae2 q {pue1 ` ve2 q.

Let us determine the stalk of M Ă at the point x P X corresponding to the prime ideal
p “ pu, v ´ 1q. Since v is invertible in Ap , we can replace the relation ue1 ` ve2 by
uv ´1 e1 ` e2 . This allows us to eliminate the component Ap e2 , and we find:
Ăx “ Mp “ pAp e1 ‘ Ap e2 q {puv ´1 e1 ` e2 q » Ap “ OX,x .
M

Similar arguments show that pM

Ăqp » Ap for every p. Indeed, for any p, either u R p or
Ă and OX have isomorphic stalks at every point.
v R p. Therefore M △

4.3 Locally ringed spaces

We would like to define a scheme to be a space which is ‘locally affine’, that is, one where
every point has a neighbourhood which is isomorphic to the spectrum of a ring. To be able to
make such a definition precise, we need a suitable category of spaces to work with. To this
end, we use the two pieces of data we have in the affine case: the topological space Spec A
and its sheaf of rings OSpec A .

Definition 4.12 (Locally ringed spaces). A locally ringed space is a pair pX, OX q where
X is a topological space and OX is a sheaf of rings on X such that all the stalks OX,x
are local rings.

To make this into a category, we need to specify what a morphism between two locally ringed
spaces is. Reflecting the above definition, a morphism betwen pX, OX q and pY, OY q should
have two components, one map between the underlying topological spaces X and Y and one
on the level of sheaves. Note that it does not make sense to talk about morphisms OY Ñ OX ,
as these sheaves live on different spaces. Instead, once a continuous map f : X Ñ Y is
specified, the sheaf map should be a map OY Ñ f˚ OX of sheaves of rings on Y . This means
that for all open subsets U Ă Y , there are ring maps
fU7 : OY pU q ÝÝÑ OX pf ´1 U q,
compatible with the restriction maps; that is, such that the following diagrams commute:
7
fU
OY pU q OX pf ´1 U q
ρU V ρf ´1 U f ´1 V (4.10)

OY pV q OX pf ´1 V q.
fV7

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The intuition again comes from the setting of affine varieties. As we saw on page ??, a
morphism of affine varieties f : X Ñ Y is precisely a continuous map such that the pullback
g ˝ f is a regular function on f ´1 pU q whenever g is regular on U . In other words, the
pullback is a ring map OY pU q Ñ OX pf ´1 U q, and the collection of all of these defines a
map of sheaves OY Ñ f˚ OX .
Note that if g is a regular function which vanishes at a point y P Y then its pullback
f 7 pgq “ g ˝ f vanishes at every point of x P f ´1 pyq in U . In other words, if g P my , then
f 7 pgq P mx for each such x and y .
For a general locally ringed space, we do not have the luxury of speaking about functions
into some fixed field k , so the sheaf map f 7 has to be specified as part of the definition of a
morphism. Moreover, the condition of ‘pulling back vanishing functions’ translates into the
following requirement. For a point x P X and y “ f pxq, the morphism f 7 induces a map of
local rings:
fx7 : OY,y ÝÝÑ OX,x , (4.11)
This map is defined as follows: pick an element sy P OY,y and represent it as the germ of a
7
section s P OY pW q over some open set W Ă Y . Then the section t “ fW psq is a section of
OX pf ´1 W q. We define fx7 psy q to be the germ of this section at x, i.e. fx7 psy q “ tx P OX,x .
This makes sense because f ´1 W contains x. By the definition of the stalk, it is clear that
this does not depend on the choice of open set W containing y .
In the case of affine varieties, the map (4.11) sends a rational function g P OY,y which
is regular at y “ f pxq to g ˝ f which is regular at x. By what we said above, fx7 sends the
maximal my into mx for every x P X . In other words, fx7 is a map of local rings.
For a general morphism of locally ringed spaces, we require that the sheaf map f 7 should
induce a map of local rings for every x P X . This is a natural condition, but it is by no means
automatic when starting from a general map of sheaves of rings OY Ñ f˚ OX .

Definition 4.13 (Morphisms of locally ringed spaces). A morphism, or simply map, of

locally ringed spaces, is a pair
pf, f 7 q : pX, OX q ÝÝÑ pY, OY q,
where
(i) f : X Ñ Y is a continuous map, and
(ii) f 7 : OY Ñ f˚ OX is a map of sheaves of rings on Y , so that for each x P X ,
with y “ f pxq the induced map on stalks
fx7 : OY,y ÝÝÑ OX,x
is a map of local rings; that is, fx7 pmy q Ă mx .

Morphisms of locally ringed spaces can be composed. Given pf, f 7 q : pX, OX q Ñ

pY, OY q and pg, g 7 q : pY, OY q Ñ pZ, OZ q, the map X Ñ Z given by the composition
g ˝ f on the level of topological spaces. The sheaf map pg ˝ f q7 is defined over an open set
U Ă Z as the composition
g7 f7
OZ pU q OY pg ´1 U q OX ppg ˝ f q´1 U q “ ΓpU, pg ˝ f q˚ OX q.

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An isomorphism of locally ringed spaces is a morphism f : X Ñ Y which admits an inverse

morphism. In other words, there is a morphism g : Y Ñ X such that g ˝ f “ idX and
f ˝ g “ idY . In more concrete terms, this boils down to: (i) f being a homeomorphism, and
(ii) and fU7 : OY pU q Ñ OX pf ´1 U q being an isomorphism of rings for every open U Ă Y .

Spec A as a locally ringed space

For a ring A, the pair pSpec A, OSpec A q is by design a locally ringed space. The set Spec A
is endowed with the Zariski topology and the structure sheaf OSpec A is a sheaf of rings with
stalks which are local rings of the form Ap . We will now show that ring maps induce maps of
locally ringed spaces.
Let ϕ : A Ñ B be a map of rings and write X “ Spec B and Y “ Spec A. We define a
morphism
pf, f 7 q : pX, OX q ÝÝÑ pY, OY q. (4.12)
as follows. On the level of topological spaces, we define f : X Ñ Y by
f ppq “ ϕ´1 ppq. (4.13)
The sheaf map f 7 : OY Ñ f˚ OX is defined using the B -sheaf construction. Consider a
distinguished open set Dpgq Ă Y “ Spec A. Then,
f ´1 pDpgqq “ Dpϕpgqq. (4.14)
so the sections are
OY pDpgqq “ Ag ,
f˚ OX pDpgqq “ OX pDpϕpgqqq “ Bϕpgq .
We define f 7 : OY Ñ f˚ OX over Dpgq by the canonical localization map
Ag ÝÝÑ Bϕpgq
a{g n ÞÑ ϕpaq{ϕpgqn . (4.15)
Note that if ϕpgq “ 0, then Bϕpgq “ 0 and this is the zero map.
These maps define a morphism of B -sheaves: if Dphq Ă Dpgq, we have a commutative
diagram

OY pDpgqq OX pDpϕpgqqq Ag Bϕpgq

OY pDphqq OY pDpϕphqqq Ah Bϕphq

Here, the vertical maps on the left are restriction maps of sheaves, while the vertical maps on
the right are localization maps of rings.
Finally, we check that for each point x P X , the induced map fx7 : OY,f pxq Ñ OX,x is
a map of local rings. Let p for the prime ideal corresponding to x, and let q “ ϕ´1 ppq for
the prime ideal corresponding to f pxq P Spec A. From the paragraph following (4.11), we
can understand the map fx7 as follows. Take an element t P OY,f pxq “ Aq . Writing it as

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t “ a{g n allows us to view it as the germ of a section a{g n P OX pDpgqq for some open set
Dpgq containing q. Note that q P Dpgq means that g R q “ ϕ´1 ppq. Then fx7 ptq is the image
of t via the composition

Ag ÝÝÑ Bϕpgq ÝÝÑ Bp “ OX,x . (4.16)

Now, if t P qAq lies in the maximal ideal, then we can write t “ a{g n with a P q “ ϕ´1 ppq.
Then ϕpaq{ϕpgqn P pBϕpgq , and hence fx7 ptq P pBp lies in the maximal ideal of OX,x . This
shows that pf, f 7 q is a morphism of locally ringed spaces.

Here is an example of a pair pf, f 7 q which is not a morphism of locally ringed spaces.

Example 4.14. Let X “ Spec Cptq and Y “ A1C “ Spec Crts. There is a natural map
f : X Ñ Y induced by the inclusion Crts Ă Cptq. On the level of topological spaces, X
consists of a single point ν , and f maps ν to the generic point η of Y . The corresponding
stalk map fν7 : OY,η Ñ OX,ν is the identity map

OY,η “ Crtsp0q “ Cptq ÝÝÑ Cptq “ OX,ν ,

which is certainly a map of local rings.

On the other hand, we could try to define a strange map g : X Ñ Y by sending ν to some
other point y P A1C corresponding to a maximal ideal pt ´ aq Ă Crts. Then pg˚ OY qpU q “
Cptq if U contains y and 0 otherwise. We can define the sheaf map g 7 : OY Ñ g˚ OX as
follows: for an open set U Ă Y , gU7 is the inclusion map ΓpU, OY q Ă Cptq if y P U , and
the zero map otherwise. This is indeed a map of sheaves of rings. However, the map on stalks
is given by

gν7 : OY,y “ Crtspt´aq ÝÝÑ Cptq “ OX,ν

and this sends the maximal ideal pt ´ aq to the unit ideal in Cptq. In other words, the element
t ´ a vanishes at y P Y , but its pullback, the image of t ´ a in Cptq does not vanish at ν .
Therefore, g is not a morphism of locally ringed spaces. △

4.4 Schemes
We have finally come to the definition of a scheme.

Definition 4.15. An affine scheme is a locally ringed space pX, OX q which is isomorphic
to pSpec A, OSpec A q for some ring A.

If X is a locally ringed space and U Ă X is an open subset, the restriction OX |U is a

sheaf on U making pU, OX |U q into a locally ringed space. Indeed, the stalks of OX |U are
the same as those of OX for every point of U . In particular, it makes sense to ask when a
locally ringed space is locally isomorphic to an affine scheme:

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Definition 4.16 (Schemes). A scheme is a locally ringed space pX, OX q which is locally
isomorphic to an affine scheme. In other words, there is an open cover tUi uiPI of
open subsets of X such that each pUi , OX |Ui q is isomorphic to some affine scheme
pSpec Ai , OSpec Ai q.

To summarize, a scheme consists of a topological space X , which has a covering consisting

of open sets of the form Spec Ai , and sheaf of rings OX which restricts to the structure
sheaves OSpec Ai . For a point x P X , the stalk OX,x is a local ring with maximal ideal mx .
The quotient field κpxq “ OX,x {mx is called the residue field. If x is contained in an affine
open set U “ Spec A and corresponds to a prime ideal p, then we have

OX,x “ Ap , mx “ pAp , κpxq “ Ap {pAp .

A morphism, or map for short, between two schemes X and Y is simply a map f between X
and Y regarded as locally ringed spaces. This also has two components: a continuous map,
which we shall denote by f as well, and a map of sheaves of rings

f 7 : OY ÝÝÑ f˚ OX ,

with the additional requirement that the induced map on stalks is a map of local rings, i.e.,
takes the maximal ideal my into mx .
In this way, the schemes form a category, which we shall denote by Sch. It contains the
category of affine schemes AffSch as a full subcategory.

4.5 Morphisms into an affine scheme

The following is a fundamental result in the theory of schemes.

Theorem 4.17. Let X be a scheme and let A be a ring. Then there is a natural bijection
HomSch pX, Spec Aq “ HomRings pA, OX pXqq. (4.17)

Proof Let Y “ Spec A and consider it as a scheme with the structure sheaf OSpec A .
Given a morphism of schemes pf, f 7 q : X Ñ Y , we evaluate f 7 over the open set U “ Y ,
to obtain a ring map

fY7 : OY pY q “ A ÝÝÑ pf˚ OX qpY q “ OX pXq.

The bijection in (4.17) is defined by sending pf, f 7 q to fY7 .

Injectivity of (4.17): Let pf, f 7 q and pg, g 7 q be two morphisms of schemes X Ñ Spec A
and assume that f 7 and g 7 induce the same ring map ϕ : A Ñ OX pXq.
Let x P X and write p Ă A for the prime ideal corresponding to y “ f pxq. If ℓ : A Ñ Ap
denotes the localization map, and ρx : OX pXq Ñ OX,x denotes the germ map, then there is

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a commutative diagram
ϕ
A OY pY q OX pXq
ℓ ρx (4.18)
fx7
Ap OY,f pxq OX,x
From the diagram, we see that
p˚q
p “ ℓ´1 ppAp q “ ℓ´1 ppfx7 q´1 pmx qq “ ϕ´1 pρ´1
x pmx qq. (4.19)
where the equality p˚q holds because fx7
is a map of local rings. As the right-hand side of
(4.19) only depends on ϕ and x, we see that p is determined by ϕ and hence f pxq and gpxq
must be equal, as they correspond to the same prime ideal in A. Therefore, f and g are equal
on the level of topological spaces. We will from now on write f for this map.
Next we show that f 7 “ g 7 , as maps of sheaves OY Ñ f˚ OX . As the sheaves involved
are on Y “ Spec A, it suffices to show that they agree over the distinguished open sets. For
h P A, consider the diagram
ϕ ϕ
OY pY q OX pXq A OX pXq
=
7 7
fDphq fDphq
OY pDphqq OX pf ´1 Dphqq Ah OX pf ´1 Dphqq

By the universal property of localization, a morphism Ah Ñ OX pf ´1 Dphqq exists if and

only if h becomes invertible OX pf ´1 Dphqq, and in which case it is unique. This means that
7 7
fDphq “ gDphq as ring maps. As this happens for every h P A, we find that f 7 “ g 7 .
Surjectivity of (4.17): Let ϕ : A Ñ OX pXq be a ring map. We will define a morphism of
schemes f : X Ñ Y such that fY7 “ ϕ.
In light of (4.13), there is a natural choice for the map on topological spaces. For x P X ,
we set f pxq to be the prime ideal in A that is the preimage of mx under the composition
ϕ ρ
A ÝÑ OX pXq ÝÝxÑ OX,x .
First we check that this assignment is continuous. For a section s P OX pXq, define the set
Dpsq “ t x P X | sx R mx u.
This set is open in X , because if sx R mx for some point x P X , then sx is invertible in
OX,x , but then s|V is also invertible in some neighbourhood V of x, and V Ă Dpsq (see
Lemma 4.18).
Consider a distinguished open set Dpgq Ă Spec A. Then
f ´1 pDpgqq “ t x P X | f pxq P Dpgq u
“ t x P X | g R pρx ˝ ϕq´1 pmx q u
“ t x P X | ρx pϕpgqq R mx u
“ t x P X | pϕpgqqx R mx u
“ Dpϕpgqq.

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Since Dpϕpgqq is open in X , and the sets Dpgq form a basis for the topology on Spec A, we
conclude that f is continuous.
By Lemma 4.18, the restriction ϕpgq|Dpϕpgqq is invertible in OX pDpϕpgqqq. Therefore, by
the universal property of localization, the ring map ϕ induces a ring map

Ag ÝÝÑ OX pDpϕpgqqq. (4.20)

Explicitly, (4.20) sends ag ´n to ϕpaq|Dpϕpgqq ¨pϕpgq|Dpϕpgqq q´n . Moreover, if Dphq Ă Dpgq,

then we have the following diagram

Ag OX pDpϕpgqqq
(4.21)

Ah OX pDpϕphqqq

which commutes by the uniqueness part in the universal property of localization.

7
We define fDpgq to be the ring map (4.20). By (4.21), these maps are compatible with the
restriction maps, and so we get a map of sheaves f 7 : OY Ñ f˚ OX .
To see that pf, f 7 q is a morphism of locally ringed spaces, we need to check that for each
x P X , the induced map fx7 : OY,f pxq Ñ OX,x is a map of local rings. That is, we need to
check that f 7 ppAp q Ă mx . But this follows by looking at the commutative diagram (4.18),
because p is exactly the inverse image of mx under the composition ρx ˝ ϕ.
Finally, if we set g “ 1 in the definition of f 7 , then ϕpgq “ 1 and (4.20) is simply given
by the original ring map ϕ : A Ñ OX pXq. This means that ϕ is induced by pf, f 7 q.

In the surjectivity part, we used the following little lemma:

Lemma 4.18. Let X be a scheme and let s P OX pU q be a section.

(i) If sx is invertible in OX,x for some point x P U , then there is an open set V
containing x such that s|V is invertible in OX pV q.
(ii) If sx is invertible in OX,x for every x P U , then s is invertible in OX pU q.

Proof (i): Let tx P OX,x be the inverse of sx , so that sx ¨ tx “ 1 in OX,x . As all elements
of OX,x are germs, we may find an open set V containing x and a section tV P OX pV q such
that ptV qx “ tx . Then ptV qx ¨ sx “ 1 in OX,x , so shrinking V if neccessary, we may assume
that sV ¨ t|V “ 1 in OX pV q.
(ii): By the first part, we may find an open covering tVi uiPI of U and sections ti P OX pVi q
such that ti ¨ s|Vi “ 1 in OX pVi q. As inverses are unique, we must have ti |Vi XVj “ tj |Vi XVj
for every i and j , and so the ti glue to a section t P OX pU q. By Locality, the section t has
the property that st “ 1 in OX pU q, because st and ‘1’ both restrict to the same section over
each Vi . Hence s is invertible in OX pU q.

In the above proof, we never used the fact that X had an affine covering consisting of
affine schemes. This means that the theorem and the proof is valid even in the case where X
is a general locally ringed space.

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Corollary 4.19. For any scheme, there is a canonical morphism θ : X Ñ Spec OX pXq.
If f : X Ñ Spec A is any morphism into an affine scheme, then there is a unique
morphism h : Spec OX pXq Ñ Spec A such that f “ h ˝ θ.

Proof The morphism θ is obtained by taking the identity map in the right-hand side of
(4.17). Morphisms f : X Ñ Spec A are in one-to-one correspondence with ring maps
A Ñ OX pXq; the factorization f “ h ˝ θ comes from the composition A Ñ OX pXq Ñ
OX pXq.
Example 4.20 (Maps to A1 and OX pXq). If A is a ring, then there is a bijection between
ring maps ϕ : Zrts Ñ A and elements of A (ϕ is determined uniquely by the image of t).
Therefore, by Theorem 4.21

HomSch pX, A1 q “ OX pXq.

In other words, an element of OX pXq is the same thing as a morphism

f : X ÝÝÑ A1 .
Thus the global sections of the structure sheaf OX do indeed correspond to some sort of
‘regular functions’ on X , not into a field k , but into the affine line over Z. △

The main theorem of affine schemes

The main consequence of Theorem 4.17 is that the category of affine schemes, AffSch, is
equivalent to the category of rings, Rings. To formulate this precisely, we note that the
assignment A ÞÑ Spec A defines a functor

Spec : Ringsop ÝÝÑ AffSch. (4.22)

For each ring map ϕ : A Ñ B , this sends ϕ to the morphism pf, f 7 q : Spec B Ñ Spec A.
It follows by the definitions that Spec ϕ ˝ Spec ψ “ Specpψ ˝ ϕq, whenever ϕ and ψ are
composable ring maps (Exericse 9.9.39).
There is also a contravariant functor Γ in the opposite direction: taking global sections of
the structure sheaf OX gives us a ring OX pXq. Furthermore, a morphism of affine schemes
f : X Ñ Y comes equipped with a map of sheaves f 7 : OY Ñ f˚ OX , which on global
sections yields a ring map

fY7 : OY pY q ÝÝÑ ΓpY, f˚ OX q “ OX pXq.

By Theorem 4.17, these two functors are mutually inverse, and give an equivalence of
categories. Thus we arrive at the following scheme-theoretic analogue of Theorem 1.29 for
affine varieties.

Theorem 4.21 (Main Theorem of Affine Schemes). The two functors Spec and Γ
are up to equivalence mutually inverse and give an equivalence between the categories
Ringsop and AffSch.

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Proof Note that there is an equality Γ ˝ Spec “ idRings . Conversely, given a morphism
of affine schemes f : X Ñ Y is induced by a unique ring map OY pY q Ñ OX pXq and
applying Spec gives us f back. Therefore Spec ˝ Γ is equivalent to idAffSch .

In summary, affine schemes X are completely characterized by their rings of global

sections OX pXq, both on the level of topological spaces and the structure sheaf. Morphisms
between affine schemes X Ñ Y are in bijective correspondence with ring homomorphisms
OY pY q Ñ OX pXq. In particular, a map f between two affine schemes is an isomorphism if
and only if the corresponding ring map is an isomorphism.

Example 4.22. There is one and only one morphism of schemes Spec Z Ñ Spec Z. Indeed,
ring maps are required to send 1 to 1, so there is only one ring map Z Ñ Z. △
Example 4.23. In order to check that a morphism of schemes f : X Ñ Y is an isomorphism,
it is not enough to check that f is injective and surjective. Also the structure sheaves have
to be isomorphic. For instance, Spec C Ñ Spec R is a homeomorphism, but the two affine
schemes are not isomorphic. For another example, see Example 1.32. △
If k is a field, then any n-tuple of polynomials f1 , . . . , fn in krx1 , . . . , xm s determines a
morphism Am n
k Ñ Ak via the ring map ϕ : kry1 , . . . , yn s Ñ krx1 , . . . , xm s which sends y1
to f1 , y2 to f2 etc. Conversely, any morphism Am n
k Ñ Ak is of this form; the polynomials
f1 , . . . , fn can be recovered as the images ϕpy1 q, . . . , ϕpyn q.
For brevity, we will continue to denote such morphisms by

f : Am ÝÝÑ An
px1 , . . . , xn q ÞÑ pf1 pxq, . . . , fn pxqq
This is a useful notation, but one should keep in mind that the indicated mapping on sets is
only valid for the ‘traditional points’ of the form pa1 , . . . , an q P k n .

4.6 Open embeddings and open subschemes

If X is a scheme and U Ă X is an open subset, the restriction OX |U is a sheaf on U making
pU, OX |U q into a locally ringed space. This is even a scheme, because if X is covered by
affines Vi “ Spec Ai , then each U X Vi is open in Vi , hence can be covered by distinguished
open subsets, which are all affine schemes. Therefore there is a canonical scheme structure on
U , and we call pU, OX |U q an open subscheme of X and say that U has the induced scheme
structure. Moreover, a morphism of schemes ι : V Ñ X is an open embedding if it is an
isomorphism onto an open subscheme of X .
When referring to ‘an open affine’ in X or ‘an open affine covering’ of X , we shall
tacitly assume that the open sets involved are given the canonical scheme structure, and so
are open subschemes. So for instance, Spec k Ă Spec krxs{px2 q is not an open affine of
Spec krxs{px2 q even though the subset is open and the scheme is affine.
Example 4.24. The open set U “ A1k ´ V pxq is an open subscheme of the affine line
A1k “ Spec krxs. Note the isomorphism U » Spec krx, x´1 s “ Spec krx, ys{pxy ´ 1q of
schemes. △

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Example 4.25 (Distinguished open subsets). More generally, each distinguished open set
Dpf q in an affine scheme Spec A is an open subscheme. It is affine, isomorphic to Spec Af .
Indeed, by Proposition 2.32 the map ι : Spec Af Ñ Spec A corresponding to the localiza-
tion map A Ñ Af is a homeomorphism onto Dpf q, and it follows readily from the definition
of the sheaf OX that the restriction OX |Dpf q coincides with the structure sheaf on Spec Af .
△

A word of warning: an open subscheme of an affine scheme might not itself be affine, as the
next example shows.

Example 4.26 (A non-affine scheme). The open subset U “ A2k ´ V pu, vq of A2k “
Spec kru, vs is not an affine scheme. This is a consequence of the fact that the restriction
map

ι7 : OA2k pA2k q ÝÝÑ OA2k pU q (4.23)

is an isomorphism. Indeed, if U were affine, the inclusion ι : U Ñ A2k , being induced

by the ring map ι7 , would be an isomorphism by the Main Theorem of Affine Schemes
(Theorem 4.21), but obviously this is not the case.
To see that the restriction map (4.23) is an isomorphism, we use the sheaf sequence (3.1)
on page 50 for the covering tDpuq, Dpvqu of U . In view of the equalities OA2k pDpuqq “
kru, vsu and OA2k pDpvqq “ kru, vsv , the sheaf sequence takes the form

α β
0 OU pU q kru, vsu ‘ kru, vsv , kru, vsuv
ι7

kru, vs

where we have also included the restriction map ι7 . Note that αpι7 pcqq “ pc, cq. The map β
sends an element f “ pauń , bv ´m q to auń ´ bv ´m , and f lies in the kernel of β precisely
when auń “ bv ´m ; or in other words, when av m “ bun . As the polynomial ring is a UFD,
we conclude that a “ cun and b “ cv m for some c P kru, vs, so that f “ pc, cq. That is, ι7
is surjective, and since it is clearly injective, it is an isomorphism. △

4.7 Closed embeddings and closed subschemes

In this section, we define closed subschemes of a scheme. Intuitively, a closed subscheme of
a scheme X is a scheme Z , which is embedded as a closed subset Z Ă X . Given that there
are many possibilities for choosing the scheme structure on the same underlying closed set,
this makes the definition slightly more subtle than the one for open subscheme.
The prototypical example to have in mind is SpecpA{aq, which as we have seen, embeds
naturally as the closed subset V paq of Spec A (Proposition 2.29 on page 36). In general, a
closed subscheme is a scheme pZ, OZ q with a morphism ι : Z Ñ X , which locally looks
like the map SpecpA{aq Ñ Spec A. We formalize this in the next definition.

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Definition 4.27 (Closed embeddings and closed subschemes). A morphism ι : Z Ñ X

is called a closed embedding if there is an affine cover tUi uiPI of X such that
(i) ι´1 pUi q is affine for every i P I .
(ii) the ring map
ι7 : OX pUi q ÝÑ OZ pι´1 Ui q
is surjective for every i.
We say that Z is a closed subscheme of X . We say two closed subschemes Z, Z 1 are
equal if there is an isomorphism ϕ : Z Ñ Z 1 such that ι “ ι1 ˝ ϕ.

In other words, X and Z are covered by affine schemes Ui “ SpecpAi q, and ι´1 pUi q “
Spec Bi , so that for each i, the induced ring map Ai Ñ Bi is surjective, which means that
Bi » Ai {ai for some ideal ai . Moreover, the morphism ι´1 Ui Ñ Ui is identified with the
morphism SpecpAi {ai q Ñ SpecpAi q.
Even if a closed subscheme Z is defined as an abstract scheme which maps into X , we
usually think of it as a closed subset of X . This is justified because the image V “ ιpZq is a
closed subset (the Ui ’s form an open cover of X , and each subset ιpZq X Ui is closed being
equal to V pai q). Moreover, we may put a structure sheaf on V by defining OV to be ι˚ OZ .
Example 4.28. The schemes Spec krxs{pxn q with n P N and k a field, give different
subschemes of A1k . Still, the underlying topological spaces are identical (a single point),
and these spectra are homeomorphic. However, they are not isomorphic as schemes, as the
underlying structure sheaves are not isomorphic. △
Example 4.29. Consider the affine 4-space A4k “ Spec A, with k a field and A “
krx, y, z, ws. Then the three ideals
I1 “ px, yq, I2 “ px2 , yq and I3 “ px2 , xy, y 2 , xw ´ yzq,
have the same radical px, yq, and hence give rise to the same closed subset V px, yq Ă A4k ,
but they give different closed subschemes of A4k . △
While the above definition can be used to specify and study a closed subscheme, it is not
immediately clear how to classify all possible closed subschemes of a given scheme, even
in the case of affine schemes. Although each ideal a Ă A gives rise to a closed subcheme
SpecpA{aq Ñ Spec A, the definition a priori allows for closed subschemes defined in terms
of other affine coverings as well. In fact, it is true that all closed subchemes of Spec A arise
from ideals, but we will need to postpone the proof of this fact until we have discussed the
neccesary material on gluing (see Corollary 9.16).

Proposition 4.30. Let X “ Spec A be an affine scheme. The map a ÞÑ SpecpA{aq

defines a one-to-one correspondence between the set of ideals of A and the set of closed
subschemes of X . In particular, every closed subscheme of an affine scheme is also affine.

While we have used V paq to denote a closed subset of Spec A, we will from now on
pragmatically use V paq to refer to both the closed subscheme associated with a and the
underlying closed subset. The intended meaning will usually be clear from the context.

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For later use, we include the following definition, which combines the two types of
embeddings we have seen:

Definition 4.31 (Locally closed embeddings). A morphism f : Z Ñ X is said to be a

locally closed embedding if it factors as
ι j
Z ÝÑ U ÝÑ X
where ι : Z Ñ U is a closed embedding and j : U Ñ X is an open embedding.

4.8 Relative schemes

For a ring A, a scheme over A or simply A-scheme, is a scheme X together with a morphism
X Ñ Spec A. Equivalently, by Theorem 4.17, it is a scheme X such that every OX pU q is
an A-algebra.
A morphism of A-schemes is a morphism f : X Ñ Y which is compatible with the two
morphisms to Spec A, that is, so that the following diagram commutes:
f
X Y
(4.24)
Spec A

In this way, the schemes over A form a category Sch{A, with the category of affine schemes
over A, AffSch{A, as a full subcategory.
The Main Theorem of Affine Schemes (Theorem 4.21) has the following analogue:

Theorem 4.32. Let A be a ring. Then the category AffSch{A of affine schemes over A
is equivalent to the opposite category Alg{A of A-algebras.

One motivation for discussing this concept comes from being able to precisely say what it
means for a scheme to be ‘defined’ over a field or a ring. If X is a scheme over a field k , then
all of the rings OX pU q are k -algebras, and for a morphism f : X Ñ Y of k -schemes, all the
ring maps fV7 : OY pV q Ñ OX pf ´1 V q are maps of k -algebras. So this is a scheme-analogue
of being an ‘affine variety over k ’.

Example 4.33. A1Q is a scheme over Q. It is however not a scheme over R. △

Example 4.34. An affine scheme Spec B is an A-scheme precisely when B is an A-algebra.
Moreover, a morphism Spec B 1 Ñ Spec B is a morphism over A if and only if the corre-
sponding ring map B Ñ B 1 is a map of A-algebras. △
Example 4.35. By Example 4.22, every scheme admits a unique morphism X Ñ Spec Z,
so every scheme is a Z-scheme in a unique way. On the level of categories, this means that
Sch{Z “ Sch. △
By the requirement (4.24), the set of morphisms between two fixed schemes X and Y

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depends on the category of schemes in which we consider them. The next example illustrates
this.
Example 4.36. The scheme X “ Spec C can be regarded as a scheme over Z, R or C. The
set of self-morphisms X Ñ X to depends on the context as follows:
(i) As a C-scheme, the only morphism X Ñ X is the identity, so HomSch{C pX, Xq
has a single element.
(ii) As an R-scheme, there are two morphisms: the identity and the one induced by
complex conjugation C Ñ C. Hence HomSch{R pX, Xq has two elements.
(iii) As a scheme over Z, the set of self-morphisms is considerably larger, as it can
be identified with the set of field automorphisms of C. In particular,
HomSch{Z pSpec C, Spec Cq
is an uncountable group.
△
There is an important generalization of this where we replace the base scheme Spec A by
a more general scheme. Given a scheme S , an S -scheme, or a scheme over S , is a scheme X
together with a morphism X Ñ S . As above, the schemes over S form a category Sch{S,
where the morphisms between two S -schemes X Ñ S and Y Ñ S is a morphism which is
compatible with the two morphisms to S .
The power of this definition comes from thinking of X as a family of schemes parametrized
by the points of S . This perspective turns out to be conceptually very useful e.g. when we
discuss fiber products and base change in Chapter 8.
Example 4.37. The Möbius strip scheme
X “ Spec Rrx, y, u, vs{pvx ´ uy, x2 ` y 2 ´ 1q
from Example 2.39 can be viewed as a scheme over R, but one can also view it as a scheme
over S “ Spec Rrx, ys{px2 ` y 2 ´ 1q. The latter perspective offers extra geometric insight,
as all the fibers of X Ñ S are affine lines. △

4.9 R-valued points

Until now, we have considered morphisms from a scheme to an affine scheme. If we instead
study the set of morphisms from an affine scheme we arrive at the concept of ‘ring-valued
points’.
A basic motivation for studying R-valued points comes from the viewpoint that spaces in
algebraic geometry are defined in terms of solutions of polynomial equations. Concretely,
given a set of polynomials f1 , . . . , fr P Zrx1 , . . . , xn s, we would like to study the set of
common solutions
f1 pt1 , . . . , tn q “ ¨ ¨ ¨ “ fr pt1 , . . . , tn q “ 0 (4.25)
Of course, one has to specify the ring in which the variables t1 , . . . , tn take their values. In
fact, since the polynomials have integer coefficients, the equations are meaningful over any
ring R. The scheme X “ Spec Zrt1 , . . . , tn s{pf1 , . . . , fr q encodes the all possible solution

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sets in a single geometric object. More precisely, the solutions of (4.25) over R will be the
set of R-valued points of X .

Definition 4.38. Let X be a scheme and let R be a ring.

An R-valued point is a morphism of schemes
Spec R ÝÝÑ X.
The set of R-valued points is denoted XpRq, that is,
XpRq “ HomSch pSpec R, Xq. (4.26)

Note that if f : X Ñ Y is morphism, then composition induces a map of sets XpRq Ñ

Y pRq. The sets XpRq also depend functorially on R: to each ring map R Ñ S there is a
corresponding morphism Spec S Ñ Spec R, which induces a map of sets XpRq Ñ XpSq
via composition. Therefore the scheme X determines a functor from the category of rings to
Sets. We will explore the link between a scheme and its associated functor in Section 12.1.
Example 4.39. For the affine n-space An “ Spec Zrt1 , . . . , tn s, we have
An pRq “ Rn .
Indeed, by Theorem 4.21 on page 73, the R-valued points f : Spec R Ñ An correspond
bijectively to ring maps
ϕ : Zrt1 , . . . , tn s ÝÝÑ R, (4.27)
and these in turn, are in bijection with the set of n-tuples pϕpt1 q, . . . , ϕptn qq P Rn ; there is
one for every element of Rn .
In particular, for fields k it holds that An pkq “ k n , which explains the notation An pkq
used in Chapter 1.
Likewise, if X Ă An is the closed subscheme defined by an ideal a “ pg1 , . . . , gr q Ă
Zrt1 , . . . , tn s, the set of R-valued points XpRq are in a one-to-one correspondence with ring
maps ϕ : Zrt1 , . . . , tn s{a Ñ R. These are in turn in bijection with ring maps ϕ as in (4.27)
that vanish on the ideal a. That is, they are in bijection with n-tuples a “ pa1 , . . . , an q P
Rn “ An pRq such that gi paq “ 0 for all i, or in other words, the solutions of the system
g1 “ ¨ ¨ ¨ “ gr “ 0 in R. △
Example 4.40. For a concrete example, consider the scheme
X “ Spec Zru, vs{pu2 ` v 2 ´ 1q.
Then the set XpRq of R-points consists of the points of the unit circle in R2 , and the Z-points
XpZq consists of the four points p˘1, 0q and p0, ˘1q. By Exercise 9.9.7, the Q-points are
given by
"ˆ ˙ *
2t 1 ´ t2 ˇˇ
XpQq “ , t P Q Ytp0, ´1qu.
1 ` t2 1 ` t2
△
The sets XpRq of R-points are important in number theory. For instance, studying the

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integer solutions to x3 ` y 3 “ z 3 , is equivalent to describing the set XpZq, where X “

Spec Zrx, y, zs{px3 ` y 3 ´ z 3 q. Sometimes, looking for solutions over other rings can
simplfy the problem. For example, one can argue that x3 ` y 3 “ z 3 has no non-trivial
solutions over Z, by looking at XpRq where R “ Zrωs where ω “ e2πi{3 is a primitive
third-root of unity. Over R, the equation can be factored x3 “ pz ` yqpz ` ωyqpz ` ω 2 yq,
and one can use the fact that R is a UFD to prove that XpZq contains only the trivial solutions
where xyz “ 0.

Example 4.41 (Non-existence of Z-points). The equation 3x2 ´ 7y 2 “ 1 has no solution in

integers x and y . Indeed, modulo 3, the equation reduces to 2y 2 “ 1 mod 3, but 2y 2 must
be 0 or 2 modulo 3. In geometric terms, the scheme

X “ Spec Zrx, ys{p3x2 ´ 7y 2 ´ 1q

has no Z-points; any Z-point of X would survive via the map XpZq Ñ XpF3 q induced by
the reduction mod 3 map Z Ñ F3 . Likewise, XpRq ‰ H is a necessary condition for the
existence of Z-points.
One says that a scheme X satisfies the Hasse principle if these conditions are also sufficent,
that is, if XpRq ‰ H and XpFp q ‰ H for all primes p implies XpZq ‰ H. The Hasse
principle holds in some cases, e.g., when X is defined by a quadratic polynomial, but it fails
in general. The Selmer curve

X “ Spec Zrx, ys{p3x3 ` 4y 3 ` 5q

has points over R and every Fp , but has none over Z. △
There is also the relative version of the notion of R-valued points. Let X be a scheme over
a ring A and let R be an A-algebra. Then an R-valued point of X is a morphism of schemes
over A
Spec R ÝÝÑ X.
The set of R-valued points is denoted by XpRq “ HomSch{A pSpec R, Xq.
Working over a base ring A is in fact very important here. The reason is that considering
all morphisms Spec R Ñ X (without the A-structure) typically leads to a set that is too
large and does not capture the geometric picture we want. For instance, for X “ Spec C, the
set XpCq has 1, 2 or an infinite number of elements depending on whether we view it as a
scheme over C, R or Z respectively (see Example 4.36).
Even though the base ring A is not present in the notation XpRq, it will generally be clear
from the context which category we are working in.

Points in schemes
Looking at maps from spectra of fields into a scheme X helps us understand the points of
X . Every point of X is a K -point for some field K . Namely, if x P X is a point, there is a
canonical map
ιx : Spec κpxq ÝÝÑ X

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which maps the only point of Spec κpxq to x. To see this, suppose x is contained in an open
affine subset U “ Spec A and corresponds to a prime ideal p Ă A. Then the residue field
is given by κpxq “ Ap {pAp , and the morphism ιx : Spec κpxq Ñ X is defined by the
composition

Spec κpxq Ñ SpecpAp q Ñ U Ñ X.

The residue field κpxq and the morphism ιx : Spec κpxq Ñ X satisfy a certain universal
property with respect to maps from spectra of fields into X :

Proposition 4.42. Let X be a scheme and let K be a field. Then there is a bijection
" ˇ *
ˇ x P X is a point, and
XpKq “ px, αq ˇ ˇ .
α : κpxq Ñ K is a map of fields

Proof Fix a point x P X , and let U “ Spec A be an open affine subscheme containing x,
so that x corresponds to a prime ideal p in A.
Write o P Spec K for the unique point in Spec K . Since Spec K has a single point, any
K -valued point f : Spec K Ñ X with image x, factors uniquely via U Ñ X . Conversely,
any K -valued point of U induces a K -valued point of X via inclusion. Therefore, we reduce
to showing that there is a bijection between K -valued points Spec K Ñ U with image x
and the set of maps of fields κpxq Ñ K .
The K -valued points of f : Spec K Ñ U are in bijection with ring maps ϕ : A Ñ K .
Under this correspondence, we have f poq “ x if and only if ϕ´1 p0q “ p, i.e., ϕ factors
uniquely as A Ñ A{p Ñ K . Since K is a field, any ring map A{p Ñ K in turn factors
uniquely via the residue field A{p Ñ κppq Ñ K . Hence the K -valued points f are are in
bijection with the maps of fields κpxq Ñ K .

4.10 Affine varieties as schemes

In Chapter 1, we defined an affine variety as an irreducible closed algebraic set V Ă Ank
where k is an algebraically closed field. The coordinate ring ApV q of V is a finitely generated
k -algebra and an integral domain. We now present the more general scheme-theoretic version:

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Definition 4.43. An affine variety over a field k is an affine scheme of the form
X “ Spec A, where A is a finitely generated k -algebra which is an integral domain. A
morphism of affine varieties over k is of the underlying k -schemes.

If we choose a presentation A “ krx1 , . . . , xn s{a, we see that any affine variety in this
sense admits a closed embedding into an affine space Ank over k . Conversely, any closed
subscheme V paq of Ank is an affine variety, as long as the defining ideal a is a prime ideal.
This definition naturally extends the classical notion when k is algebraically closed. In
this setting, then any affine variety V Ă An pkq as defined in 1 gives rise to an affine variety
in the new sense by taking X “ Spec ApV q. The scheme X is irreducible as ApV q is an
integral domain. Furthermore, the Nullstellensatz implies that the k -valued points of X are in
bijection with the points of V , allowing us to view V as a subset of X . Under this inclusion,
the Zariski topology on V is the restriction of the Zariski topology on X .
The scheme X however contains many additional points. These correspond to the non-
maximal ideals of ApV q, or equivalently, the subvarieties of V of dimension at least 1.
The assignment V Ñ X extends to morphisms. Morphisms of ‘classical’ affine varieties
V Ñ W correspond to maps of k -algebras ApW q Ñ ApV q, and these in turn are in bijection
with morphisms of k -schemes X Ñ Y . This means that we have defined a fully faithful
functor from the category of affine varieties AffVar{k to the category of affine schemes over
k , AffSch{k .
By Example 4.9, the structure sheaf on X coincides with the sheaf of regular functions (as
in Chapter 1). Over an open set U Ă X , OX pU q consists of the rational functions f P KpXq
which are regular at every point of U .

Example 4.44. The following schemes are affine varieties:

AnC , Spec Rrx, ys{px2 ` y 2 q, Spec Fp rx, y, zs{py 2 ´ xzq.

The following schemes are not:

Spec Z, Spec Crx, ys{px2 ` y 2 q, Fp rx, ys{py 2 q.

△
The next proposition tells us that for a morphism of affine varieties, closed points map to
closed points, as in the classical situation:

Proposition 4.45. Let f : X Ñ Y be a morphism of affine varieties over a field k . If

x P X is a closed point, then f pxq is closed in Y .

Proof Let X “ Spec B and Y “ Spec A, where A and B are finitely generated k -
algebras, and suppose f is induced by a map of k -algebras ϕ : A Ñ B . The point x
corresponds to a maximal ideal m in B , and κpxq “ B{m is a finite extension of k , by
Theorem 1.8. Let p “ ϕ´1 pmq Ă A be a prime ideal corresponding to f pxq. The map ϕ
induces an injective map of k -algebras:

A{p ãÑ B{m “ κpxq.

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g1´1 V

X12 V

τ X

X21 g2´1 V
X2

Figure 4.1 Gluing two schemes together

Since κpxq is a finite field extension of k , the domain A{p also a finite-dimensional k -algebra.
To show that p is maximal, it suffices to show that A{p is a field.
Take any nonzero a P A{p. Multiplication by a defines an injective k -linear map A{p Ñ
A{p. Because A{p is finite-dimensional, this map is also surjective. Hence, there exists
b P A{p such that ab “ 1. Hence A{p is a field and p is a maximal ideal.

4.11 Gluing two schemes together

By definition, a scheme is a space which is locally built from affine open subschemes. Over
the next few sections, we will explain how to conversely construct schemes by gluing together
schemes along open subschemes. These gluing techniques are fundamental in scheme theory.
As we will see in Chapter 5, this gives a vast collection of new examples of schemes. The
gluing of schemes is also an important part in many general existence proofs, such as the
construction of the fiber product.
We begin by examining the most basic case, where we glue together two schemes along a
common open subscheme.
Let X1 and X2 be schemes, with open subschemes X12 Ă X1 and X21 Ă X2 respectively,
and let τ : X21 Ñ X12 be an isomorphism.
We construct the underlying topological space X as the quotient space

X “ pX1 \ X2 q { „

where „ is the equivalence relation defined by x „ τ pxq for all x P X21 Ă X2 . Let
q : X1 \ X2 Ñ X denote the quotient map, and define gi “ q|Xi : Xi Ñ X for i “ 1, 2.
Then the maps gi are open, and their images Ui “ gi pXi q form an open cover of X .
An open set V Ă X is open if and only if gi´1 V is open in Xi for i “ 1, 2. We define the

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sections of OX over V as compatible pairs of sections from X1 and X2 :

! ˇ )
OX pV q “ ps1 , s2 q P OX1 pg1´1 V q ˆ OX2 pg2´1 V q ˇ s1 |g1´1 V XX12 “ τ 7 ps2 |g2´1 V XX21 q
ˇ

For V Ă U , the restriction OX pU q Ñ OX pV q sends ps1 , s2 q to ps1 |g1´1 V , s2 |g2´1 V q. It is

not hard to check that OX is a sheaf and that pX, OX q is a locally ringed space. Moreover,
the maps gi : Xi Ñ X are open embeddings onto Ui Ă X for i “ 1, 2, so every point of
X has an open neighbourhood which is an affine scheme. (We will prove a more general
statement below).
The main example to keep in mind is when X1 and X2 are both affine, say X1 “ Spec A
and X2 “ Spec B , and they are glued together along two distinguished open subsets Dpaq
and Dpbq for some a P A and b P B . The gluing map τ is specified by a ring isomorphism
between the localizations

ϕ : Aa ÝÝÑ Bb .

We picture this by the following diagram

τ
Spec A Ą Spec Aa “ Dpaq » Dpbq “ Spec Bb Ă Spec B

A convenient fact is that everything about X can be computed in terms of A, B and the ring
map ϕ. For instance, the global sections have the following description:

OX pXq “ t pf, gq P A ˆ B | g{1 “ ϕpf {1q in Bb u. (4.28)

Many interesting examples arise from this basic construction (see the examples at the end of
this chapter.)

4.12 Gluing sheaves

We will now study the more general situation, where we glue together an arbitrary collection
of schemes. For this, we also need general results on gluing sheaves and morphisms of
sheaves.
Let X be a topological space X and let tUi uiPI be an open cover of X . For convenience,
we will denote the pairwise and triple intersections by

Uij “ Ui X Uj and Uijk “ Ui X Uj X Uk , i, j, k P I.

The simplest case of gluing involves morphisms of sheaves. The following proposition gives
the precise conditions under which this can be done:

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Proposition 4.46 (Gluing conditions for maps for sheaves). Let F and G be sheaves
on X . Suppose we are given: for each i P I , a morphism of sheaves
ϕi : F|Ui Ñ G|Ui ,
such that for all i, j P I , the restrictions agree on overlaps:
ϕi |Uij “ ϕj |Uij . (4.29)
Then there exists a unique morphism of sheaves ϕ : F Ñ G satisfying ϕ|Ui “ ϕi for all
i P I.

Proof Let V Ă X be an open set and let s P FpV q be a section. Then the open sets
Vi “ Ui X V form a cover of V . Consider the sections ϕi ps|Vi q P GpVi q. By (4.29), we have

ϕi psq|Vij “ ϕi ps|Vij q “ ϕj ps|Vij q “ ϕi psq|Vij for all i, j P I.

Since G is a sheaf, the sections ϕi ps|Vi q glue uniquely to a section in GpV q which we define
as ϕpsq. This assignment is additive and compatible with restrictions, so ϕ defines a map of
sheaves. Moreover, by construction, ϕ|Ui “ ϕi for all i.
For uniqueness, suppose ϕ and ψ are two morphisms of sheaves so that ϕpsq|Ui “ ψpsq|Ui
for all i P I , then ϕpsq “ ψpsq by the Locality axiom for G , and consequently ϕ “ ψ .

For gluing sheaves, the set-up is as follows: given a sheaf Fi on each Ui , the goal is to
construct a global sheaf F on X that restricts to Fi for every Ui . A necessary condition for
such an F to exist is that the Fi ’s should be isomorphic over the intersections Uij . In fact, by
specifying the precise conditions that these isomorphisms must satisfy (the gluing data), we
get not just a necessary but also a sufficient condition.

F0 F1
U0

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Proposition 4.47 (Gluing conditions for sheaves). Let tUi uiPI be a covering of X .
Suppose we are given, for each i P I , a sheaf Fi on Ui , and for each i, j P I an
isomorphism
τji : Fi |Uij Ñ Fj |Uij ,
satisfying the three conditions
(i) τii “ idFi
(ii) τji “ τij´1
(iii) On the triple intersection Uijk , we have τki “ τkj ˝ τji
Then there exists a sheaf F on X and isomorphisms
νi : F|Ui Ñ Fi
such that νj “ τji ˝ νi over each intersection Uij . The sheaf F is unique up to isomor-
phism.

The three conditions (i)–(iii) parallel the three requirements for a relation to be an equiva-
lence relation; the first reflects reflectivity, the second symmetry and the third transitivity.
To motivate these a bit further, note that if we have managed to construct F and νi , the
isomorphisms τji “ νj ˝ νi´1 appear as the composition
Fj |Uij » F|Uij » Fi |Uij
Isomorphisms of this form automatically satisfy (i)–(iii). For instance, to verify (iii):
τkj ˝ τji “ pνk ˝ νj´1 q ˝ pνj ˝ νi´1 q “ νk ˝ νi´1 “ τki .
Proof If W Ă X is an open set, we will write Wi “ Ui X W and Wij “ Uij X W .
A section of F over an open set V is given by a collection of compatible sections si P
Fi pVi q. Precisely, for each i, j P I , we want to identify sj and τji psj q over Vij . So we define
␣ ˇ ( ź
FpV q “ psi qiPI ˇ τji psi |Vij q “ sj |Vij Ă Fi pVi q. (4.30)
iPI

The restriction maps are induced componentwise: if W Ă V , the map FpV q Ñ FpW q sends
psi qiPI to psi |Wi qiPI . This is well-defined, because the τji are compatible with restrictions,
which ensures that τji psi |Wij q “ sj |Wij if τji psi |Vij q “ sj |Vij . We have therefore defined a
presheaf on X . We next check the two sheaf axioms.
Locality: let s “ psi q P FpV q be a section, and suppose that s|Vα “ 0 over every open
set in a cover tVα uαPΛ of V . Then also si |Ui XVα “ 0 in Fi pVα X Ui q for all α and i. As
Vα X Ui forms an open cover of V X Ui , and Fi is a sheaf on Ui , this means that si “ 0 in
FpV X Ui q. Since this holds for every i, we see that s “ 0 in FpV q.
Gluing: Let sα P FpVα q be a set of compatible sections over the opens of a covering
tVα uαPΛ of V . This means that sα and sβ are equal when restricted to Vαβ “ Vα X Vβ . For a
fixed i P I , we then have a compatible collection of sections sαi P FpUi X Vα q, which, since
Fi is a sheaf, glue to a unique element si P FpUi q. We have τij psj q “ si in FpVij q because
this holds when restricted to each Vα X Uij , and since sα P FpVα q. The tuple s “ psi q
therefore defines an element of FpV q, which by construction restricts to sα on each Wα .
Note that we haven’t used the condition (iii) yet. It will be needed in order to construct

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the isomorphisms νi : F|Ui Ñ Fi . To avoid getting confused by the indices, we work with a
fixed index α P I . Suppose V Ă Uα is an open set. The projection map

να : F|Uα pV q ÝÝÑ Fα pV q, psi qiPI ÞÑ sα

induces a well-defined sheaf map να : F|Uα ÝÝÑ Fα . We proceed to verify that the να give
the desired isomorphisms.
To begin with, we note requirement νβ “ τβα ˝ να is fulfulled. Indeed, by the compatibility
condition in (4.30), we have for V Ă Uαβ ,

τβα pνα psqq “ τβα psα q “ sβ “ νβ .

να is injective: if s “ psi q P FpV q is a section such that sα “ 0 P Fα pV q, then we also
have
si “ si |Viα “ τiα psα q “ 0
for all i P I , and hence s “ 0.
να is surjective: take any section σ P Fα pV q over some V Ă Uα and define s “
pτiα pσ|Viα qqiPI . Note that for every i, j P I , we have
τji pτiα pσ|Vjiα qq “ τjα pσ|Vjiα q
Therefore, s satisfies the condition in (4.30), and defines an element of FpV q. As ταα pσ|Vαα q “
σ by the first gluing condition, we also have να psq “ σ .
Example 4.48. If X is the scheme obtained by gluing together X1 “ Spec A and X2 “
Spec B , Proposition 4.47 tells us that giving a sheaf F on X is equivalent to specifying (i) a
sheaf F1 on X1 ; (ii) a sheaf F2 on X2 ; (iii) a sheaf isomorphism

ν12 : F2 |Dpbq ÝÝÑ F1 |Dpaq ,

where we use the isomorphism τ to identify Dpbq and Dpaq. In the special case that F1 “ M
Ă
and F2 “ Nr for modules M and N over A and B respectively, it is equivalent to specify an
isomorphism of Aa -modules
v12 : Aa ÝÝÑ Bb .
△

4.13 Gluing schemes

When we talk about gluing schemes, the set-up is as follows. We are given a collection
of schemes tXi uiPI , and in each of the schemes Xi we are given a collection of open
subschemes Xij , one for each j P I . The goal is to produce a new scheme X by gluing
together all the Xi ’s along these open subschemes. This is done by identifying the open
sets Xij Ă Xi and Xji Ă Xj using scheme isomorphisms τji : Xij Ñ Xji . If we let
Xijk “ Xik X Xij (these correspond to the triple intersections before the gluing has been
done), we require that τji pXijk q “ Xjik . Notice that Xijk is an open subscheme of Xi .

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X12 X13
τ12 τ31
X123

X21 X31
X231
X312
X
X23 τ23
X2 X32 X3

There are three gluing conditions, similar to the ones we saw for sheaves, which must be
satisfied for the gluing to be possible.

Proposition 4.49 (Gluing conditions for schemes). Suppose that we are given: a col-
lection of schemes tXi uiPI ; for each i, j an open subschemes Xij Ă Xi and scheme
isomorphisms τji : Xij Ñ Xji satisfying
(i) τii “ idXi
´1
(ii) τij “ τji
(iii) τij takes Xijk into Xjik and τki “ τkj ˝ τji over Xijk .
Then there exists a scheme X with open embeddings gi : Xi Ñ X onto an open
subscheme Ui “ gi pXi q Ă X such that
‚ tUi uiPI forms an open cover of X .
‚ For each i, j P I , gi pXij q “ Ui X Uj and the following diagram commutes:

τij
Xij Xji
gi gj

Ui X Uj

The scheme X is uniquely characterized by these properties up to isomorphism.

For the latter, we rely on the gluing technique for sheaves explained in Proposition 4.47.
The fact that X is locally affine will follow immediately once the embeddings gi are in place,
because the Xi ’s are schemes and therefore locally affine.
Proof We construct the scheme X in two steps: first the underlying topological space X
and then the structure sheaf. š
Define the topological space X as the quotient of the disjoint union i Xi by the equiv-
alence relation generated by x „ τji pxq for x P Xij . The three gluing conditions enusure
that this is an equivalence relation. The first requirement means that the relation is reflexive,
the second that itšis symmetric, and the third ensures it is transitive. Here X has the quotient
topology: if π : i Xi Ñ X denotes the quotient map, a subset U of X is open if and only
´1
if π pU q is open.
Topologically, the maps gi : Xi Ñ X are simply the maps induced by the open inclusions

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ď
π ´1 pgi pU qq “ τji pU X Xij q,
j

we conclude that each gi is an open map, hence a homeomorphism onto its image.
Write Ui “ gi pXi q, Uij “ gpXi q X gpXj q and Uijk “ Ui X Uj X Uk . On Xij , we have
7
the isomorphisms τji : OXj |Xij Ñ OXi |Xij , the sheaf maps of the scheme isomorphisms
τji : Xij Ñ Xji . These satisfy τii7 “ id, τji7 “ pτij7 q´1 and τki
7
“ τji7 ˝ τkj
7
(on Xijk ) by the
gluing conditions for the morphisms τji . Applying Proposition 4.47 the structure sheaves
OXi glue to a sheaf of rings OX . By construction, this sheaf of rings restricts to OXi on
each of the open subsets Xi , and therefore the stalks are local rings. So pX, OX q is a locally
ringed space which is locally affine, hence a scheme.
We leave it to the reader to check the uniqueness statement in the proposition.

Finally, we consider conditions under which we can glue morphisms of schemes

Proposition 4.50 (Gluing morphisms of schemes). Let X and Y be schemes and let
tUi uiPI be an open cover of X . Given morphisms fi : Ui ÝÝÑ Y such that
fi |Ui XUj “ fj |Ui XUj
for each i and j , there exists a unique morphism
f : X ÝÝÑ Y
such that f |Ui “ fi for every i P I .

Proof On the level of topological spaces, we set f pxq “ fi pxq if x P Ui . This is well-
defined because fi pxq “ fj pxq for x P Ui X Uj , and continuity follows from the continuity
of each fi .
Next we define the sheaf map f 7 . If V Ă Y is an open set, we construct the ring map
f : OY pV q Ñ OX pf ´1 V q as follows. Take any section s P OY pV q. Using the sheaf maps
7

fi7 over Ui , we get sections ti “ fi7 psq in OX pf ´1 V X Ui q. As fi7 and fj7 restrict to the same
map on Uij , we have

ti |f ´1 V XUij “ tj |f ´1 V XUij

in OX pf ´1 V X Uij q. The ti therefore glue together to a section t P OX pf ´1 V q, and we

can define f 7 psq to be t. This assignment is compatible with restriction maps, and define a
sheaf of rings f 7 and that fV7 “ pfi q7V psq when V Ă Ui . The pair pf, f 7 q is therefore a map
of locally ringed spaces because it is locally given by the fi .
The uniqueness of f follows from the fact that morphisms of schemes are determined by
their restrictions to any open cover, and the restrictions f |Ui are equal to fi by design.

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The affine line with two origins

Consider the scheme X obtained by gluing together two copies X1 and X2 of the affine line
A1k “ Spec krus over a field k along their common open subset X12 “ Spec kru, u´1 s with
the identity morphism ϕ : kru, u´1 s Ñ kru, u´1 s on the open set. The resulting scheme X
is covered by two A1k ’s which overlap outside the origin. However, as the gluing process does
nothing over the origins of each A1k , there are now two points in X that replace the origin. X
is called the affine line with two origins.

0 A1
01 X
0 A 1
02

This scheme is not affine. To see this, we use the sheaf sequence, which takes the form

0 ΓpX, OX q ΓpA1k , OA1k q ‘ ΓpA1k , OA1k q ΓpX12 , OX12 q

“
“

ρ
krus ‘ krus kru, u´1 s

where now ρpppuq, qpuqq “ ppuq ´ qpuq, and it follows that ΓpX, OX q » krus. Moreover,
both inclusions ι : A1k Ñ X induce the identity map ι7 : krus Ñ krus on global sections. If
X were affine, this would imply that ι : A1k “ Spec krus Ñ X is an isomorphism, which is
clearly not the case (it is not surjective, since the image misses one of the two origins).

Semi-local rings
Semi-local rings are rings with finitely many maximal ideals. In the next two examples we
give a few examples of such rings and how they can be described as local rings glued together.

Example 4.51 (Semi-local rings). Consider the two rings Zp2q and Zp3q . These are both
discrete valuation rings with with a common fraction field Q and maximal ideals p2q and
p3q respectively. As described in Example 2.14 on page 31, X1 “ Spec Zp2q consists of two
points: the closed point x1 corresponding to p2q, and the generic point η1 corresponding to
p0q. Likewise, Spec Zp3q consists of two points x2 and η2 . The generic points given the open
embeddings Spec Q Ñ Xi for i “ 1, 2. Hence we can glue together X1 and X2 along the
two generic points and we obtain a scheme X with one open point η and two closed points
x1 and x2 .
By (4.28), we can compute OX pXq as follows:

OX pXq “ t pa, bq P Zp2q ˆ Zp3q | a “ b in Q u » Zp2q X Zp3q .

The ring Zp2q X Zp3q is a semi-local ring with the two maximal ideals p2q and p3q. By the
Main Theorem of Affine Schemes (Theorem 4.21) there is a map X Ñ Spec Zp2q X Zp3q ,
and it is left as an exercise to show that this is an isomorphism. △

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p3q
p2q η p3q η
p2q p5q

Example 4.52 (More semi-local rings). More generally, if P “ tp1 , . . . , pr u is a finite set
of distinct prime numbers, we can consider the schemes Xp “ Spec Zppq for p P P and
glue them together to form a new scheme X . Indeed, each Xp has an open point ηp , which
is the image of the canonical open embedding Spec Q Ñ Xp , and we glue Xp and Xq by
identifying ηp and ηq using the identity map on Spec Q.
We may compute the global sections of the structure sheaf of the new scheme using the
sheaf sequence
ś ś
0 Ñ ΓpX, OX q pPP ΓpXp , OX q p,qPP ΓpXp X Xq , OX q
“

“
ś ś
pPP Zppq ρ p,qPP Q.

Here, the map ρ sends an r-tuple pap qpPP to the sequence

Ş pap ´ aq qp,qPP , and it follows that
the kernel of ρ is equal to the intersection AP “ pPP Zppq . This is a semi-local ring whose
maximal ideals are the ppqAP ’s for p P P . There is a canonical morphism X Ñ Spec AP ,
and again we leave it to the reader to verify that this is an isomorphism. △

4.14 Exercises
Exercise 4.14.1. Prove the uniqueness part in Proposition 4.49.
Exercise 4.14.2. Let X and Y be schemes and let B be a basis for the topology on X .
Suppose that there is a collection of morphisms fU : U Ñ Y , one for each U P B , such that
if V P B satisfies V Ă U , we have
fU |V “ fV .
Show that there exists a unique morphism of schemes f : X Ñ Y such that f |U “ fU .
Exercise 4.14.3 (A finite non-affine scheme). Consider a set X consisting of three elements,
x, y, z . Define a topology on X where the open sets are defined as H, txu, tx, yu, tx, zu,
and X itself. On X , define a sheaf of rings OX where OX ptxuq corresponds to the field
Qptq, while for the open sets OX ptx, yuq, OX ptx, zuq, and OX pXq we assign the localized
ring Qrtsptq . Show that pX, OX q is a scheme which is not isomorphic to an affine scheme.
H INT: An affine scheme is isomorphic to Spec OX pXq.
Exercise 4.14.4 (The cuspidal cubic). Let k be an algebraically closed field, and consider
the morphism
f : A1k ÝÝÑ Spec krx, ys{px3 ´ y 2 q
t ÞÑ pt2 , t3 q.
induced by the ring map ϕ : krx, ys ÞÑ krts sending x to t2 and y to t3 .

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a) Show that f is a homeomorphism.

b) Let p P A1k be the origin. Compute the map fp7 : OX,p0,0q Ñ OA1k ,p and show
that it is not an isomorphism. Conclude that the cuspidal cubic is not even
’locally isomorphic’ to A1k near the origin.
? ?
Exercise 4.14.5. Consider the ring A “ Zr ´5s and the ideal ? a “ p2, 1 ` ´5q. Show
that a is not principal as follows. For an element α “ a ` b ´5, define the norm of α as
N pαq “ a2 ` 5b2 .
a) Show that A{a » Z{2.
b) Show that N pxyq “ N pxqN pyq for all x, y P A.
c) Show that N pxq “ 1 if and only if x is a unit.
d) Show that there is no element x with N pxq “ 2.
e) Show that if a “ pxq, then x must be a unit, so that a “ A. Conclude that a
cannot be principal.

Exercise 4.14.6. Let X be a scheme whose underlying topological space is a finite and
discrete set. Show that X is affine.

Exercise 4.14.7. Let f : X Ñ Spec A be a morphism of schemes. Show that the closure of
the image f pXq is given by V pKer ϕq where ϕ : A Ñ OX pXq is the ring map inducing f .

Exercise 4.14.8. Let k be a field and let X “ Spec A, where A “ krx, y, z, ws{pxz, xw, yz, ywq.
a) Show that X is the union of two affine planes intersecting at the origin.
b) Let U Ă X be the open subset obtained by removing the origin. Define the
subset S Ď A by

S :“ tf P A | @p P U | f pxq ‰ 0 for all x P U u

where mp is the maximal ideal of the local ring OX,p at the point p P U .
Prove that the natural homomorphism S ´1 A Ñ OX pU q is not an isomor-
phism.
c) Explain why the analogous statement in part does hold for the union of two
affine lines intersecting at a single point.

Exercise 4.14.9 (The reduction of a scheme). Let X be a scheme ? and let ? B be the basis of
affine open subschemes of X . For a ring R, we let Rred “ R{ 0, where 0 is the ideal of
nilpotent elements in R.
a) Show that V ÞÑ OX pV qred defines a B -sheaf of rings on X . If X “ Spec A,
? sheaf is isomorphic to the sheaf associated to the A-module
show that this
Ared “ A{ 0.
b) Let OXred denote the sheaf associated to the B -sheaf from a). Show that
pX, OXred q is a scheme, which is isomorphic to SpecpAred q if X “ Spec A is
affine.
c) Show that there is a closed embedding ι : Xred Ñ X .
d) Show that Xred and ι satisfies the following universal property: for any reduced
scheme Y and a morphism f : Y Ñ X , there is a unique morphism g : Y Ñ
Xred so that f “ ι ˝ g .

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e) If f : X Ñ Y is a morphism, show that there is a unique morphism fred : Xred Ñ

Yred so that fred ˝ rX “ rY ˝ fred . Show that assignments X ÞÑ Xred and
f ÞÑ fred defines a functor Sch to RedSch which is adjoint to the inclusion
functor RedSch Ñ Sch, where RedSch is the full subcategory of Sch whose
objects are the reduced schemes.
Exercise 4.14.10. Let f : X Ñ Y be a morphism of schemes. Show that the following are
equivalent.
(i) f is dominant.
(ii) f pDq is dense for one dense subset D Ă X .
(iii) f pDq is dense for all dense subsets D Ă X .
If X is irreducible, they are also equivalent to
Y is irreducible and the generic point of X maps to the generic point of Y .
Exercise 4.14.11. Let X be an integral scheme and let f P KpXq. Show that the set of
points p P X such that f P OX,p is open.
Exercise 4.14.12. Fill in the details for the proof of Proposition ??.
Exercise 4.14.13. Let X be an integral scheme and let f : X 99K Y be a rational map. Show
that there is a unique maximal open set U for which the f is defined. This is another way of
resolving the ambiguity in the definition of a rational map. H INT: Proposition 4.50.
Exercise 4.14.14. Let X Ă P2k denote the Klein quartic curve defined by the equation
x3 y ` y 3 z ` z 3 x “ 0. Compute the number of k -points on X for k “ F2 , F4 and F8 .
Exercise 4.14.15. Let X be a scheme. Show that the map p ÞÑ tpu defines a bijection
X ÝÝÑ tirreducible closed subsets of Xu
Exercise 4.14.16. Let X “ Spec A and let M be an A-module. Show that M Ă is an OX -
Ăq is an OX pU q-module, and the module structure is compatible
module, that is, ΓpU, M
with restriction maps. H INT: Show this first for U “ Dpf q, then cover a general open by
destinguished open sets.
Exercise 4.14.17. Let f : X Ñ Y be a morphism. Show that if g P OY pY q corresponds
to the morphism g : Y Ñ A1 , then f 7 pgq P OX pXq corresponds to the composition
X Ñ Y Ñ A1 .
Exercise 4.14.18. Show that A1C embeds as a closed subscheme of A2R . Can A1R be a closed
subscheme of some AnC ?
Exercise 4.14.19. Let f : Z Ñ X be a morphism and let U Ă X be an open subscheme
such that f pZq Ă U (as sets). Show that there exists a unique morphism g : Z Ñ U such
that f “ ι ˝ g where ι : U Ñ X is the inclusion map.
Exercise 4.14.20. Consider X “ Spec A, where A “ Rru, vs{pu2 ` v 2 ` 1q. Show that
XpRq “ H, but XpCq is infinite. Show that A contains infinitely many maximal ideals, so
the underlying topological space of X is infinite.

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Projective space

5.1 Projective space

Traditionally, the projective n-space over a field k is defined as the quotient space
` ˘
Pn pkq “ k n`1 ´ t0u { „ (5.1)

where we say that two pn ` 1q-tuples pa0 , . . . , an q and pb0 , . . . , bn q are equivalent if there is
some λ P k ˆ so that bi “ λai for all i. Geometrically, this means that Pn pkq parameterizes
the 1-dimensional subspaces in k n`1 , or in other words, the lines through the origin.
The equivalence class of an pn ` 1q-tuple px0 , . . . , xn q is usually written px0 : ¨ ¨ ¨ : xn q.
So for instance, p1 : 3q “ p´1 : ´3q and p1 : 2 : 3q “ p2 : 4 : 6q. The x0 , . . . , xn are
called homogeneous coordinates. It is important to note that x0 , . . . , xn are not coordinates
in the usual sense, as they are not even functions. However, the ratios xj {xi define functions
on the subset
Ui “ t px0 : ¨ ¨ ¨ : xn q P Pn pkq | xi ‰ 0 u.
Note that Pn pkq is covered by these sets:
n
ď
n
P pkq “ Ui
i“0

For each i “ 0, . . . , n, there is a bijection

ϕi : An pkq ÝÝÑ Ui (5.2)

py1 , . . . , yn q ÞÝÑ py1 : ¨ ¨ ¨ : yi´1 : 1 : yi : ¨ ¨ ¨ : yn q.
The inverse is defined by sending a point px0 : ¨ ¨ ¨ : xn q P Ui with xi ‰ 0 to the point
ˆ ˙
x0 xi´1 xi`1 xn
,..., , ,... P An pkq. (5.3)
xi xi xi xi
Thus Pn pkq is a union of n ` 1 affine spaces An pkq with affine coordinates given by the
ratios in (5.3). We can define a topology on Pn pkq by defining a set W Ă Pn pkq to be open
if and only if W X Ui is open in Ui for all i “ 0, . . . , n.
The ratios x0 {xi , . . . , xn {xi also serve as natural candidates for defining regular func-
tions on Pn pkq. A regular function f : Ui Ñ k , however it is defined, should pull back
to a regular function Dpxi q Ă An`1 pkq. This pullback corresponds to an element g P
krx0 , . . . , xn , x´1 ˆ
i s which is invariant under scaling xi ÞÑ txi for t P k . When k is infinite,

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such polynomials must be homogeneous of degree 0, or equivalently, they are polynomials in

the ratios x0 {xi , . . . , xi {xi .
Note that the set Pnk ´ U0 consists of all the points px0 : ¨ ¨ ¨ : xn q where x0 “ 0. This can
be naturally be identified with Pn´1 pkq via the map py0 : ¨ ¨ ¨ : yn´1 q ÞÑ p0 : y1 : ¨ ¨ ¨ : yn q.
We may write
Pn pkq “ U0 Y Pn´1 pkq.

This way of thinking of Pn pkq, as adding to An pkq a Pn´1 pkq ‘at infinity’ is often a more
useful way of thinking about the points of Pn pkq.

Example 5.1. The projective line P1 pkq is a union

P1 pkq “ U0 Y U1

We identify U0 “ t px0 : x1 q | x0 ‰ 0 u with A1 pkq using the map z ÞÑ p1 : zq, so

informally z “ x1 {x0 defines an affine coordinate over U0 . Likewise, U1 is identified with
A1 pkq using the map w ÞÑ pw : 1q and w “ x0 {x1 is the affine coordinate over U1 . The
intersection U0 X U1 consists of the points px0 : x1 q where both x0 and x1 are non-zero.
Over this open set, the coordinates z and w are related by w “ z ´1 . One can then view
P1 pkq as a space obtained by gluing together two copies of A1 pkq, using the homeomorphism
z ÞÑ z ´1 over the subset A1 pkq ´ t0u.
Note that P1 pkq “ U0 Y tp0 : 1qu, so we can think of P1 pkq as adding to A1 pkq a ‘point
at infinity’. △

p1 : 0q
U0
U0 “ A1 pkq

P1 pkq
U1 “ A1 pkq U1

p0 : 1q

Figure 5.1 Gluing two affine lines to get P1 pkq

As Pn pkq is defined from An`1 pkq ´ tp0, . . . , 0qu the closed sets of Pn pkq are closely
related to closed sets in An`1 pkq, which are in turn defined by ideals in krx0 , . . . , xn s.
However, because of the equivalence relation, one cannot directly speak of the zero set of
any polynomial F px0 , . . . , xn q in Pn pkq. For instance, on P1 pkq the polynomial x20 ´ x1
takes the value 0 for px0 , x1 q “ p1, 1q, but not at px0 , x1 q “ p2, 2q. However, if F is
homogeneous, that is, all the terms have the same degree, then the zero set of F is well-
defined, as F pλx0 , . . . , λxn q “ λd F px0 , . . . , xn q where d is the degree of F . Then it makes
sense to define the zero set

Z` pF q “ t p P Pn pkq | F ppq “ 0 u.

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More generally, if a Ă krx0 , . . . , xn s is a homogeneous ideal, e.g., generated by homoge-

neous polynomials, we can define
Z` paq “ t p P Pn pkq | F ppq “ 0 for all F P a homogeneous u.
Every closed set of Pn pkq is of this form (see Exercise ??).
Example 5.2. The projective plane P2 pkq is covered by 3 open sets
P2 pkq “ U0 Y U1 Y U2
and each Ui can be identified with an affine plane A2 pkq. For instance, U0 is identified with
A2 pkq with affine coordinates x “ x1 {x0 and y “ x2 {x0 and U1 is A2 pkq with affine
coordinates u “ x0 {x1 and v “ x2 {x1 . Over the intersection U0 X U1 these are related by
u “ x´1 and v “ x´1 y .
We have P2 pkq “ U0 Y P1 pkq, where P1 pkq sits inside P2 pkq as the closed set tx0 “ 0u.
Thinking about P2k as A2k with the ‘line at infinity added’ leads to a space where the theory
of intersections is much nicer than in affine space. For instance, consider two paralell lines
L1 , L2 in A2k defined by the linear equations
L1 : ay1 ` by2 ` c “ 0
L2 : ay1 ` by2 ` d “ 0
where c ‰ d. If we identify A2k with the open set U0 Ă P2k , the closures L1 and L2 are given
by the equations
L1 : ax1 ` bx2 ` cx0 “ 0
L2 : ax1 ` bx2 ` dx0 “ 0
These lines do intersect, namely in the point p0 : ´b : aq, which lies in P2 ´ U0 . In fact, any
two lines in P2 pkq, that is, closed subsets defined by a linear equation, will always intersect
in P2k . This is the most classical example illustrating that projective varieties have better
geometric properties than affine varieties. △

P2
A2 p0 : 1 : 0q

p1 : 0 : 0q p0 : 0 : 1q
A2

Figure 5.2 The projective plane

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Projective space as a scheme

We would now like to construct a scheme Pn , so that for every field k , the k -points of Pn is
given by the set (5.1), that is, the 1-dimensional subspaces in k k`1 . In fact, the construction
works over any base ring A. The basic idea is to construct PnA from n ` 1 copies of affine
n-space AnA using similar gluing formulas in the previous section.
Fix a ring A and variables x0 , . . . , xn . For each i “ 0, . . . , n, define the ring
„ ȷ
x0 xn
Ri “ A ,..., . (5.4)
xi xi
Since Ri is a polynomial ring in n variables, each Ui “ Spec Ri is isomorphic to an affine
space AnA . We will also, for 0 ď i, j, k ď n consider the rings
„ ȷ „ ȷ
x0 xn xi x0 xn xi xi
Rij “ A ,..., , and Rijk “ A ,..., , , .
xi xi xj xi xi xj xk

If we consider these rings as subrings of the ring Arx0 , x´1 ´1

0 , . . . , xn , xn s, we have for each
pair i, j an equality of subrings
Rij “ Rji .

This follows from the identities xl {xi “ xl {xj ¨ xj {xi , for all i, j and l. Similarly,

Rijk “ Rjik “ Rjki .

Note that we may view Rij as the localization of Ri in the element xj {xi . Hence Uij “
Spec Rij can be identified with the distinguished open subscheme Dpxj {xi q Ă Ui . Since
Rij “ Rji , using the identity maps τij : Uij Ñ Uji as gluing maps, we see that the gluing
conditions are satisfied, and the Ui glue together to a scheme, which we denote by PnA .

Definition 5.3. The scheme PnA is called the projective n-space over A.
The projective space over Z is called the projective n-space, and it is denoted by Pn .

Note that all rings Ri are A-algebras, so each Ui is a scheme over A and comes with a
structure map Ui Ñ Spec A. The same is true for the localizations Rij , and we see that the
structure maps glue together to a morphism π : PnA Ñ Spec A, making PnA an A-scheme.
If Spec A is irreducible, then so is PnA . This is because PnA contains U0 » AnA as a dense
open subset and AnA is irreducible if A is. Likewise, PnA is reduced if A is, because it has the
same local rings as AnR , and AnA is reduced if A is. This means that PnA is integral if A is an
integral domain.

Example 5.4. The projective line P1A over A is constructed by gluing two copies of the affine
line A1A , U0 “ Spec Arus and U1 “ Spec Aru´1 s, along U01 “ Spec Aru, u´1 s which is
a distinguished open set in both U0 and U1 . △

Example 5.5. The projective plane P2A is constructed by gluing together the three affine
planes Ui “ D` pxi q “ Spec Ri for i “ 0, 1, 2. It is sometimes convenient to rewrite these
charts using the ‘U0 -coordinates’, i.e., writing x “ x1 {x0 and y “ x2 {x0 . We can then

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express the other ratios in terms of x and y . For instance x2 {x1 “ x´1 . With this convention,
the three affine opens become
U0 “ Spec Arx, ys, U1 “ Spec Arx´1 , yx´1 s, U2 “ Spec Ary ´1 , xy ´1 s.
△
There is a morphism of schemes
π : An`1
A ´ V px0 , . . . , xn q ÝÝÑ PnA . (5.5)
which can be seen as a scheme-analogue of the quotient map k n`1 ´ t0u Ñ Pn pkq.
To define it, we work locally over Ui . The inclusions Ri Ă Arx0 , . . . , xn , x´1
i s determine
morphisms of affine schemes
πi : Spec Arx0 , . . . , xn , x´1
i s ÝÝÑ Ui .
Note that the scheme of the left can be identified with the distinguished open set Dpxi q Ă
An`1
A , and the collection of these form an affine cover of Dpx0 q Y ¨ ¨ ¨ Y Dpxn q “ AA
n`1
´
V px0 , . . . , xn q. To check that the morphisms πi glue, we need only check that they agree
over the intersections. This follows by applying Spec to the following diagram:

Arx0 , . . . , xn , x´1
i s Arx0 , . . . , xn , x´1 ´1
i , xj s Arx0 , . . . , xn , x´1
j s

” ı ” ı
xi xj
Ri Ri xj
“ Rj xi
Rj

The projective space over Z is especially important, because for every field k , the k -valued
points of PnZ coincides with the projective space Pn pkq as defined in the introduction.

Proposition 5.6. For any field k , we the k -valued points of Pn is given by (5.1), i.e.,
` ˘
Pn pkq “ k n`1 ´ t0u { „ . (5.6)

Proof The image of a k -point ι : Spec k Ñ Pn is a single point in Pn , so it must be

contained in one of the Ui . Therefore, the k -points of Pn pkq is the union of the k -points of
the Ui . Identifying Ui pkq “ An pkq “ k n , we define τ : k n`1 ´ t0u Ñ Pn pkq by the map
ˆ ˙
a0 ai´1 ai`1 an
τ pa0 , . . . , an q “ ,..., , ,..., P Ui pkq if ai ‰ 0
ai ai ai ai
Then τ is well-defined, surjective, and τ pa0 , . . . , an q “ τ pb0 , . . . , bn q precisely when
pb0 , . . . , bn q “ λ ¨ pa0 , . . . , an q for some λ P k ˆ .
More generally, if R is any ring, and s “ ps0 , . . . , sn q is an pn ` 1q-tuple of elements
which generate the unit ideal in R, there is a corresponding R-point
ιa : Spec R ÝÝÑ Pn . (5.7)
We define this morphism using the gluing theorem for morphisms. By our assumption,
the distinguished open sets Dps0 q, . . . , Dpsn q cover Spec R. Over the distinguished open

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sets Dpsi q “ Spec Rsi , we have ring maps ϕi : Zrx0 {xi , . . . , xn {xi s Ñ Rsi defined by
xk {xi ÞÑ sk {si which make the following diagrams commute:
” ı ” ı ” ı
x0
Z xi
, . . . , xxni Z x0
xi
, . . . , xxni , xxji Z x0
xj
, . . . , xxnj

Rsi Rsi sj Rsj

It follows that the scheme maps Spec Rsi Ñ Ui glue to the desired morphism (5.7).
Observe that if λ P R is a unit, then pλs0 , . . . , λsn q determines the same R-point as
ps0 , . . . , sn q. This is because multiplying each si by a unit do not change the expressions
sk {si used in the construction of ϕi .
It is natural to ask whether the formula Pn pRq “ pRn ´ t0uq { „ holds for all rings R,
that is, whether every R-valued point is of ‘homogeneous form’. While this is true if R is a
local ring (Exercise 6.5.27) or R “ Z (Exercise 6.5.28), it does not hold in general. In fact, it
fails for an interesting reason, as the following example shows:
?
Example
? 5.7. Let us consider the R-valued points of P1 for R “ Zr ´5s. The ring
Zr ´5s is famously not a UFD, as
? ?
2 ¨ 3 “ p1 ` ´5qp1 ´ ´5q. (5.8)

We will exploit this observation to define a non-trivial R-point of P1R .

Note that 2 and 3 generate the unit ideal of R, so Spec R is covered by the two distin-
guished open sets Dp2q “ Spec Rr 21 s and Dp3q “ Spec Rr 13 s, and these two intersect along
? ?
the distinguished open set Dp6q “ Spec Rr 16 s. Let α “ 12 p1 ` 5q and β “ 13 p1 ´ 5q
and consider the ring maps ϕ0 : Zrxs Ñ Rr 12 s and ϕ1 : Zrx´1 s Ñ Rr 13 s sending x ÞÑ α
and x´1 ÞÑ β . By the factorization (5.8), we see that αβ “ 1, so the following diagram
commutes:
ϕ0
Zrxs Rr 12 s

Zrx, x´1 s Rr 16 s

ϕ1
Zrx´1 s Rr 13 s

Applying Spec, we see that the morphisms Specpϕ0 q and Specpϕ1 q glue to a morphism
Spec R Ñ P1 . The corresponding R-point in P1 pRq is not of the?form pa : bq with a, b P R
would correspond to p1 ` ´5 : 2q (this is what it is
generating the unit ideal. If it were, it?
over Dp2q). However, the ideal p1 ` ´5, 2q is not the unit ideal in R, in fact it is not even
a principal ideal in R. (For a proof of this, see Exercise 4.14.5.)
What makes this example work is the fact that R is not a UFD and has many non-principal
ideals. We will consider this example later in Chapter 17 when we discuss invertible sheaves
and maps to projective space. △

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The structure sheaf of PnA

The following computation is very important.

Proposition 5.8. For each ring A, we have ΓpPnA , OPnA q “ A.

Proof By the sheaf sequence associated to the covering U0 , . . . , Un , a global section of

OPnA consists of elements si P ΓpUi , OPnA q that agree on the overlaps Ui X Uj . Over Ui we
have ΓpUi , OPnA q “ Ri and a section si is a polynomial of the form
ˆ ˙
x0 xn
gi ,..., (5.9)
xi xi
The compatability condition on the overlaps means that
ˆ ˙ ˆ ˙
x0 xn x0 xn
gj ,..., “ gi ,..., . (5.10)
xj xj xi xi
The only way (5.10) can hold is that gi and gj are of degree 0 as polynomials in the ratios
xk {xi and xk {xj . That is, gi and gj are equal to the same element in A. Conversely, if we
choose any element a P A and set gi “ a for every i “ 0, . . . , n, then the sections clearly
glue to a global section of OPnA .
In particular, for a field k , the space of global sections of OPnk is just the ‘constants’, k .
When k “ C, this can be seen as an analogue of Liouville’s theorem, that the only global
analytic functions on CP1 are the constants. This example also shows that the group OX pXq
may not give much information about X for general schemes.
We note that we also have got yet another example of a scheme which is not affine: if P1C
were affine, it would have to be isomorphic to Spec C according to Theorem 4.21 on page 73.
But this is clearly not the case, as P1C contains infinitely many closed points. P1C is in fact
related to the first example of a non-affine scheme of Example 4.26 on page 75, namely the
affine plane A2C “ Spec Cru, vs with the ‘origin’ V pu, vq removed, see (5.5).

Closed subschemes of PnA

In this section, we will consider closed subchemes of the projective space PnA . We say that
a polynomial F P Arx0 , . . . , xn s is homogeneous of degree d if the monomials in the xi
appearing in F all have degree d. For such elements, we can consider their ‘dehomogenization’
with respect to the variable xi , defined by
ˆ ˙ „ ȷ
´d x0 x1 xn x0 xn
Fpiq “ xi F px0 , . . . , xn q “ F , ,..., P Ri “ A ,..., .
xi xi xi xi xi
Conversely, if f P Ri is a polynomial of degree d in the x0 {xi , . . . , xn {xi , then the polyno-
mial
ˆ ˙
x0 xi´1 xi`1 xn
F px0 , . . . , xn q “ xdi f ,..., , ,...
xi xi xi xi
will be homogeneous of degree d.

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The homogenization dehomogenization operations will in most cases be inverses. There are
exceptions however: any power xd0 will dehomogenize to 1, and there is no way of recovering
xd0 without
” ı d. The best we can say is that there is a bijection between polynomials
knowing
in A xx0i , . . . , xxni of degree d and homogeneous polynomials in Arx0 , . . . , xn s of degree d
not divisible by xi .
More generally, if a Ă R is a homogeneous ideal, we define the dehomogenization of a
with respect to xi by
„ ȷ
x0 xn
apiq “ t Fpiq P A ,..., | F P a u.
xi xi
It is not difficult to check that if a is a prime ideal, then so is apiq for each i “ 0, . . . , n.
Consider the subschemes defined by the apiq :
Xi “ SpecpRi {apiq q ÝÝÑ SpecpRi q.
If Fpiq and Fpjq denote the dehomogenizations with respect to xi and xj , we have
ˆ ˙d „ ȷ „ ȷ
xj xi xj
Fpiq “ Fpjq in Ri “ Rj
xi xj xi
This implies that the two ideals apiq and apjq become equal in the localization Ai rxi {xj s “
Aj rxj {xi s. Consequently, the subschemes Xi coincide over the intersections Uij , and conse-
quently they glue together to a closed subscheme X “ V` paq Ă PnA .

Definition 5.9. We say that an A-scheme X is projective over A if it is isomorphic to

one of the form V` paq for some homogeneous ideal a Ă Arx0 , . . . , xn s.

Example 5.10. For d P N, the homogeneous polynomial F “ xd0 ` xd1 ` xd2 defines a closed
subscheme X “ V` pF q Ă P2A . When A “ k is a field, X is covered by three affine charts,
each isomorphic to the plane curve V pxd ` y d ` 1q Ă A2k , so we may picture as a plane
curve of degree d. The geometry of this scheme is however more intricate for other rings,
such as A “ Z or A “ Crts. △
Example 5.11. If a “ px0 , . . . , xn q, then the corresponding closed subscheme is empty.
Indeed, over Ui , the dehomogenized ideal contains xi {xi “ 1, so ?a defines the empty scheme
in each Ui . The same argument applies to any ideal a such that a “ px0 , . . . , xn q. For this
reason, px0 , . . . , xn q is called the irrelevant ideal.
This also implies that two ideals can correspond to the same closed subscheme. For
instance, in P1k the two ideals px0 q and px20 , x0 x1 q “ px0 q X px20 , x1 q both define the point
p0 : 1q. △
Example 5.12 (Homogeneous coordinates). For an pn ` 1q-tuple pa0 , . . . , an q of elements
in A which generate the unit ideal, we define a closed subscheme defined by the ideal a
generated by the 2 ˆ 2-minors of the matrix
ˆ ˙
x0 x1 . . . x n
. (5.11)
a0 a1 . . . an

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That is, the homogeneous ideal a “ pai xj ´ aj xi |0 ď i, j ď nq. One can check that the
corresponding closed embedding is exactly the morphism ιa : Spec A Ñ PnA defined in (5.7)
(see Exercise 6.5.8). △

Proposition 5.13. Every closed subset Z Ă PnA is of the form V` paq for some homoge-
neous ideal a Ă Arx0 , . . . , xn s.

Proof By construction, a subset Z is closed if and only if Z X Ui for each i “ 0, . . . , n.

As each Ui “ Spec Ri is affine, we have Z X Ui “ V pai q for some radical ideal ai Ă Ri .
Let us define
deg F
a “ t F P Arx0 , . . . , xn s | F homogeneous and x´
i F P ai for all i “ 0, . . . , n u.
Then a is homogeneous. Let us check that Z “ V` paq.
Let p P Z be a point. We may assume that p P Z X U0 “ V pa0 q. Therefore f ppq “ 0 for
every f P a0 . Consider F P a of degree d. Then x´d ´d
0 F P a0 , so px0 F qppq “ 0. As p P U0 ,
we must have F ppq “ 0. Hence p P V` paq.
Conversely, let p P V` paq. As before, we may assume that p P U0 . By assumption, for
deg F
every F P a, we have F ppq “ 0 and hence x´ 0 F P a0 . Therefore, p P V pa0 q “ Z XU0 .
This shows that p P Z , so V` paq “ Z .
If I Ă Arx1 {x0 , . . . , xn {x0 s is an ideal, we define the homogenization of I as the ideal
a Ă Arx0 , . . . , xn s obtained by homogenizing all the elements of I . Geometrically, the
subscheme V` paq Ă PnA represents the closure of V pIq Ă Ui in PnA .

A family of sheaves on PnA

The projective space PnA carries a special family of sheaves, known as the twisting sheaves
OPnA pmq, where m is an integer. These sheaves will play an important role in the study of
projective schemes.
These sheaves are closely related to homogeneous polynomials. While homogeneous
polynomials of degree m ě 1 in krx0 , . . . , xn s do not define regular functions on Pnk , they
do define global sections of the sheaf OPnk pmq. Intuitively, over an open set U Ă Pnk , the
sections of OPnk pmq can be identified with the set of quotients
f px0 , . . . , xn q
where deg f ´ deg g “ m
gpx0 , . . . , xn q
where f, g are homogeneous and gppq ‰ 0 for all p P U .
To define these sheaves by gluing, we use the covering of PnA by Ui “ Spec Ri where
Ri “ Ar xx0i , . . . , xxni s, as before. Over Ui , we define the Ri -module
„ ȷ
x0 xn m
Mi “ A ,..., xi
xi xi
Note that since xm m m
i “ pxi {xj q xj , there is an equality of A-modules
„ ȷ „ ȷ
x0 xn xi m x0 xn xj m
A ,..., , x “A ,..., , x (5.12)
xi xi xj i xj xj xi j

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In other words, we may identify the localizations

pMi qxj {xi “ pMj qxi {xj (5.13)
x
as modules over the ring Ri r xxji s “ Rj r xji s.
Note that Mi is a free Ri -module of rank 1, with xm i serving as a generator. For each i
and j , one can pass from Mi to Mj by multiplying by pxj {xi qm :
ˆ ˙ ˆ ˙m ˆ ˙
x0 xn m xi x0 xn
g ,..., xi “ g ,..., xm
j . (5.14)
xi xi xj xj xj

By the equality (5.13), the two sheaves M Ăi and M Ăj on Ui restrict to the same sheaf on
Uij “ SpecpAi rxi {xj sq, namely the tilde of the module (5.12) (see Exercise 9.9.37). As the
isomorphisms are identity maps, the gluing conditions are satisfied, and the sheaves M
Ăi over
n
Ui glue to a sheaf on PA . We denote this sheaf by OPA pmq.
n

In the special case m “ 0, we have Mi “ Ai and the sheaf OPnA p0q “ OPnA is simply the
structure sheaf of PnA . The following computation generalizes Proposition 5.8:

Proposition 5.14. ΓpPnA , OPnA pmqq can be identified with the free R-module of homo-
geneous polynomials of degree m in the x0 , . . . , xn .

Proof By the sheaf sequence associated to the covering U0 , . . . , Un , a global section of

OPnA pmq consists of sections si P ΓpUi , OPnA pmqq that agree on the overlaps Ui X Uj . Over
Ui we have ΓpUi , OPnA pmqq “ Mi and a section si is an element of the form
ˆ ˙
x0 xn
gi ,..., xm
i P Mi . (5.15)
xi xi
The compatability condition on the overlaps implies that
ˆ ˙ ˆ ˙
x0 xn m x0 xn
gj ,..., xj “ gi ,..., xm
i . (5.16)
xj xj xi xi
If m ă 0, then clearly there is only one tuple satisfying (5.16), namely where g0 “ ¨ ¨ ¨ “
gn “ 0.
If m ě 0, we see that the gj are determined by g0 . Comparing degrees in (5.16), we see
that g0 must have degree ď m as a polynomial in the x1 {x0 , . . . , xn {x0 . This means that
F “ g0 xm0 is a homogeneous polynomial of degree m in x0 , . . . , xn with coefficients in
A, and si “ gi xm
i “ F for all i “ 0, . . . , n. Conversely, any such F determines a valid
pn ` 1q-tuple ps0 , . . . , sn q.

5.2 The Proj construction

In this section, we discuss a generalization of the construction of PnA , known as the Proj-
construction. Starting with any graded ring
à
R“ Rd (5.17)
dě0

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we will construct a scheme Proj R, called the projective spectrum. See Appendix ??? for
background and notation on graded rings and modules.
The construction is somewhat parallel to that in the spectrum of a ring in Chapter 2.
In both constructions, the topological space is defined in term of prime ideals in the ring
and the structure sheaf is defined in term of localizations of the ring. There are however
several notable differences between the two constructions. For instance, Proj R does not
depend functorially on the ring R, in the sense that graded ring maps do not always induce
maps between the projective spectra. Also, different graded rings R may yield isomorphic
projective spectra.

Definition 5.15. Let R be a graded ring. The projective spectrum, Proj R is defined as
the set of homogeneous prime ideals of R that do not contain the irrelevant ideal R` , that
is,
␣ (
ProjpRq “ p P Spec R ´ V pR` q | p homogeneous .

The reason for excluding primes containing the irrelevant ideal R` comes from Example
5.11. In that example, when R “ krx0 , . . . , xn s with the standard grading, the irrelevant
ideal px0 , . . . , xn q corresponds to the origin p0, . . . , 0q in An`1 pkq, which does not give a
well-defined point of Pn .
The set Proj R is endowed with the Zariski topology where the closed sets are of the form
V` paq “ t p P Proj R | a Ă p u
with a a homogeneous ideal. The three topology axioms follow from the identities in the
following lemma.

Lemma 5.16. Let a, b and tai uiPI be homogeneous ideals. Then:

Ă b, thenŞV` pbq Ă V` paq.
(i) If a ř
(ii) V` p ai q “ V` pai q.
(iii) V` p?abq “ V` pa X bq “ V` paq Y V` pbq.
(iv) V` p aq “ V` paq.

Proof Note that sums, products and radicals remain homogeneous when the involved ideals
are homogeneous. The proofs of the four statements are exactly the same as in Lemma 2.3
for Spec R (the arguments there remain valid under the additional constrains that the prime
ideals are homogeneous and do not contain the irrelevant ideal).
Just as for the spectrum of a ring, Proj R has a collection of distinguished open sets, which
form a basis for the topology.
For each f P R which is homogeneous of positive degree, we define the distinguished
open set D` pf q as
D` pf q “ tp P Proj R | p S f u.
In other words, D` pf q is the set of homogeneous prime ideals in R that do not contain
the irrelevant ideal R` , and do not contain f . It is clear that D` pf q is an open set, being the
complement of the closed set V` pf q.

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Lemma 5.17. Let R be a graded ring.

(i) For any two homogeneous elements f and g of positive degree, we have
D` pf q X D` pgq “ D` pf gq.
(ii) The sets D` pf q, where f ranges over homogeneous elements of R of
positive degree, form a basis for the topology on Proj R.

Proof These statements follow as in the affine case. For (i), for a homogeneous prime ideal
p, the conditions f R p and g R p are equivalent to f g R p.
For (ii), note that a closed set V` paq is equal to the intersection of the V` pf q’s for the
homogeneous f P a X R` . Taking complements we see that any open set Proj R ´ V` paq
is the union of the corresponding D` pf q’s.

In analogy with the formulas for defining PnA , we will use the degree 0 localizations of R
to define the structure sheaf on Proj R. More precisely, for a nonzero homogeneous element
f P R, the ring Rf is naturally a Z-graded ring, by defining

degpa{f n q “ deg a ´ n deg f.

Thus it makes sense to consider the subring of degree 0 elements pRf q0 .

Recall that for a distinguished open set Dpf q of an affine scheme Spec A, there is a
canonical homeomorphism between Dpf q and Spec Af which sends a prime p P Dpf q to
the prime ideal pAf . In analogy with this, we will show below that the map p ÞÑ ppRf q0
defines a homeomorphism between D` pf q and Spec pRf q0 , where pRf q0 denotes the degree
0 part of the localization Rf .
If f, g P R are homogeneous elements of positive degree, the canonical localization map
Rf Ñ Rf g induces a map of rings

pRf q0 ÝÝÑ pRf g q0 (5.18)

The ring on the right can be viewed as a localization of the ring on the left in the degree 0
element g deg f f ´ deg f . Indeed, in pRf qu , both f and g are invertible, so we may identify
pRf qu “ Rf g . As the element g deg f f ´ deg f has degree 0, this identification preserves the
Z-grading. Looking in degree 0 part, we find

ppRf q0 qgdeg f f ´ deg g “ pRf g q0 . (5.19)

Proposition 5.18. For each nonzero homogeneous f P R` , the map

ϕppq “ ppRf q0
defines a homeomorphism
ϕ : D` pf q ÝÝÑ SpecpRf q0 . (5.20)

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The homeomorphisms (5.20) are compatible in the sense that the following diagram

D` pf gq D` pf q
ϕf g ϕf (5.21)

SpecpRf g q0 SpecpRf q0 .

commutes for each f, g P R` homogeneous.

Proof Note that ϕ is defined as the composition

D` pf q Ñ Dpf q » SpecpRf q Ñ SpecpRf q0 . (5.22)

Therefore, ϕ is continuous. We next show that it is bijective.

The following observation will be useful. If x P Rnd , and p is a homogeneous prime ideal
such that p S f , then
a ¨ f ´n P ϕppq ðñ a P p. (5.23)

This follows because af ´n P ϕppq is equivalent to a{1 P pRf , and this is equivalent to
a P p, because f R p.
Injectivity of ϕ: If ϕppq “ ϕpp1 q, and a P Rn , then (5.23) shows that ad P Rnd is
contained in p if and only it is contained in p1 . But as p is prime, it contains ad if and only if
it contains a, and the same applies to p1 . Hence p “ p1 ,
À of ϕ: let q Ă pRf q0 be a prime ideal. We consider the homogeneous prime
Surjectivity
ideal p “ ně0 pn where
" *
d ´n
pn “ a P Rn | a ¨ f Pq .

In other words, pn consists of the homogenous elements of degree n that end up in q when
‘dehomogenized’.
The first thing to check is that p is a prime ideal. Once we know it is an ideal, it will be
homogeneous by definition. It is clear that it is closed under multiplication by elements of R:
if t P Rm , and ad f ń P q, then tad f ´mń P q in pRf q0 , and hence ta P pn`m . It is also
an additive subgroup: assume we are given two elements a, b P pn . By definition, we have
ad ¨ f ń P q and bd ¨ f ń P q. Expanding pa ` bq2d ¨ f ´2n , we obtain a sum where every
term is a multiple of either ad ¨ f ń or bd ¨ f ń . Therefore pa ` bq2d ¨ f ´2n P q, and hence
pa ` bqd ¨ f ń P q, because q is prime. Consequently p is closed under addition.
To show that p is prime, it suffices to check that ab P p implies either a P p or b P p for a
and b homogeneous. That a P Rm and b P Rn satisfy ab P p means that pabqd f ´pm`nq P q.
Since f is a unit in Rf , it holds that pabqd P q, and hence either a P q or b P q. Therefore we
have either ad f ´m P pm or bd f ń P pn , and so p is prime.
Next we verify that ϕppq “ q. This follows from the implications

af ń P q ðñ ad f ńd P q ðñ a P pnd ðñ af ń P ϕppq (5.24)

which follow from (5.23) and the primeness of p and q.

Hence ϕ is bijective. To prove that ϕ is a homeomorphism, it suffices to show that it is

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open. Since the open sets of the form D` pf gq form a basis for the open sets contained in
D` pf q, it suffices to show that for any homogeneous element g P R, we have
ϕpD` pf gqq “ Dpg d f ´ deg f q.
For p P D` pf q, the following series of equivalences hold because pRf is a prime ideal:
p P D` pf gq ðñ g R p
ðñ g d f ´ deg g R ppRf q0 “ ϕppq.
ðñ ϕppq P Dpg d f ´ deg g q
Hence ϕpD` pf gqq “ Dpg d f ´ deg g q.
Finally, we show that the diagram (5.21) commutes. The lower horizontal map is induced
by the localization map ℓ : Rf Ñ Rf g . If p P D` pf gq is a homogeneous prime ideal such
that p S f and p S g , then we have ℓ´1 ppRf g q “ pRf . Since ℓ preserves the gradings, we
have
ℓ´1 ppRf g X pRf g q0 q “ pRf X pRf q0 .
This shows that p maps to pRf X pRf g q0 via both compositions in (5.21).
In light of (5.19), we may identify SpecpRf g q0 with a distinguished open subset of both
SpecpRf q0 and SpecpRg q0 . We now glue SpecpRf q0 to SpecpRg q0 along SpecpRf g q0
using the ring maps
pRf q0 ÝÑ pRf g q0 ÐÝ pRg q0 .
Here the ‘triple intersections’ can be identified with the affine scheme SpecpRf gh q0 . The
gluing conditions for schemes are satisfied (taking the gluing isomorphisms to be the identity
maps), and hence we obtain a scheme, which we denote by Proj R. The resulting structure
sheaf has the property that for each f P R homogeneous of positive degree,
OProj R pD` pf qq “ pRf q0 . (5.25)
The scheme Proj R is in a natural way a scheme over Spec R0 . Indeed, for each open set U ,
the group OProj pU q is an R0 -algebra (this is clear if U “ D` pf q since OpD` pf qq “ pRf q0
is an R0 -algebra, and the general case follows by covering U by D` pf q’s). In particular,
OProj R pProj Rq is an R0 -algebra, so applying Theorem 4.17 to R0 Ñ OProj R pProj Rq we
get a canonical morphism π : Proj R Ñ Spec R0 . When restricted to a distinguished open
D` pf q, this morphism can be identified with the canonical map SpecppRf q0 q Ñ SpecpR0 q.
Example 5.19. For a ring A and each non-negative integer n, we have
PnA “ Proj Arx0 , . . . , xn s.
To see this, note that for R “ Arx0 , . . . , xn s, then D` pxi q “ pRxi q0 “ Spec Ri with Ri “
Arx0 {xi , . . . , xn {xi s and the intersections D` pxi q X D` pxj q are equal to D` pxi xj q “
Spec pRxi xj q0 . Thus Proj R is obtained as exactly the same gluing construction as PnA . △
Example 5.20. The projective 0-space P0A “ Proj Arts is particularly simple: since the
irrelevant ideal is generated by t, we have
Proj Arts “ D` ptq “ SpecpArt, t´1 sq0 “ Spec A.

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Over a field, we recover the expected geometric picture where P0k “ Spec k is a point. △

Example 5.21. Consider R “ krx0 , x1 s{px0 x1 q with the natural grading. Geometrically,
Spec R ´ V px0 , x1 q represents the union of the x0 - and x1 -axes, excluding the origin.
Therefore, we expect Proj R to consist of only two points. Besides the irrelevant ideal
R` “ px0 , x1 q, there are only two homogeneous prime ideals, px0 q and px1 q. Hence,
Proj R indeed consists of two points. △

5.3 Functoriality
In contrast to the Spec-construction, the Proj-construction is not entirely functorial. While
a map of graded rings ϕ : R Ñ S induces a map on homogeneous prime ideals via p ÞÑ
ϕ´1 ppq, this does not always yield a morphism. The issue is that the preimage ϕ´1 ppq of a
homogeneous prime ideal p Ă S may contain the irrelevant ideal R` .
To remedy this, we define the base locus of ϕ as the closed set

V pϕpR` qq Ă Proj S.

This consists of precisely the prime ideals p in S for which ϕ´1 ppq Ą R` . This means that
for prime ideals p R V pϕpR` qq, the preimage ϕ´1 ppq defines a point of Proj R.

Proposition 5.22. Let ϕ : R Ñ S be a map of graded rings. Then there is a morphism of

schemes
F : Proj S ´ V pϕpR` qq ÝÝÑ Proj R.
which on the level of topological spaces is given by p ÞÑ ϕ´1 ppq.

Proof Note that the complement U “ Proj S ´ V pϕpR` qq is open in Proj S and has a
canonical scheme structure. By the above remarks, the mapping p ÞÑ ϕ´1 ppq which defines
F is well-defined. It is also continuous, because it is the restriction of the continuous map
Spec S Ñ Spec R.
Next, we define the map of sheaves of rings F 7 : OProj R Ñ F˚ OU . We define this map
using the B -sheaf construction on the open sets D` pf q. Note that as f runs through the
homogeneous elements of R` , the D` pϕpf qq cover U . Moreover,

F ´1 pD` pf qq “ D` pϕpf qq (5.26)

Indeed, for p P U :

F ppq P D` pf q ðñ f R ϕ´1 ppq ðñ ϕpf q R p ðñ p P D` pϕpf qq

We have OProj R pD` pf qq “ pRf q0 and pF˚ OU qpD` pf qq “ pSϕpf q q0 . We define the map
F 7 over D` pf q to be the degree part 0 part of the localization

pRf q0 ÝÝÑ pSϕpf q q0 . (5.27)

By the diagram below, these ring maps are compatible with the restriction maps, and define a

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map of sheaves OProj R Ñ F˚ OU .

pRf q0 pSϕpf q q0

pRf g q0 pSϕpf gq q0

If we restrict pF, F 7 q to a distinguished open set D` pf q Ă Proj S , we obtain the morphism

of affine schemes
SpecpSϕpf q q0 ÝÝÑ SpecpRf q0 .
In particular, this implies the map induced by F 7 on local rings is a map of local rings. (In
fact, the induced map is given by pRϕ´1 ppq q0 ÝÝÑ pSp q0 .) Hence pF, F 7 q is a morphism of
schemes.

Example 5.23 (Projection from a linear subspace). If we consider the graded ring map given
by the inclusion Zrx0 , . . . , xr s ãÑ Zrx0 , . . . , xn s, the base locus equals the subscheme
V` px0 , . . . , xr q. We get a corresponding morphism
Pn ´ V` px0 , . . . , xr q ÝÝÑ Pr .

On the level of k -points, the induced map Pn pkq ´ V` px0 , . . . , xr q Ñ P1 pkq is just the
projection pa0 : ¨ ¨ ¨ : an q ÞÑ pa0 : . . . , ar q. △

Closed embeddings
Let a be a homogeneous ideal in the graded ring R. The quotient map ϕ : R Ñ R{a is
a map of graded rings, and it satisfies ϕpR` q “ pR{aq` . To see this, note that a prime
q P ProjpR{aq contains pR{aq` if and only if ϕ´1 pqq contains ϕ´1 ppR{aq` q “ R` . The
base locus is therefore empty, and we obtain a morphism

ι : Proj R{a ÝÝÑ Proj R

whose image is V` paq.
We claim that ι is a closed embedding. It will suffice to check this over each D` pf q
for f P R homogeneous. Over this open set, we have ι´1 D` pf q “ D` pϕpf qq, and the
restriction of ι to ι´1 D` pf q is identified with the morphism

Spec ppR{aqf q0 ÝÝÑ Spec pRf q0

induced by the degree zero part of the localization Rf Ñ pR{aqf . Since this map is surjective,
ι|ι´1 D` pf q is a closed embedding.
In fact, under some mild assumptions on the graded ring R, every closed embedding into
Proj R arises in this way. We will prove this in Chapter ??.

Example 5.24. Let R “ krx0 , . . . , xn s with the standard grading, and let a Ă R be a
homogeneous ideal. Consider the closed subscheme Y “ V` paq Ă Pnk “ ProjpRq. For

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each i “ 0, . . . , n, the distinguished open subset D` pxi q Ă Pnk is isomorphic to Ank via the
map
„ ȷ
x0 xn
pRxi q0 » k ,..., .
xi xi
The ideal paRxi q0 in pRxi q0 is generated by the dehomogenizations of elements of a with
respect to xi , i.e.,
␣ (
paRxi q0 “ fpiq | xdi f P a for some d P N ,
The intersection Y X D` pxi q is then defined by
Y X D` pxi q “ V ppaRxi q0 q Ă D` pxi q » Ank .
Thus Y coincides with the closed subscheme V` paq defined earler. △

Veronese subrings
Let R be a graded ring and let n be a positive integer. The Veronese subring
à
Rpnq “ Rnd
dě0

pnq
is a graded ring. The inclusion R Ă R induces a morphism
vn : ProjpRq ÝÝÑ ProjpRpnq q.
To see that the base locus is empty, note that the irrelevant ideal of Rpnq is generated by
pnq
all elements in R whose degree is positive and divisible by n. Then ϕpR` q defines the
empty set, since any prime p Ă R such that R` X Rpnq Ă p must contain all of R` : for any
a P R` , we have an P R` X Rpnq , which forces a P p as well.

Proposition 5.25. The morphism vn is an isomorphism.

Proof Both ProjpRq and ProjpRpnq q are covered by distinguished open sets of the form
D` pf n q where f P R` is homogeneous. Over such an open set, the morphism vn is induced
by the inclusion
pnq
pRf n q0 ÝÝÑ pRf n q0
This inlcusion is actually an equality: if g{f ns P pRf n q0 , then deg g “ ns ¨ deg f , so
g P Rpnq and g{f ns belongs to the left-hand side.
This means that vd restricts to an isomorphism of schemes over an open covering of
ProjpRpdq q, and hence it is an isomorphism.

More examples
An important special case is when R “ Arx0 , . . . , xn s and S “ Ary0 , . . . , ym s with
ϕ : R Ñ S is defined by
ϕ : Ary0 , . . . , ym s ÝÝÑ Arx0 , . . . , xn s; yi ÞÑ fi

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where f0 , . . . , fm are homogeneous polynomials of the same degree d. In that case, the base
locus of ϕ is the closed subscheme V` pf0 , . . . , fm q. Strictly speaking, to make ϕ into a map
of graded rings, we need to adjust the grading by defining degpyi q “ d for each i. However,
doing this does not change the Proj by Proposition 5.25.
We will for brevity write
Pm
A ´ V` pf0 , . . . , fn q ÝÝÑ PnA
px0 : ¨ ¨ ¨ : xm q ÞÝÑ pf0 pxq : ¨ ¨ ¨ : fn pxqq
for the morphism defined by f0 , . . . , fn , even if the indicated map is only valid on the level
of k -points, when A “ Z or A “ k .
In good cases, the subscheme V` pf0 , . . . , fn q is empty, and we obtain an actual morphism
Pm n
A Ñ PA . This happens if and only if the radical of pf0 , . . . , fn q contains irrelevant ideal
px0 , . . . , xn q.
Example 5.26. Let n P N and consider the morphism
f : P1A ÝÝÑ P1A
px0 : x1 q ÞÝÑ pxn0 : xn1 q
Here the base locus is indeed empty, as the radical of pxn0 , xn1 q is equal to the irrelevant ideal
px0 , x1 q. The morphism f is called the n-th power map. If k is a field, the map on k -points
is given by pa : bq ÞÑ pan : bn q. In general, f is a finite morphism, as over the two standard
affines it coincides with the morphism A1A Ñ A1A induced by u ÞÑ un . △

The Veronese embedding

Example 5.27. Consider the morphism
ι : P1A ÝÝÑ P2A
px0 : x1 q ÞÝÑ px20 : x0 x1 : x21 q
This is actually a closed embedding. Over the open set D` py0 q, we have ι´1 pD` py0 qq “
D` px0 q and the morphism ι is given by
„ ȷ „ ȷ
x1 y1 y2
Spec A ÝÝÑ Spec A , (5.28)
x0 y0 y0
induced by the ring map y1 {y0 ÞÑ px0 x1 q{x20 “ x1 {x0 and y2 {y0 ÞÑ x21 {x20 . As this ring
map is surjective, we obtain a closed embedding. Similarly, ι is a closed embedding over
D` py1 q.
Note the relation px0 x1 q2 “ px20 qpx21 q between the three monimials. This translates into
the equation y12 ´ y0 y2 “ 0 for the image of ι. In fact, ι is exactly the closed subscheme
associated with the homogeneous ideal a “ py12 ´ y0 y2 q. △
Example 5.28 (Rational normal curves). More generally, the d ` 1 monomials of degree d,
xd0 , xd´1
0 x1 , . . . , xd1 define a closed embedding
ν : P1A ÝÝÑ PdA .

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onto the closed subscheme V` paq Ă PnA defined by the ideal a generated by the 2 ˆ 2-minors
of the matrix
ˆ ˙
y0 y1 . . . yd´1
. (5.29)
y1 y2 . . . yd
When A “ k is a field, this closed subscheme is called the rational normal curve of degree d.
The case d “ 3 is perhaps especially interesting: it is the twisted cubic curve, which is the
projective version of the curve appearing in Example 1.33. It is the curve in P3k defined by
the 2 ˆ 2 minors of the matrix
ˆ ˙
y0 y1 y2
.
y1 y2 y3
△
Example 5.29 (The rational quartic curve). Consider the morphism ι : P1k Ñ P3k defined by
the four monomials
x40 , x30 x1 , x0 x31 , x41
Note that the monomial x20 x21 is missing. By looking in the four affine charts of P3k one can
check that ι is a closed embedding. Writing y0 , y1 , y3 , y4 for the coordinates on P3k , the image
X “ ιpP1k q is defined by ideal
` ˘
I “ y1 y3 ´ y0 y4 , y33 ´ y1 y42 , y0 y32 ´ y12 y4 , y13 ´ y02 y3
Geometrically, X arises as the projection of the rational normal curve in P4k from the closed
point p0 : 0 : 1 : 0 : 0q. △
Example 5.30 (Veronese varieties). The morphism v2 : P2k Ñ P5k defined by

px0 : x1 : x2 q ÞÑ px20 : x0 x1 : x0 x2 : x21 : x1 x2 : x22 q,

defines an isomorphism between P2k and the Veronese surface V` paq Ă P5k . Here a is the
ideal generated by the 2 ˆ 2-minors of the matrix
¨ ˛
y0 y1 y2
˝y1 y3 y4 ‚.
y2 y4 y5

Under this embedding, lines in P2k are mapped into conics on V .

V
P2
v2

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Weighted projective spaces

A rich source of examples is given by the weighted projective spaces, which are defined in
terms of polynomial rings with non-standard gradings:

Definition 5.31. For a field k and integers d0 , . . . , dn P N, we define the weighted

projective space over k as
Ppd0 , . . . , dn q “ Proj krx0 , . . . , xn s
where degpxi q “ di for each i.

While this definition resembles that of a traditional projective space (where all di are equal
to 1), the weighted projective spaces give a surpringly diverse and rich class of examples.
Example 5.32 (The weighted projective space Pp1, 1, dq). Consider the graded ring R “
krx, y, zs where degpxq “ degpyq “ 1 and degpzq “ d, and let X “ Pp1, 1, dq “ Proj R.
The scheme X is covered by the affine open subsets D` pxq, D` pyq, and D` pzq.
The first two open subsets are easily understood: we have pRx q0 “ kry{x, z{xd s and
pRy q0 “ krx{y, z{y d s, both isomorphic to the polynomial ring in two variables, so that
D` pxq and D` pyq are each isomorphic to A2k . The third open subset is more subtle: the
degree 0 part of Rz is the k -subalgebra generated by the monomials xd´i y i z ´1 for 0 ď i ď d,
so that
pRz q0 “ krxd´i y i z ´1 | 0 ď i ď ds.
This ring is not a polynomial ring when d ą 1, and in fact D` pzq is singular if d ě 2.
The inclusion S “ krx, ys ãÑ R defines a rational map f : X 99K P1k whose base locus is
the closed point p “ V` px, yq. On the open subset D` pxq, the map f is given by projection
onto the first factor, i.e., by the map py{x, z{xd q ÞÑ y{x, and similarly on D` pyq it is given
by px{y, z{y d q ÞÑ x{y . The gluing of D` pxq and D` pyq over D` pxyq is determined by
the transition function z{xd “ px{yqd z{y d , which shows that X ´ tpu is isomorphic to the
line bundle Ld from Section 6.3. Thus, X is obtained from this line bundle by contracting
the section to the point p. △
Example 5.33. The weighted projective plane Pp1, 2, 3q is the Proj of the ring R “
krx0 , x1 , x2 s where the variables x0 , x1 , x2 have degrees 1, 2, 3 respectively. The affine
opens are
„ ȷ
x1 x2
pRx0 q0 “ k 2 , 3
x x
„ 02 0 ȷ
x0 x0 x2 x21
pRx1 q0 “ k , ,
x1 x21 x31
„ 3 ȷ
x x0 x1 x31
pRx2 q0 “ k 0 , , 2 .
x2 x2 x2
This means that D` px0 q “ SpecpRx0 q0 is isomorphic to an affine plane A2k . The open sets
D` px1 q and D` px2 q are not. For instance,
pRx2 q0 » krU, V, W s{pU W ´ V 3 q.

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We can embed Pp1, 2, 3q into a standard projective space by looking at the Veronese subring
generated by degree 6 elements
Rp6q “ krx60 , x31 , x22 , x40 x1 , x30 x2 , x20 x21 , x0 x1 x2 s.
The graded surjection kry0 , . . . , y6 s Ñ Rp6q defined by y0 ÞÑ x60 , y1 ÞÑ x31 , . . . , y6 ÞÑ
x0 x1 x2 induces a closed embedding
ι : Pp1, 2, 3q ÝÝÑ P6k .
△
One of the advantages of weighted projective spaces is that they provide natural ambient
spaces for projective schemes. Given a graded ring R generated by homogeneous elements
x0 , . . . , xn of degrees d0 , . . . , dn respectively, there is a graded surjection
R0 rx0 , . . . , xn s ÝÝÑ R
which induces a closed embedding Proj R Ñ Ppd0 , . . . , dn q over R0 . Here is a concrete
example:
Example 5.34. Recall the hyperelliptic curves from Section 6.4. While the equation
y 2 “ f px0 , x1 q
where f is homogeneous of degree 2d, does not define a closed subscheme of P2k , it does
define a closed subscheme in the weighted projective space Pp1, 1, dq “ Proj krx0 , x1 , ys
with degpyq “ d. In fact, every hyperelliptic curve constructed earlier embed as closed
subschemes of Pp1, 1, dq defined by this equation.
The affine covering of Pp1, 1, dq consists of three open sets D` px0 q, D` px1 q, and D` pyq.
However, for the hyperelliptic curve X defined by y 2 “ f px0 , x1 q, only the first two charts
are needed since X is entirely contained in D` px0 q Y D` px1 q “ Pp1, 1, dq ´ V` px0 , x1 q.
△
? Ş
Exercise 5.3.1. Let a be a homogeneous ideal. Then a “ pĄa p where the intersection
?
is taken over all the homogeneous ideals that contain a. H INT: If x R a, then pick a
homogeneous prime ideal p so that xn R p for all n, and choose p maximal with respect to
this property. Show that p is a prime ideal.
5.3.2. Let a and I be homogeneous ideals in the graded ring R. Show the following:
Exercise ?
a) If I “ R` , then V` paq “ V` pa X Iq. Hence, when constructing the closed
sets V` paq, it suffices to work with
?ideals contained in the irrelevant ideal.
b) V` paq “ H if and only if R` Ă a. H INT: Use Exercise 5.4.11.
c) Show that the Zariski topology on ProjpRq Ă SpecpRq is the induced topology
from SpecpRq. H INT: Any ideal a has a corresponding ’homogenization’, the
ideal generated by all homogeneous components of the elements in a.
Exercise 5.3.3. Let R be a graded ring and let S Ă R be a subring such that for some
N P N, Sd “ Rd for all d ě N . Show that ProjpSq » ProjpRq.
Exercise 5.3.4. Let ϕ : R Ñ S be a map of graded rings.

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a) Assume that Rd Ñ Sd is surjective for all sufficiently large d. Show that

V pϕpR` qq “ H.
b) If Rd Ñ Sd is an isomorphism for all sufficiently large d, show that Proj S Ñ
Proj S is an isomorphism.
Exercise 5.3.5. Let k be an infinite field. Show that a Laurent polynomial f P krx0 , . . . , xn , x´1
i s
is invariant under the scaling operation xi ÞÑ txi if and only if it belongs to the subring
krx0 {xi , . . . , xn {xi s.
Exercise 5.3.6. Consider the projective twisted cubic curve Y Ă P3k as described in Example
5.28. Show that Y X D` py0 q can be identified with the affine twisted cubic, as defined in
Chapter 1. Describe the ‘points at infinity’ V py0 q X Y .

5.4 Exercises
Exercise 5.4.1. Let a Ă krx, y, zs be the ideal pxy, xz, yzq. Show that A “ R{a is graded
ring and describe each homogeneous component An .
Exercise 5.4.2. Consider the weighted projective space Ppp, qq “ Proj R, where R “
krx0 , x1 s with degpx0 q “ p and degpx1 q “ q . Show that
Ppp, qq » P1k .
H INT: Consider the Veronese subring Rpdq where d “ pq , and define a ring map ϕ : kru, vs Ñ
Rpdq by u ÞÑ xq0 , v ÞÑ xp1 .
Exercise 5.4.3. A polynomial ring krt0 , . . . , tn s can be given a non-standard grading by
declaring the degree of each ti to be any given natural number di . For instance, give R “
krt0 , t1 s a grading by letting deg t0 “ 2 and deg t1 “ 3.
a) Describe the homogeneous pieces Rn of degree n.
b) Let krus have standard grading and define a map ϕ : R Ñ krus by the assign-
ments t0 ÞÑ u3 and t1 ÞÑ u2 . Show that ϕ is a map of graded rings.
c) Describe the kernel and the cokernel of ϕ as graded modules.
Exercise 5.4.4. Show that an ideal a in a graded ring R is homogeneous if and only if it is
generated by homogeneous elements.
Exercise 5.4.5. Let R be a graded ring which is not necessarily positivelyřgraded. Assume
that a homogeneous element f of R is expressed as a combination ř f “ ai gi where the
gi ’s are homogeneous. Show that f may be expressed as f “ i bi gi , where each bi is
homogeneous of degree deg f ´ deg gi . H INT: Homogeneous components are unique.
Exercise 5.4.6.?Let R be a graded Ş and a, b and tai uiPI be homogeneous ideals. Show
ř ring
that the ideals a, ab, pa : bq, ai , i ai are homogeneous.
Exercise 5.4.7. Let R be a graded ring and let f P Rd be homogeneous of degree d. Show
that there is an isomorphism of rings
pRf q0 “ Rpdq {pf ´ 1q. (5.30)

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Exercise 5.4.8. Let R be a graded ring and p a homogeneous prime ideal. Show that pRp q0
is a local ring with maximal ideal equal to
m “ t f g ´1 | f P p, g P Sppq and deg f “ deg g u.
where Sppq is the multiplicative subset of homogeneous elements in R ´ p.
Exercise 5.4.9. Let R be a graded ring and p a homogeneous ideal in R. Show that p is
prime if and only if xy P p implies x P p or y P p for all homogeneous elements x and y .
Exercise 5.4.10. Let R and S be graded rings and ϕ : R Ñ S a map of graded rings. Show
that the inverse image ϕ´1 p of an ideal p Ă S is homogeneous whenever p is.
? Ş
Exercise 5.4.11. Let a be a homogeneous ideal. Then a “ pĄa p where the intersection
?
is taken over all the homogeneous ideals that contain a. H INT: If x R a, then pick a
homogeneous prime ideal p so that xn R p for all n, and choose p maximal with respect to
this property. Show that p is a prime ideal.
Exercise 5.4.12. Let R be a graded ring and let f and tfi uiPI be homogenous elements from
R all of positive degree. Show that the distinguished open sets D` pfi q cover D` pf q if and
only if a power of f lies in the ideal generated by the fi ’s.
Exercise 5.4.13. Let R be a graded ring and let he π : Proj R Ñ Spec R0 be the structure
map. Show that for each f P R0 , the inverse image π ´1 Dpf q is isomorphic to Proj Rf .
Exercise 5.4.14. Let R be a 1-dimensional graded ring, with R0 “ k a field, and assume that
R is finitely generated as a k -algebra. Show that Proj R is a finite set. H INT: the maximal
ideal R` contains all homogeneous prime ideals.
Exercise 5.4.15. If R is a graded integral domain, show that the function field of X “ Proj R
is given by
" *
g
KpXq “ | g P R, h P R, deg g “ deg h Ă kpRq (5.31)
h
Exercise 5.4.16. Show that Proj R is empty if and only if every element in R` is nilpotent.
Exercise 5.4.17. Let R a graded ring. Show R is Noetherian if and only if R0 is Noetherian
and R` is finitely generated.
Exercise 5.4.18. Find a non-Noetherian graded ring R such that
a) Proj R is Noetherian.
b) R is not of finite type over a field k , but Proj R is.
c) R is not an integral domain, but Proj R is an integral scheme.
Exercise 5.4.19. Consider the Q-scheme X “ ProjpQrx, y, zs{p2x2 ` y 2 ´ 5z 2 qq. Show
that XpQq “ H.
Exercise 5.4.20. Consider the projective scheme
X “ Proj Zrx, y, zs{p2x2 ` 2y 2 ´ 3z 2 q.
Describe the scheme-theoretic fibers of the canonical map π : X Ñ Spec Z. Which ones are
irreducible, reduced, integral? What are their dimensions?

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Exercise 5.4.21 (Cremona transformation). Let A be a ring and consider the map of graded
rings ϕ : Zru0 , u1 , u2 s Ñ Zrx0 , x1 , x2 s defined by the three assignments ui Ñ xj xk where
the indices satisfy ti, j, ku “ t1, 2, 3u.
Determine the base locus Bspϕq and describe the k -points of V pBspϕqq when k is a field.
Exercise 5.4.22. Show that the inverse of the Veronese map νd is not induced by a map of
graded rings R Ñ Rpdq .
Exercise 5.4.23. Show that Proj krx0 , x1 s{px20 , x21 q “ H.
Exercise 5.4.24. Show that Pp1, . . . , 1, dq is isomorphic to the cone over the Veronese
variety Vn,d .
Exercise 5.4.25. Let tti u be a finite set of generators for the graded ring R and let di “
deg ti .
a) Let D be the least common multiple of the di and set Di “ D{di . Show that
the Veronese ring RpDq is generated by elements of degree D.
b) Show that Proj R embeds as a closed subscheme of the weighted projective
space PR0 pd0 , . . . , dn q over R0 .
Exercise 5.4.26. Let x and y be two points in Pnk . Prove there is an open affine U Ă Pnk
containing both x and y .
Exercise 5.4.27. Show that equation (19.4) holds.
Exercise 5.4.28 (The weighted projective space Pp1, 1, pq). Let R be as in the Example 5.32
above, and let A “ krx, y, ws with the usual grading. Furthermore, let α : R Ñ A be the
map of graded rings that sends z to wp , while leaving x and y unchanged.
a) Show that α is a map of graded rings and induces a morphism π : P2k Ñ Proj R.
b) Describe the fibers of π over closed points in case k is algebraically closed.
Exercise 5.4.29. Let R “ krx, y, zs be the polynomial ring with grading given by deg x “ 1,
deg y “ 2 and deg z “ 3, and consider Proj R (which also is denoted Pp1, 2, 3q). The aim
of the exercise is to describe the three covering distinguished subschemes D` pxq, D` pyq
and D` pzq.
a) Show that pRx q0 “ kryx´2 , zx´3 s and that D` pxq » A2k .
b) Show that pRy q0 » krx2 y ´1 , z 2 y ´6 , xzy ´2 s. Show that the map of graded
rings kru, v, ws Ñ pRy q0 given by the assignments x ÞÑ yx´2 , v ÞÑ z 2 y ´6
and w ÞÑ xzy ´2 induces an isomorphism kru, v, ws{pw2 ´ uvq » pRy q0 .
Hence D` pyq is a hypersurface in A3k . Show it is not isomorphic to A2k .
H INT: Check the local ring at the origin.
c) Show that Rz “ krx3 z ´1 , y 3 z ´2 , xyz ´1 s and that the map kru, v, ws Ñ
pRz q0 defined by the assignments x ÞÑ x3 z ´1 , v ÞÑ y 3 z ´2 and w ÞÑ xyz ´1
induces an isomorphism kru, v, ws{pw3 ´ uvq » pRz q0 . Show that it is not
isomorphic to A2k .
d) Show that the map R Ñ krU, V, W s sending x ÞÑ U , y Ñ V 2 and z ÞÑ W 3
induces a map P2k Ñ Proj R, and describe the fibers over closed points.
Exercise 5.4.30. Let k be a field. Show that:

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a) Any closed subscheme of Ank of dimension n ´ 1 is a hypersurface.

b) Any closed subscheme of Pnk of dimension n ´ 1 is a hypersurface.
Exercise 5.4.31. Let R “ krx0 , x1 s where k is a field, and consider the morphism
π : Spec R ´ V px0 , x1 q Ñ P1k “ Proj R
defined in (5.5).
a) Show that π maps a k -point pa, bq to pa : bq
b) Show that π maps each height 1 prime p “ pf px0 , x1 qq to the generic point in
P1k .
Exercise 5.4.32. Let R “ krx, y, zs{pxz, yz, z 2 q. Consider the map ϕ : R Ñ R given by
ϕpxq “ x, ϕpyq “ 0, ϕpzq “ 0.
a) Show that ϕ induces a morphism f : Proj R Ñ Proj R.
b) Show that f is the identity morphism.
In particular, different graded ring maps can induce the same map on Proj’s.
Exercise 5.4.33. Let R “ Crx0 , x1 , x2 s and let the symmetric group S3 act on R by
permuting the variables.
a) Show that the invariant polynomials of R are generated by s1 “ x0 ` x1 ` x2 ,
s2 “ x0 x1 ` x0 x2 ` x1 x2 and s3 “ x0 x1 x2 .
b) Show that s1 , s2 , s3 define a morphism π : P2C Ñ Pp1, 2, 3q.
c) Describe the fibers of π .
d) Show that Pp1, 2, 3q is homeomorphic to the quotient space P2 {S3 .
Exercise 5.4.34
À (Nakayama’s lemma for graded modules). Let R be a graded ring and
let M “ dPZ M d be a finitely generated graded R-module. Let x1 , . . . , xn be a set of
homogeneous elements. Show that x1 , . . . , xn generate M if and only if their classes generate
M {R` M as a R0 -module.
Exercise 5.4.35. Let R be a graded ring and let p1 , . . . , pr P Proj R be points. Show that
there exists a homogeneous f P R` such that D` pf q contains all the pi . H INT: Prime
avoidance.
Exercise 5.4.36. Let U “ P2k ´ tp0 : 0 : 1qu.
a) Show that OP2k pU q » k .
b) Is U “ Proj R for a k -algebra R?
Exercise 5.4.37. Let R be a Noetherian, normal graded integral domain.
a) Show that Proj R is a normal scheme. H INT: Show that each pRf q0 is normal.
b) Find a graded ring R so that Proj R is normal, but R is not.
Exercise 5.4.38. Let R and R1 be two graded rings with degree 0 part A “ R0 “ R01 . Show
that the tensor product R bA R1 is naturally a graded ring with a decomposition
à à
R bA R 1 “ Ri bA Rj1 ,
ně0 i`j“n

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Exercise 5.4.39 (Proj and base change). Let B be a R0 -algebra. Show that there is a canonical
isomorphism
Proj R ˆR0 Spec B » ProjpR bR0 Bq.
H INT: There is an isomorphism Rf bR0 B » pR bR0 Bqf b 1 , defined by sending x{f n b b
to px b bq{pf b 1qn .
Exercise 5.4.40 (Proj and fiber products). Let R and R1 be graded rings with R0 “ R01 “ A.
Consider the following subring S of R bA R1 :
à
S“ Ri bA Ri1
iě0

a) Let f and g be homogeneous elements of positive degree in R and R1 respec-

tively. Show that there is a canonical isomorphism of A-algebras

θf,g : pRf q0 bA pRg q0 pSf b g q0 ,

defined by xf ´a b yg ´b ÞÑ pxf b b yg a qpf b gq´pa`bq .

b) Show that there is a canonical isomorphism between

τf,g : D` pf b gq » D` pf q ˆA D` pgq.

where D` pf b gq Ă Proj S .
c) Show that there is a natural isomorphism
»
Proj S Proj R ˆA Proj R1 .
d) In the case R and R1 are polynomial rings with the standard grading, show
that the isomorphism in (c)) identifies with the Segre embedding as defined in
Section XXX.

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More examples

6.1 Toric varieties

In this section, we explore a large class of examples, known as toric varieties. Like projec-
tive spaces, toric varieties are defined in terms of algebras generated by monomials. The
correspondence between monomials an lattice points in Rn sets up an interesting interplay
between algebra and combinatorics, which allows us to do computations with toric varieties
using combinatorial methods (and in some cases, solve combinatorial problems using tools
from algebraic geometry). We will confine ourself to discussing affine and projective toric
varieties. We refer to the standard texts ? and ? for more information about this rich subject.

Affine toric varieties

Let k be a field and consider the ring of Laurent polynomials krx˘1 ˘1
1 , . . . , xn s. We will
use multiindex notation for monomials, that is, if m “ pm1 , . . . , mn q P Zn , then xm “
xm mn
1 ¨ ¨ ¨ xn .
1

Given a subset S Ă Rn , we define the cone of S to be the non-negative span of S , that is,
" *
ConepSq “ λ1 v1 ` ¨ ¨ ¨ ` λs vs | vi P S, and λi ě 0, @i “ 1, . . . , s .

For a cone C , we define the following subalgebra generated by the monomials corresponding
to the lattice points in C :
à
krCs “ k xm . (6.1)
mPCXZn

The toric variety associated to C is the affine scheme X “ Spec krCs.

We will usually assume that C is rational polyhedral, meaning that it is generated by a
finite set of integer points S Ă Zn . This implies that there are finitely many lattice points
m1 , . . . , mr P C so that every point in C X Zn can be expressed as a non-negative Z-linear
combination of the m1 , . . . , mr . On the level of rings, this means that the following map of
k -algebras
ϕ : kry1 , . . . , yr s ÝÝÑ krCs; yi ÞÑ xmi . (6.2)
is surjective, and hence induces a closed embedding
ι : X ÝÝÑ Ark .
To find the ideal defining X inside Ark , we need to find the kernel of ϕ. Write A for the
n ˆ r matrix with the vectors m1 , . . . , mr as column vectors.
120

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Proposition 6.1. The kernel of ϕ is given by the ideal

IA “ pyu ´ yv | u ´ v P Ker Aq (6.3)

Proof It is clear that IA Ă Ker ϕ. For the reverse inclusion, we use a monomoial order
argument. More precisely, we will consider the lexicographic ordering ă on monomials in
kry1 , . . . , yn s, so that for instance
y12 ą y1 y2 ą y1 y3 ą y22 ą y2 y3 ą y32 .
If g P Ker ϕ, we can write it as
ÿ
g “ cu yu ` cv yv .
vău
u
where cu y ‰ 0 is the leading term with respect to ă. Applying ϕ, we get
ÿ
0 “ ϕpgq “ cu xAu ` cv xAv .
vău

This is an identity of polynomials in krx˘1 ˘1

1 , . . . , xm s, so there must be cancellations between
Au
the monomials. In particular, the term cu x must cancel with some other terms in the sum.
This means that there exists some v with v ă u such that Au “ Av , or in other words,
u ´ v P Ker A. Replacing g with g ´ cu pxu ´ xv q, we obtain a polynomial which (2.2,4.4)has a
leading term which is strictly smaller than that of g with respect to ă. Note that xu ´ xv
belongs to the ideal IA . Continuing in this manner, we eventually obtain the zero polynomial,
which means that g is an element of IA .
Hence IA is a prime ideal defined by binomials. To find a finite generating set, a few more
computations are usually needed.
(0,2.2)

2 2

1 1

(2.2,0)

1 2 1 2 (2.2,0)
(0,0) (0,0)

Figure 6.1 Two cones in R2 . On the left, the cone generated by p1, 0q and p0, 1q. On
the right, the cone generated by p1, 0q and p1, 2q.

Example 6.2. If C is the first quadrant, generated by p1, 0q, p0, 1q, then krCs “ krx, ys.
The corresponding toric variety is A2k . △
Example 6.3. If C is the cone generated by p1, 0q, p1, 2q, then krCs “ krx, xy, xy 2 s.
Note that krx, xy, xy 2 s » kru, v, ws{pv 2 ´ uwq, so the corresponding toric variety is the
quadratic surface v 2 “ uw in A3k . △

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Example 6.4. Note that C “ Rn itself is a cone. The corresponding toric variety is the
algebraic torus
T “ Spec krx˘1 ˘1
1 , . . . , xn s

The reason behind the name is the following. When k “ C, the set of C-points T pCq is
identified with pCˆ qn , which in the Euclidean topology is homotopy equivalent to a product
pS 1 qn . Thus, for n “ 2, this is the classical torus S 1 ˆ S 1 .
If X is any toric variety, then T embeds as a distinguished open subscheme in X . Indeed,
if v P C is any vector in the interior of C , then the localization krCsxv is isomorphic to
krx˘1 ˘1
1 , . . . , xn s. Thus toric varieties can be viewed as ‘compactifications’ of the torus T . △

Projective toric varieties

A lattice polytope in Rn is the convex hull of a finite set of points in Zn . In other words, there
are m1 , . . . , mr P Zn such that
" ÿr *
P “ λ1 m1 ` ¨ ¨ ¨ ` λr mr | λi “ 1 and λ1 , . . . , λr ě 0 .
i“1

Starting with a lattice polytope P , we can define a collection of cones as follows. For each
v P P X Zn , let P ´ v “ t x P Rn | x ` v P P u denote the translate of P by ´v, so that
the vertex v ends up at the origin. Then the cone
Cv “ ConepP ´ vq
is a cone spanned by the elements of P as viewed from the vertex v. This is pictured in the
figure below.

Cw
0
v P Cv v´w

w w´v

In particular, to the polytope P , we can attach k -algebras krCv s, one for each v P P X Zn .
As P has only finitely many vertices, the cones Cv are each spanned by finitely many
points with integer coordinates. This implies that each krCv s is a finitely generated k -algebra.

Lemma 6.5. If v and w are two vertices of P , then

k rCv sxw´v “ k rCw sxv´w . (6.4)

Proof Let v1 , . . . , vr be the vertices of P , ordered so that v “ v1 and w “ v2 . Then

the cone Cv is the non-negative span of the vectors v2 ´ v1 , . . . , vr ´ v1 , and Cw is the

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non-negative span of the vectors v1 ´ v2 , v3 ´ v2 , . . . , vr ´ v2 . Clearly, these spans are

the same up to translation by v2 ´ v1 , so we get
Cv ` Rpw ´ vq “ Cw ` Rpw ´ vq. (6.5)
The equality (6.4) follows, because both sides are generated by the same monomials.
Consider the affine schemes
Xv “ SpecpkrCv sq
where v P P is a vertex. If w P P is another vertex, the affine scheme
Xv,w “ SpecpkrCv sxw´v q
is identified with a distinguished open set in Xv (defined by the monomial xw´v ). The gluing
conditions for gluing the Xv are then satisfied. The resulting scheme, XP , is called the toric
variety associate to the polytope P .
Example 6.6 (The projective line). Let P “ r0, 1s be the unit interval in R. Then P has
two vertices 0 and 1. For the vertex 0, which is already at the origin, the cone S0 is the
non-negative span of P , that is, S0 “ r0, 8q. Likewise S1 “ p´8, 0s. This gives the
following two k -algebras:: A0 “ krxs and A1 “ krx´1 s. The corresponding toric variety is
P1k . △
Example 6.7 (Projective space). Let P Ă R2 be the triangle with vertices at 0, e1 and
e2 . There are three cones: S1 “ tx ě 0, y ě 0u, S2 “ tx ď 0, x ` y ď 0u and
S2 “ ty ď 0, x ` y ď 0u. These are depicted in Figure 6.7.

S1
S2

Figure 6.2 Constructing the projective plane

The cones S1 , S2 , S3 give the following three k -algebras

A1 “ krx, ys, A2 “ krx´1 , x´1 ys, A3 “ kry ´1 , xy ´1 s.
In light of Example ??, we recoginze the corresponding toric variety as the projective plane
P2k .
More generally, if P Ă Rn is the standard n-simplex, that is, the polytope with vertices at
the origin and the n standard basis vectors 0, e1 , . . . , en , then the corresponding toric variety
is the projective space Pnk .
△
Example 6.8. Consider the polytope P given by the unit square, that is, the convex hull of
the four points p0, 0q, p1, 0q, p1, 1q, p0, 1q.

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Then the four cones S1 , . . . , S4 are the four quadrants in R2 . Hence

A1 “ krx, ys A2 “ krx´1 , ys
A3 “ krx´1 , y ´1 s A4 “ krx, y ´1 s

The corresponding toric variety is P1k ˆk P1k . We will define products of general schemes in
Chapter 8. Note that the polytope P is itself a product: P “ r0, 1s ˆ r0, 1s, and r0, 1s is the
polytope defining P1k . This is a general fact about products of toric varieties (see Exercise
???). △

Example 6.9 (Hirzebruch surfaces). Let r be an integer and consider the polytope P defined
by the four points p0, 0q, p0, 1q, p1, 1q and pr ` 1, 1q. This is pictured below for r “ 1. The
case r “ 0 was considered in the previous example.

This gives the following k -algebras:

A1 “ krx, ys A2 “ krx´1 , x´r ys

A3 “ krx´1 , xr y ´1 s A4 “ krx, y ´1 s

The corresponding toric variety Fr is called a Hirzebruch surface.

There are four vertices, so four cones. Two of these are shown below:

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We compute that

A1 “ krx, ys A2 “ krx´1 , x´1 ys

A3 “ krx´1 , xy ´1 s A4 “ krx, y ´1 s
△
Each of the rings krCv s and their localizations are subrings of krx˘1 ˘1
1 , . . . , xn s,
and the
morphisms T Ñ Spec krCv s glue to an open embedding T Ñ X .
The toric variety XP admits a closed embedding into projective space. More precisely, let
m1 , . . . , mr be the lattice points of P , and assume that P satisfies the following property: for
each i, the semigroup Cmi X Zn is generated by the elements m1 ´mi , m2 ´mi , . . . , mr ´
mi . This condition is not too restrictive, as it is satisfied for sufficently large scalings of P ,
and XP remains unchanged under scaling P . Moreover, it is automatically satsfied if P has
dimension at most 2.
In this setting, krCmi s is generated by the monomials xm1 ´mi , . . . , xmr ´mi , and we can
define a surjective map of k -algebras
„ ȷ
y1 yr yj
ϕi : k ,..., ÝÝÑ krCmi s; ÞÑ xmj ´mi (6.6)
yi yi yi
and hence a closed embedding Spec krCmi s Ñ Ui “ Spec kr yy1i , . . . , yyri s. It is straightfor-
ward to check that these embeddings glue to an embedding
f : X ÝÝÑ Pr´1
k .

Alternatively, one can consider the morphism

f0 : Spec krx˘1 ˘1
1 , . . . , xn s ÝÝÑ Pr´1
defined by the monomials xm1 , . . . , xmr . Then X is isomorphic to the Zariski closure of
image of f0 in Pr´1
k .
Another nice property of the toric varieties XP is that they provide natural ways to
‘compactify’ affine hypersurfaces. If f P krx1 , . . . , xn s is a nonzero polynomial, we can
consider the associated Newton polytope Pf Ă Rn by taking the convex hull of the points in
Zn corresponding to monomials appearing in f . This is best explained through an example
Example 6.10. Consider the polynomial
f “ 1 ` x2 y ` xy 2
Then the Newton polytope P “ Pf is the convex hull of the vertices p0, 0q, p2, 1q, p1, 2q.
The toric variety associated to P is covered by the three affine charts
R1 “ krx2 y, xy, xy 2 s “ kru1 , u2 , u3 s{pu22 ´ u1 u3 q
R2 “ krx´2 y ´1 , x´1 , x´1 ys “ krv1 , v2 , v3 s{pv23 ´ v1 v3 q
R3 “ krx´1 y ´2 , y ´1 , xy ´1 s “ krw1 , w2 , w3 s{pw23 ´ w1 w3 q
The hypersurface given by V pf q Ă T “ Spec krx˘1 , y ˘1 s is an open set in a hypersurface
Y Ă XP . Over the open set U1 “ Spec R1 this is simply given by the linear equation

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1 ` u1 ` u3 “ 0. Over U2 “ Spec R2 it is given by the ‘dehomogenization’ x´2 y ´1 ` 1 `

x´1 y “ v1 ` 1 ` v3 and so on. △
While any affine hypersurface can be embedded into some projective space by homogeniz-
ing, the toric varieties XP tend to be ‘more economical’, as they take into account only the
monomials appearing in f .
(1,2)
2

(2,1)
1

(0,0) 1 2

Figure 6.3 A cubic surface. Figure 6.4 The polytope P

Exercise 6.1.1. Let P Ă R3 be a the convex hull of the lattice points 0, e1 , e2 , e1 ` e2 ` 3e3 .
Show that P has only 4 lattice points and that P does not satisfy the condition mentioned on
page 125.
Exercise 6.1.2. In the notation of Proposition 6.1, let T “ tu1 , . . . , ur u be a Z-basis for
Ker A and let
1
IA “ pyu` ´ yu´ | u P Sq
where we decompose u “ u` ´ u´ P Zn in terms of its non-negative and non-positive
1
entries. Show that IA is the saturation of IA with respect to the maximal ideal at the origin,
1 8
i.e., IA “ IA : py1 , . . . , ym q .

6.2 The blow-up of the affine plane

In this section we will construct the blow-up of A2k at the origin, by gluing together two affine
planes.
Consider the morphism π : A2k ´ tp0, 0qu Ñ P1k from (5.5). On k -points, the morphism is
given by pa, bq ÞÑ pa : bq, where pa : bq are the homogeneous coordinates on P1k . It will be
helpful to think geometrically, so that P1 pkq parameterizes lines through the origin in A2 pkq.
Then π sends a point pa, bq ‰ p0, 0q to the unique line passing through p0, 0q and pa, bq In
particular, all the points on the line bx ´ ay “ 0 get mapped to the point pa : bq P P1k .
From this perspective, it is not surprising that π cannot be extended to a morphism
A2k Ñ P1k . In fact, there is not even an extension as a continuous map, because π takes
different values along lines bx ´ ay “ 0 and p0, 0q is contained in all of them.
While π cannot be extended to A2k , it is possible to embed A2k ´ tp0, 0qu as an open set in
a scheme X that admits a morphism to P1k extending π . In fact, there is a scheme X which is
in a certain sense ‘minimal’ among all such schemes, called the blow-up of A2k at the origin.

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More formally, the blow-up X is a scheme admitting two morphisms, p : X Ñ A2k and
q : X Ñ P1k so that
(i) The scheme theoretic fiber E “ p´1 p0, 0q over the origin p0, 0q is isomorphic
to P1k .
(ii) p defines an isomorphism betwen X ´ E and A2k ´ tp0, 0qu.
(iii) The morphism q ˝ p´1 , defined over X ´ E , coincides with π .
This is pictured in the diagram below (the dashed arrow indicates that π is only defined on an
open set):

X
p q

π
A2k P1k

Before we construct the scheme X , let us describe what the set of k -points of X looks like.
The k -points of X can be thought of as pairs pp, rLsq where p P A2 pkq is a k -point and
L Ă A2 pkq is a line containing p. In terms of coordinates, we can define this as
" *
ˇ
Xpkq “ ppx, yq, pu0 : u1 qq u1 x ´ u0 y “ 0 Ă A2 pkq ˆ P1 pkq.
ˇ (6.7)

In other words, Xpkq is the graph of the mapping px, yq ÞÑ px : yq.

We can verify that the properties (i), (ii), (iii) hold as follows. Firstly, note that p´1 p0, 0q “
p0, 0q ˆ P1 pkq is a copy of the projective line P1 pkq, as the equation u1 x ´ u0 y “ 0 puts no
conditions on u0 and u1 . On the other hand, if px, yq is a k -point in the open set A2k ´tp0, 0qu,
then px : yq is a well-defined point in P1 pkq, and p´1 px, yq “ ppx, yq, px : yqq. Therefore,
p defines a bijection over the set A2 pkq ´ tp0, 0qu. Now composing the inverse of p´1 with
the second projection sends px, yq to px : yq, so (iii) follows as well.

Figure 6.5 The blow-up of the plane at a point

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Note that (6.7) is just as set of k -points. We haven’t yet defined a scheme structure on the
blow-up. However, this set provides a clue for constructing X as a scheme.
We consider the usual construction of P1k via gluing V0 “ Spec krus to V1 “ Spec krvs
using the identification v “ u´1 over the overlaps. In light of (6.7), we then define
X0 “ Spec B0 ; B0 “ krx, y, us{py ´ uxq. (6.8)
and over the open set V1 , we define
X1 “ Spec B1 ; B1 “ krx, y, vs{px ´ vyq.
Note that B0 » krx, us and B1 » kry, vs, so X0 and X1 are both isomorphic to affine
planes.
Furthermore, the map u ÞÑ v ´1 defines an isomorphism of rings
pB0 qu “ krx, y, u, u´1 s{py ´ uxq » krx, y, v, v ´1 s{pvy ´ xq “ pB1 qv .
This means that we may identify the distinguished open Dpuq Ă X0 with the distinguished
open Dpvq Ă X1 , and hence glue X0 and X1 together to form a new scheme X .
We next define the morphisms p and q . Note that there is a morphism X0 Ñ V0 induced by
the ring map krus Ñ B0 . Likewise, there is a morphism X1 Ñ V1 induced by krvs Ñ B1 .
These morphisms are compatible with the gluing isomorphism u ÞÑ v ´1 , so they glue to a
morphism X Ñ P1k . Simularly, the two ring maps krx, ys Ñ B0 and krx, ys Ñ B1 induce
a morphism p : X Ñ A2k .
The k -points of the scheme X are precisely given by the set (6.7). To see this, consider
a morphism ι : Spec k Ñ X . Such a morphism must have image contained in either X0
and X1 . Assuming, say, ι maps to pa, b, cq P X1 pkq, then b “ ac, which corresponds to
the point ppa, bq, p1 : cqq in the set (6.7). Conversely, any pair pa, bq, pc0 : c1 q satisfying
c1 a ´ c0 b “ 0, we can assume without loss of generality that c0 “ 1. Then b “ ac1 , and the
pair defines the k -point pa, b, c1 q in X0 .
Let us consider the fiber of the morphism q : X Ñ P1 . On the level of k -points, if
pu0 : u1 q P P1 pkq is fixed, the k -points of the fiber q ´1 pu0 : u1 q consists of the points
px, yq, pu0 : u1 q such that u1 x ´ u0 y “ 0, that is, the points on the line u1 x ´ u0 y “ 0 in
A2 pkq. In fact, the scheme-theoretic fibers are all isomorphic to affine lines. To see this, take
a point p P P1k , which we may assume lies in the open set V0 “ Spec krus and corresponds
to a prime ideal p Ă krus. Then the scheme-theoretic fiber is given by
Spec krx, y, us{ ppy ´ uxq ` pq “ Spec krx, us{p » A1kppq
The fibers of the morphism p : X Ñ A2k are also interesting. First off all, the scheme theoretic
fiber E over the origin p0, 0q, which corresponds to m “ px, yq Ă krx, ys, is glued together
by the two affine schemes
E0 “ Spec krx, y, us{py ´ ux, x, yq “ Spec krus
and
E1 “ Spec krx, y, vs{pvx ´ y, x, yq “ Spec krvs.
Hence, the scheme theoretic fiber E is isomorphic to P1k .

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We claim that p restricts to an isomophism of schemes

X ´ E ÝÝÑ A2k ´ tp0, 0qu. (6.9)

Indeed, note that E X X0 Ă X0 is the closed subscheme V pxq inside X0 . Likewise, E X X1

is given by V pyq in X1 . This means that X ´ E is covered by the two affine subsets
Dpxq Ă X0 and Dpyq Ă X1 . Note that p maps Dpxq Ă X0 isomorphically to Dpxq Ă A2k .
This follows because it is induced by ring isomorphism

krx, x´1 , ys Ñ pB0 qx “ krx, x´1 , y, us{py ´ uxq

Likewise p restricts to an isomorphism over Dpyq, proving the claim.
While the discussion on k -points required working over a field, the scheme X constructed
via gluing two copies of the plane works over any ring. So for instance, there is a blow-up of
A2Z at the origin px, yq.
One can also define the blow-up of Ank at the origin, or in fact, along any linear subspace.
While the details can be worked out, they are more naturally understood in the context of the
Proj-construction, as we will see in Example ??.

6.3 Line bundles on P1

The sheaf OP1k pmq on the projective line P1k , which we constructed in Example 5.1, has a
geometric alter ego, the so-called line bundle Lm . This is a scheme with a morphism

π : Lm P1k ,

Each fiber of π is an affine line A1k (hence the name ‘line bundle’). In this section we shall
construct these schemes explicitly and study some of them in detail.
For simplicity, we will work over a field k use the standard covering of P1k by U0 “
Spec krus and U1 “ Spec kru´1 s glued along their intersection, U0 XU1 “ Spec kru, u´1 s.
Recall that the sheaves OP1k pmq are obtained by gluing OU0 and OU1 together by means
of the multiplication by um map on OU0 XU1 . The new schemes Lm will be constructed
essentially by the same gluing process, but schemes and not sheaves, will be glued together.
Two copies of A2k , V0 “ Spec kru, ss and V1 “ Spec kru´1 , ts, will be glued together using
the isomorphism

Dpuq “ Spec kru, u´1 , ts » Spec kru, u´1 , ss “ Dpu´1 q,

which is induced by the isomorphism of k -algebras ρ : kru, u´1 , ss Ñ kru, u´1 , ts that
sends s to um t and u to u.
The situation is described with the following commutative diagram of ring maps:
ρ
kru, ss kru, u´1 , ss » kru, u´1 , ts kru´1 , ts

krus kru, u´1 s kru´1 s,

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where the maps other than ρ are the inclusions. Applying Spec, we get the following diagram
of affine schemes:

A2 “ V 0 Dpuq » Dpu´1 q V1 “ A2k

U0 U0 X U1 U1 .

The gluing conditions are trivially fulfilled (only a single morphism is involved), and hence
we obtain a scheme Lm . It admits a morphism π : Lm Ñ P1 since the lower row gives the
gluing data for P1k . Note that if x P P1 is a closed point, say x P U0 , then the fiber π ´1 pxq
is isomorphic to the affine line A1kpxq . As noted above, this is the reason for the term ‘line
bundle’: intuitively Lm is a family of affine lines parameterized by the base space P1k .

C
Ln

There is a copy of P1k embedded in Lm which is called the zero section of Lm ; that is, there
is a closed embedding ι : P1k Ñ Lm whose image is a closed subscheme C Ă Lm that meets
each fiber π ´1 pxq “ A1kpxq in the origin. Intuitively, this subscheme is defined by one of
the equations s “ 0 or t “ 0 in each fiber. More precisely, C is given by C X V0 “ V psq
and C X V1 “ V ptq. In the ring ΓpV0 X V1 , OLm q, the relation s “ um t holds, and as u is
invertible in ΓpV0 X V1 , OLm q, the principal ideals psq and ptq are equal. The two closed
subschemes V psq X V0 X V1 and V ptq X V0 X V1 coincide, and V psq and V ptq can be
patched together to a subscheme C .
We claim that C is a section of the morphism π ; that is, that π ˝ ι “ idP1k . As V psq “
Spec kru, ss{psq “ Spec krus as a subscheme of V0 , and V ptq “ Spec kru´1 , ts{ptq “
Spec kru´1 s inside V1 , we see that C » P1k . Consider the composition of the maps

krus kru, ss kru, ss{psq “ krus,

where the first map is the canonical inclusion and corresponds geometrically to π|V0 , and the
second is the canonical quotient map and corresponds to the inclusion ι0 : V psq “ C XV0 Ñ
V0 . Clearly, it holds that π ˝ ι0 “ idU0 . In a similar manner, it follows that π|V1 ˝ ι1 “ idU1 ,
hence π ˝ ι “ idP1k and C is a section.

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A few particular cases

The schemes Lm give an interesting class of examples of schemes, and we will come back to
them several times in the book. For now let us study some of them in more detail.
Example 6.11 (The line-bundle L0 ). The scheme L0 is glued together of two copies of A2k
with the help of the inclusions

kru, ts kru, u´1 , ts kru´1 , ts.

In addition to π , the bundle L0 admits a morphism L0 Ñ A1k obtained by gluing together the
two maps Spec kru, ts Ñ Spec krts and Spec kru´1 , ts Ñ Spec krts. The scheme L0 is
identified with the ‘fiber product’ P1 ˆk A1k (fiber products will be study in detail in Chapter
8), and is the scheme associated with the product variety P1 pkq ˆ A1 pkq. △
Example 6.12 (The line-bundle L1 ). The scheme L1 is isomorphic to the complement of
a closed point P in the projective plane, i.e. Y “ P2k ´ tP u. Indeed, choose coordinates
x0 , x1 and x2 in the projective plane and consider the two distinguished open subschemes
V0 “ Spec krx1 {x0 , x2 {x0 s and V1 “ Spec krx0 {x1 , x2 {x1 s. Their union in P2k equals
the complement of the closed point P “ p0 : 0 : 1q. Renaming the variables u “ x0 {x1 ,
s “ x2 {x1 and t “ x2 {x0 , we find that V0 “ Spec kru, ss and V1 “ kru´1 , ts, and the
identity x2 {x1 “ x0 {x1 ¨ x2 {x0 turns into the equality s “ ut, which is precisely the gluing
data for L1 .
Geometrically the morphism P2k ´ tP u Ñ P1k is given by ‘projection from the point P ’.
The fibers are the lines in P2k through P (with the point P removed), and the zero section
equals V px2 q (the line ‘at infinity’). △
Example 6.13 (The line-bundle L´1 ). We have in fact seen the scheme L´1 before: it is
isomorphic to the blow-up of A2k at the origin. Recall that the blow-up X comes equipped
with a map q : X Ñ P1k , which is described in detail at the end of Section 6.2. One checks
without much difficulties that the gluing maps used for forming q are the same as for making
L´1 . The zero-section C corresponds to the exceptional divisor E in the blow-up. See also
Exercise 6.5.17 below. △
Example 6.14 (The line-bundle L´2 ). The scheme L´2 is quite interesting. It is the so-called
desingularization of a quadratic cone. The quadratic cone is the subscheme Q “ V py 2 ´ xzq
of A3k , which is equal to Spec R with R “ krx, y, zs{py 2 ´ xzq. We claim that there is a
surjective morphism σ : L´2 Ñ Spec R, which is an isomorphism outside the curve C . (The

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morphism σ is helpful for understanding the quadratic cone. In the terminology of Chapter
11, Q has a ‘singularity’ at the origin, whereas L´2 is ‘nonsingular’.)
We shall construct σ by giving the restrictions σi to each of the two opens V0 and V1 that
make up L´2 . Recall that V0 “ Spec kru, ss and V1 “ Spec kru´1 , ts with gluing map
Spec kru, u´1 , ss » Spec kru, u´1 , ts given by the assignment s ÞÑ u´2 t. The maps σi
are Spec’s of the ring maps ϕ0 : R Ñ kru, ss and ϕ1 : R Ñ kru´1 , ts coming from the
assignments

ϕ0 : x ÞÑ s, y ÞÑ us, z ÞÑ u2 s
ϕ1 : x ÑÞ u´2 t, y ÞÑ u´1 t, z ÞÑ t.

It holds that ϕ0 py 2 ´xzq “ pusq2 ´upusq “ 0 and ϕ1 py 2 ´xzq “ pu´1 tq2 ´u´1 pu´1 tq “
0, so the ϕi ’s are well-defined. The σi ’s are compatible with the transitions function and
can be glued together to the desired map σ : L´2 Ñ P1k . Indeed, one easily checks that the
diagram
R
ϕ0 ϕ1

kru, u´1 , ss ρ kru, u´1 , ts

commutes. (For instance, ρpϕ0 pxqq “ ρpsq “ u´2 t “ ϕ1 pxq.)

C L´2

y 2 “ xz

Let us analyse the fibers of the morphism σ . We begin by figuring out what happens over the
open set V0 “ Spec kru, ss, where σ restricts to the map

σ0 : Spec kru, ss Ñ Q

corresponding to ϕ0 . Consider the maximal ideal m “ px, y, zq Ă R of the origin. The fiber
over m corresponds to prime ideals in p Ă kru, ss containing mkru, ss “ ps, su, su2 q “ psq.
In other words, the fiber equals the closed set σ ´1 pV pmqq “ V psq. This means that the
whole ’u-axis’ V psq in A2 “ Spec kru, ss is collapsed onto the origin in Q. Likewise, the
’u´1 -axis’ in A2 “ Spec kru´1 , ts is collapsed to the origin. This means that the whole
zero-section C in L´2 is mapped to the origin. In fact, C is the only subscheme of L´2 which
is collapsed in this manner:

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»
Proposition 6.15. The map σ restricts to an isomorphism L´2 ´ C Ñ Q ´ tpu, where
p is the origin in Q.

Proof The complement Q ´ tpu of the origin is covered by the two distinguished open
sets Dpxq and Dpzq (note that Dpyq “ Dpy 2 q “ Dpxzq by the quadratic relation defining
R). Likewise, the complement L´2 ´ C of the zero-section is covered by the distinguished
open subsets Dpsq Ă V0 “ Spec kru, ss and Dptq Ă V1 “ Spec kru´1 , ts. It holds
that σ0´1 pV pxqq “ V psq Ă Spec kru, ss, and this means that the restriction σ|V0 “ σ0
maps Dpsq onto Dpxq. In fact, using the identification Dpxq “ Spec Rx , and the identity
Rx “ pkrx, y, zs{py 2 ´ xzqqx » krx, ysx , we see that σ0 is the map
Spec kru, sss Ñ Spec krx, ysx
induced by the ring map such that x ÞÑ s and y ÞÑ us. This is an isomorphism because we
have inverted s. Hence σ|V0 is an isomorphism over Dpxq. A symmetric argument shows
that σ|V1 is an isomorphism over Dpzq. All together, σ is an isomorphism outside C .

6.4 Double covers

Let A be a base ring, and let f P R “ Arx1 , . . . , xn s be a non-zero polynomial. This defines
a closed subscheme X of the affine space An`1
A “ Spec Rrys given by
X “ Spec Rrys{py 2 ´ f q.
The ring map R Ñ Rrys{py 2 ´ f q induces a morphism σ : X Ñ AnA . We call X , with the
map σ , the double cover of AnA “ Spec R associated to f .
The name ‘double cover’ comes from the fibers of f . Consider the case where A “ k is
an algebraically closed field. Consider a k -point of Ank corresponding to the maximal ideal
m “ px1 ´ a1 , . . . , xn ´ an q. Then the fiber of σ over m is given by

Specpkrys{ppy 2 ´ f q ` mqq » Spec krys{py 2 ´ f pa1 , . . . , an qq.

If f pa1 , . . . , an q ‰ 0, then krys{py 2 ´ f pa1 , . . . , an qq » krys{py ´ bq ˆ krys{py ` bq
where b is a square root of f pa1 , . . . , an q. Hence, if the characteristic of k is not equal to 2,
the fiber consists of two points. If the characteristic is equal to 2, or f pa1 , . . . , an q “ 0 the
fiber consists of a single point, but with ‘multiplicity 2’: it is isomorphic to Spec krys{qpy 2 q.
We will also consider double covers of projective spaces by gluing together the double
coverings we just constructed. We begin with the case of P1 .

Hyperelliptic curves
Let k be a field and consider a polynomial of degree n

ppxq “ an xn ` ¨ ¨ ¨ ` a1 x ` a0

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We will write n “ 2g ` 1 or n “ 2g ` 2 depending on whether n is odd or even. We will

consider the double cover of A1k defined by the equation y 2 “ ppxq. More formally, consider
the two affine schemes X1 “ Spec A and X2 “ Spec B , where
A “ krx, ys{py 2 ´ ppxqq and B “ kru, vs{pv 2 ´ u2g`2 ppu´1 qq.
Here
u2g`2 ppu´1 q “ an ` an´1 u ¨ ¨ ¨ ` a1 un´1 ` a0 un (6.10)
if n is even, and
u2g`2 ppu´1 q “ an u ` an´1 u2 ¨ ¨ ¨ ` a1 un´1 ` a0 un`1 (6.11)
if n is odd.
Consider the ring map ϕ : Bu Ñ Ax defined by ϕpuq “ x´1 and ϕpvq “ x´g´1 y . This
is well-defined, as the caluculation
ϕpv 2 ´ u2g`2 ppu´1 qq “ x´2pg`1q y 2 ´ x´2g´2 ppxq “ x´p2g`2q py 2 ´ ppxqq
shows that the defining ideal for Bu maps into the one defining Ax . The map ϕ is an
isomorphism, with inverse defined by x ÞÑ u´1 and y ÞÑ u´g´1 v .
Therefore, the two distinguished open sets Dpxq “ Spec Ax and Dpuq “ Spec Bu
are isomorphic and we can glue them together to a scheme X . The scheme X is called
a hyperelliptic curve. In the case g “ 1, the curve X is an example of an elliptic curve.
As a set, the scheme X consists of the points of Spec A, along with the points of Spec B

y 2 “ ppxq

Figure 6.6 A hyperelliptic curve

outside Dpuq “ Dpxq, that is, the points of V puq Ă Spec B . The number of points in V puq
depends on whether n is even or odd. If n is odd, then the equation (6.11) implies that V puq
consists of a single point, corresponding to the maximal ideal pu, vq. If n is even, then
V puq “ V pu, v 2 ´ an q
which consists of two points if an has a square root in k and a single point otherwise.
The scheme X admits a morphism π : X Ñ P1k to the projective line. Consider the two
inclusions krxs Ă A and krus Ă B . Under the isomorphism ϕ : Bu Ñ Ax above, krus is
mapped into krxs and u maps to x´1 , so there is a commutative diagram:
uÞÑx´1
krus krxs

ϕ
Bu Ax .

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The two inclusions give maps X1 Ñ U0 “ Spec krxs Ă P1k and X2 Ñ U1 “ Spec krus Ă
P1k , where U0 and U1 are joined together to a P1k according to the rule x Ø u´1 . By the
observation above, this is compatible with the way X1 and X2 are joined together, and the
two maps glue to the morphism π .
The morphism π is also called a double cover of P1k . Over each affine, π restricts to a
double cover of A1A . Therefore, if A is an algebraically closed field, the fibers remain either
two points, or one point with ‘multiplicity 2’ as in the previous example.
Notice that the gluing map defining X is very similar to the one involved in the construction
of the schemes Lm from Section 6.3. In fact, X is a closed subscheme of L´g´1 . Indeed,
L´g´1 is obtained by gluing U1 “ Spec krx, ys and U2 “ Spec kru, vs and X1 and X2
are naturally closed subschemes of U1 and U2 respectively. As the isomorphism defining
L´g´1 is exactly by the same formula as ϕ, we find that X1 and X2 glue together to a
closed subscheme of L´g´1 . The gluing isomorphisms are moreover compatible with the
two morphisms to P1k , in the sense that the following diagram commutes:

X L´g´1

P1k

Higher-dimensional double coverings

The above construction generalizes in a straightforward manner to higher-dimensional pro-
jective spaces. We will even consider projective spaces over any ring A.
Let A be a ring and let R “ Arx0 , . . . , xn s with the usual grading. Let f P R be a
homogeneous polynomial of degree 2d, and for each 0 ď i ď n let
„ ȷ ˜ˆ ˙2 ˆ ˙¸
x0 xn y L y x0 xn
Si “ A ,..., , d ´f ,..., (6.12)
xi xi xi xdi xi xi

For each pair i, j , we define Sij “ Si rxi {xj s. Then we have equalities Sij “ Sji , by the
following identity

ˆ ˙2d ˜ˆ ˙2 ˆ ˙¸ ˆ ˙2 ˆ ˙
xi y x0 xn y x0 xn
´f ,..., “ ´f ,..., .
xj xdi xi xi xdj xj xj
As in the example of projective space, the Spec Si ’s glue together along the open subschemes
Spec Sij ’s to a scheme X . Moreover, keeping the notation Ri from the previous section,
the morphisms Spec Si Ñ Spec Ri , induced by the inclusions Ri Ñ Si , glue together to a
morphism π : X Ñ PnA .

6.5 Exercises
Exercise 6.5.1. Consider Pn pkq as defined on page 94. Show that every closed subset of
Pn pkq is of the form Z` paq for some homogeneous ideal a.

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V pf q P2

Figure 6.7 A double cover of the projective plane.

Exercise 6.5.2. Let Fpr denote the field with pr elements.

a) Show that #An pFpr q “ prn .
b) Show that #Pn pFpr q “ prn ` prpn´1q ` ¨ ¨ ¨ ` pn ` 1.
Exercise 6.5.3. Let X “ Spec A be an affine scheme over a field k . Show that every
morphism P1k Ñ X is constant, i.e. it factors through some k -valued point of X .
Exercise 6.5.4. Show that P1A is not affine for any ring A. H INT: The canonical map
P1A Ñ Spec A is never an isomorphism.
Exercise 6.5.5. Describe the closed subsets of P1C , P1R and P1Q .
Exercise 6.5.6. Prove that Pnk is irreducible for any n ě 0.
Exercise 6.5.7. Show that the Z-valued points of P1Z is given by t pa : bq | a, b P Z u.
Exercise 6.5.8. Let A be a ring and let a0 , . . . , an be elements generating the unit ideal
in A. Show that the closed subscheme defined by (5.11) corresponds to the A-valued point
ιa : Spec A Ñ PnA defined in (5.7). H INT: If π : PnA Ñ Spec A is the structure map, show
that π ´1 Dpai q “ PnAa and reduce to the case where a0 is invertible.
i

Exercise 6.5.9. Let k be a field and let F P krx0 , x1 s be a nonzero homogeneous polynomial
of degree d. Show that the subscheme Z ř “ V` pF q consists of d points counted with
multiplicity. More precisely, show that d “ xPZ dimk OZ,x .
Exercise 6.5.10. Let X be the affine line with two origins, as defined in Section ??.
a) Imitate the construction of the sheaves OP1k pnq on P1k to form a collection of
sheaves OX pmq on X , one for each integer m.
b) Show that OX pmq and OX pnq are not isomorphic unless m “ n. H INT:
Consider the behaviour of sections at the two origins.
Exercise 6.5.11. Verify the claims in Examples 4.51 and 4.52 above that X is isomorphic
respectively to Spec Z2 X Z3 and to Spec AP . H INT: Use the uniqueness statement in
Proposition 4.49 on page 88.
Exercise 6.5.12. Glue Spec Zp2q to itself along the generic point to obtain a scheme X . Show
that X is not affine. H INT: Show that OX pXq “ Zp2q .

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Exercise 6.5.13. Show that for any ring A, ΓpPnA , OPnA q “ A.

Exercise 6.5.14. Compute the space OX pXq of global sections of the blow-up X and
describe the canonical map X Ñ Spec OX pXq.
Exercise 6.5.15. Imitate the construction above to define the blow-up of An along a codi-
mension 2 linear space V px, yq.
1
Exercise 6.5.16. Let k be a field and consider the scheme GL2 “ Spec krx11 , x12 , x21 , x22 , det s
where det “ x11 x22 ´ x12 x21 .
a) Find a non-constant morphism f : GL2 Ñ P1k .
b) Show that f is surjective. Deduce that the image of an affine scheme may not
be contained in an affine scheme in general.
c) Describe the fibers of f .
Exercise 6.5.17. Check that L´1 is indeed the blow-up constructed in Section 6.2.
Exercise 6.5.18. Show that for m ě 0, the scheme L´m admits a morphism σ : L´m Ñ Y
contracting the zero-section C to a point.
Exercise 6.5.19. For the canonical morphism π : Lm Ñ P1k , show that
à
π˚ OLm “ OP1k p´imq.
iě0

Exercise 6.5.20. When k is an algebraically closed field the k -points of Lm are described by
expressions resembling homogeneous coordinates.
a) Show that the k -points of Lm are precisely the equivalence classes of triples
px0 : x1 | tq,
where x0 , x1 , t P k , with px0 , x1 q ‰ p0, 0q under the relation
px0 : x1 | tq “ pαx0 : αx1 | αm tq,
for α P k a non-zero scalar.
b) Show that the zero section is the set of points of the form px0 : x1 | 0q, and that
if m ě 0 and ppx0 , x1 q is a homogeneous polynomial of degree m, then the
map P1 pkq Ñ Lm pkq given by the assignment
px0 : x1 q ÞÑ px0 : x1 | qpx0 , x1 qtq
is a well-defined section of Lm pkq Ñ P1 pkq (at least in a set-theoretic sense).
Exercise 6.5.21. Define f : L´m pkq Ñ Am`1 pkq by
px0 : x1 | tq ÞÑ ptxm m´1
0 , tx0 x1 , . . . , tx0 xm´1
1 t, xm
1 q

Show that this map is well-defined and collapses the zero-section to the origin. Define and
describe a scheme version of this map.
Exercise 6.5.22. Assume that k is algebraically closed. Let a2g`1 “ 1 and a1 “ ´1 and
ai “ 0 for the other indices. Determine the image of Dpxq and Dpuq in P1k . Find all points
in P1k where the fiber of the double covering f does not consist of exactly two points. How
many are there?

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Exercise 6.5.23. Let k be an algebraically closed field. Show that the k -points of Fm pkq are
in a one-to-one correspondence with the equivalence classes of quadruples
px0 : x1 | y0 : y1 q
under the equivalence relation
px0 : x1 | y0 : y1 q „ pαx0 : αx1 | αm βy0 : βy1 q
for non-zero scalars α and β .
Exercise 6.5.24. The different Hirzebruch surfaces are closely related, as this exercise shows.
a) Show that for some point P there is a map Fm ´ P Ñ Fm´1 that induces an
isomorphism on the complement of two fibers.
b) Show that for some point P there is a map Fm´1 ´ P Ñ Fm that induces an
isomorphism on the complement of two fibers.
H INT: On k -points, these are px0 : x1 | y0 : y1 q ÞÑ px0 : x1 | y0 : x1 y1 q with P “ p1 : 0 |
0 : 1q and px0 : x1 | y0 : y1 q ÞÑ px0 : x1 | x1 y0 : y1 q with P “ p1 : 0 | 1 : 0q.
Exercise 6.5.25. Show that the open subschemes Lm ´ C´m and Lm ´ Cm of respectively
Lm and L´m are isomorphic over P1k . Show that gluing them together gives Fm .
Exercise 6.5.26. Let X “ P1k . Show that any element OX pXq corresponding to a map
X Ñ A1 factors via a ”constant map” Spec k Ñ A1 .
Exercise 6.5.27. Let R be a local ring. Show that Pn pRq “ Pn pRq “ pRn ´ 0q { „ .
H INT: The maximal ideal must land in some Ui ; show that the other points must be contained
there as well.
Exercise 6.5.28. Show that the Z-points of P1 are in bijection with the set of pairs pa, bq P Z2
with a, b coprime up to multiplication by ´1.
Exercise 6.5.29. A cyclic cover of AnA is a scheme described by the equation y r “
f px1 , . . . , xn q in An`1 n
A . Generalize (6.12) to define cyclic covers of PA .

Exercise 6.5.30. Let P be a lattice polytope and let Q Ă P be a face of P , that is, there is a
linear form ℓ on Rn so that P Ă tℓ ě 0u and P X tℓ “ 0u “ Q. Show that there is a closed
embedding XQ Ñ XP . Describe the corresponding embeddings for the examples P1k and
P2k .
Example 6.16. Let K be a field and let tAi uiPI be a collection of subrings of K . Suppose
that for each pair i, j P I , there is an element gij P Ai such that for all i, j, k P I :
(i) pAi qgij “ pAj qgji
(ii) pAi qgij gik “ pAj qgji gjk “ pAk qgki gkj
where the equalities are as subrings of K . Show that the affine schemes Ui “ Spec Ai can
be glued together to a scheme X . △

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Basics of scheme theory

139

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Properties of schemes

7.1 Reduced schemes

A ring A is said to be reduced if it has ? elements. In general, if A is a ring, the
?no nilpotent
reduction of A is the ring Ared “ A{ 0 where 0 is the ideal of nilpotent elements of A.
We define a scheme X to be reduced if, for every x P X , the local ring OX,x is reduced,
that is, it has no nilpotent elements. While the condition is on stalks, in fact every OX pU q is
reduced:

Lemma 7.1. A scheme X is reduced if and only if for every open U Ă X , the ring
OX pU q has no nonzero nilpotent elements.

Proof If X is reduced, and s P OX pU q is a nonzero element, then it as non-zero germ in

at least one local ring OX,x . Therefore, if s is nilpotent then so is its germ, and hence OX,x
would then not be reduced. Conversely, let x P X be a point and let s P OX,x be any element.
We may write s as the germ of some section t P OX pU q. By assumption t is not nilpotent,
and hence s is not nilpotent either.
Example 7.2. An affine scheme X “ Spec A is reduced precisely when A is a reduced ring.
So for instance Spec Z and Ank are reduced, but SpecpZ{18q and Spec krxs{px3 q are not. △
For any scheme X , there is an associated reduced scheme Xred called the reduction of X .
The two schemes X and Xred have the same underlying topological space, but the structure
sheaves are different; OXred is obtained from OX by locally modding out by nilpotent
elements.
More precisely, consider the basis B of affine open sets U Ă X . We define a B -presheaf
by setting
OXred pU q “ OX pU qred . (7.1)
If V Ă U are two open affines, and s P OX pU q is a nilpotent element, then so is
the restriction s|V . It follows that the restriction maps of OX induce restriction maps
OXred pU q Ñ OXred pV q. Moreover, by Exercise 4.14.9 the B -sheaf axioms are satisfied, so
this gives rise to a sheaf of rings on X which we denote by OXred . We let Xred denote the
locally ringed space pX, OXred q.
If X “ Spec A, the reduction of X is simply the scheme SpecpAred q. We saw a glimpse
of this scheme in Corollary 2.31.
In general, if X is a scheme, then also Xred is a scheme, because if Ui “ Spec Ai is
an affine cover of X , then Xred is covered by the affine schemes pVi qred “ SpecpAi qred .

141

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Moreover,
? Xred is naturally a closed subscheme of X , as for each i, the ring pAi qred “
Ai { 0 is a quotient of Ai .

Non-reduced schemes appear naturally when we consider intersections of subschemes.

Suppose for instance we have an affine scheme X “ Spec A and two closed subschemes
Y and Z , defined by ideals a and b in A, respectively. Then we define the scheme-theoretic
intersection of Y and Z to be the closed subscheme of Spec A defined by the ideal a ` b.
Note that there is no reason to expect that the ideal a ` b is a prime ideal in A, even if a and b
are. Moreover, the scheme-theoretic intersection fail to be both reduced and irreducible even
if both Y and Z are are. This is very natural and important: the scheme-theoretic intersection
more accurately captures the multiplicities of an intersection, e.g. as in Bezout’s theorem.
This point is yet another reason for transitioning from varieties to schemes.
Example 7.3. Let X “ A2k “ Spec krx, ys, and consider the two subschemes Y “
V py ´ x2 q and Z “ V pyq. The scheme-theoretic intersection of these is given by the ideal
py ´ x2 , yq “ px2 , yq, which is not a radical ideal in krx, ys. The nilpotent elements of
krx, ys{px2 , yq “ krxs{px2 q in some sense account for the ‘tangency’ of the intersection
X XY. △
y

y “ x2

Example 7.4. Here is a similar example in A3k . Consider

X “ Spec krx, y, zs{pz ´ xy 2 q
which is a closed subscheme of A3 (a cubic surface). The intersection of X with the plane
defined by z “ 0 is given by the ideal I “ pz, xy 2 q, whose primary decomposition is
pz, xy 2 q “ pz, y 2 q X px, zq.
The intersection Spec krx, y, zs{I therefore is the union of the lines y “ z “ 0 and
x “ z “ 0. Being defined by the non-radical ideal pz, y 2 q, the component along the former
has ‘multiplicity 2’, which reflects the fact that the plane is tangent to X along that line. So
the intersection is neither irreducible nor reduced. △

7.2 Integral schemes

A scheme X is defined to be integral if OX pU q is an integral domain for every affine open
set U Ă X . As the name suggests, an affine scheme X “ Spec A is integral if and only if A
is an integral domain. This follows from the following proposition:

Proposition 7.5. A scheme X is integral if and only if it is irreducible and reduced.

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Figure 7.1 The surface z “ xy 2

Proof Assume that X is integral. To show X is reduced, let x P X be a point and choose
an affine open U “ Spec A containing it so that x corresponds to a prime ideal p. Then by
assumption, A is an integral domain, so the local ring OX,p “ Ap has no zerodivisors. For
irreducibility: if X is not irreducible, then there exist two nonempty open set U, V such that
U X V “ H. But then, by the sheaf sequence we have OX pU Y V q “ OX pU q ˆ OX pV q,
which is not an integral domain.
Conversely, suppose that X is irreducible and reduced. Let U “ Spec A Ă X be an
affine open set. Suppose that f, g P OX pU q “ A are elements so that f g “ 0; we have to
show that f “ 0 or g “ 0. Note that U “ Spec A “ V p0q “ V pf gq “ V pf q Y V pgq. As
U “ X is irreducible, we must have, say V pf q “ U . This means that f is nilpotent in A.
But as X is reduced, there are no nilpotents of A, and so f “ 0.

Example 7.6. Ank and Spec Z are integral schemes. The schemes SpecpCrxs{x2 q and
SpecpZ{60q are not. △

Many of the examples in Chapter 1 correspond to non-integral schemes (e.g., Example

1.17 and Exercise 1.8.39).

7.3 Function fields

An important property of integral schemes X is that they have a function field, KpXq,
analogous to the field of rational functions on an affine variety (as discussed in Chapter 1).
To define KpXq, we note that any integral scheme X has a unique generic point η . This
point is the unique point which dense in X , that is, it belongs to every open non-empty subset
of X (see Exercise 9.9.21).

Definition 7.7. For an integral scheme X , we define the function field, or the field of
rational functions, to be the local ring at the generic point η P X :
KpXq “ OX,η ,

Concretely, if U “ Spec A is any affine open subset in X , then A is an integral domain,

and η corresponds to the zero ideal p0q of A. Then the local ring OX,η is canonically identified
with the field of fractions KpAq of A.

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Example 7.8. The function field of Spec Z is equal to OSpec Z,p0q “ Zp0q “ Q. △
Example 7.9. The function field of Ank “ Spec krx1 , . . . , xn s is equal to the field kpx1 , . . . , xn q
of rational functions in x1 , . . . , xn . △
Example 7.10 (The quadratic cone). The quadratic cone X “ Spec krx, y, zs{py 2 ´ xzq
is an integral scheme, as y 2 ´ xz is irreducible. To compute the function field KpXq, we
express z as z “ x´1 y 2 , and we find
KpXq “ kpx, y, zq{py 2 ´ xzq » kpx, yq.
△
Example 7.11. The ‘affine line with two origins’ X is both irreducible and reduced. The
function field is isomorphic to K “ kpuq. The two local rings OX,01 and OX,02 are equal
as subrings of K ; they are both equal to kruspuq . This is somewhat unsettling: any rational
function which is regular at 01 is automatically regular at 02 and it takes the same value there.
This is related to the property of ‘separatedness’, which we will discuss in Chapter ??. △
We showed in that each OX pU q is a subring of KpXq when X was an integral affine
scheme. This extends to integral schemes in general.

Proposition 7.12. Let X be an integral scheme.

(i) For an open set U Ă X , there is a natural inclusion OX pU q Ă KpXq.
(ii) For a point p P X , there is a natural inclusion OX,p Ă KpXq.
With these identifications, we have
č
OX pU q “ OX,p Ă KpXq. (7.2)
pPU

Proof (i): For V “ Spec A affine, the germ map OX pV q Ñ OX,η corresponds to the
inclusion A Ă KpAq. This implies that for an arbitrary U Ă X , the germ map OX pU q Ñ
OX,η is injective: if s P OX pU q satisfies sη “ 0, then for any affine V Ă U , the factorization
OX pU q Ñ OX pV q ãÑ OX,η forces s|V “ 0 and hence s “ 0 by the Locality axiom. Hence
each OX pU q embeds as a subring of KpXq via the germ map. The statement (ii) then follows
by taking direct limits of the OX pU q Ă KpXq for U containing p.
We now prove the equality (7.2). The ‘Ă’-inclusion is clear. For the reverse inclusion,
suppose f P KpXq lies in OX,p for all p P U . For each p P U , we may pick an affine open
neighbourhood Vp of p and sections hp P OX pVp q so that the germ of hp at η is equal to f .
By the injectivity of OX pVp X Vq q Ñ OX,η , the hp must agree on the overlaps Vp X Vq , and
hence they glue to a section of OX pU q which maps to f .
If p P X is a point and f P KpXq is a rational function, we say that f is defined at p if
f P OX,p . The subset of points U Ă X where f is defined is open (Exercise 4.14.11). The
set X ´ U is called the indeterminacy locus of f .
Example 7.13. If A is an integral domain, then a rational function a{b P KpAq is defined on
the open set Dpbq Ă Spec A. However, it may be defined on a larger open set (see Example
1.23). △

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A rational map f : X 99K Y is a morphism defined on some open dense subset U Ă X .

To avoid the ambiguity in the domain of definition, we say that two morphisms f : U Ñ Y
and g : V Ñ Y define the same rational map if f |W “ g|W for some dense open subset
W Ă U X V . This defines an equivalence relation on pairs pU, f q, and formally, a rational
map is an equivalence class of such pairs.

Proposition 7.14. For an integral scheme X , there is a bijection

KpXq “ trational maps f : X 99K A1 u.

Proof This is essentially a consequence of Theorem 4.17, which implies that for an open
set U Ă X there is a bijection

HomSch pU, A1 q “ OX pU q.

Taking the direct limit over all open sets U (which all contain the generic point η ), the left
side gives the set of rational maps to A1 , whereas the right side gives OX,η “ KpXq.

7.4 Dominant morphisms

We say that a morphism of schemes f : X Ñ Y is dominant if the image of f is a dense
subset of Y . This means that any non-empty open set of Y intersects the image f pXq Ă Y .
Note that this is a purely topological condition.

Example 7.15. The morphism Spec krx, ys{pxy ´ 1q Ñ Spec krxs is dominant. More
generally, any open embedding U Ñ X of a dense open subset is dominant. △

Example 7.16. The map Spec krx, ys Ñ Spec kru, vs given by pu, vq ÞÑ px, xyq is
dominant, as the image is Dpvq Y tV pu, vqu (see Example 1.35) △

Example 7.17 (Dominant morphisms into affine schemes). A morphism of affine schemes
f : Spec B Ñ Spec A, induced by a ring map ϕ : A Ñ B , is dominant if and only if every
element of Ker ϕ is nilpotent. To see this, note that by (iii) of Proposition 2.29, the closure
of f pSpec Bq “ f pV p0qq equals V pϕ´1 p0qq “ V pKer ϕq. So f pSpec Bq is dense if and
only if V pKer ϕq “ Spec Aa . This condition holds precisely when Ker ϕ Ă p for all p, or
equivalently when Ker ϕ Ă p0q.
The same argument applies more generally to morphisms into an affine scheme f : X Ñ
Spec A. By Theorem 4.17, f is induced by a unique ring map ϕ : A Ñ OX pXq and Exercise
4.14.7 shows that the closure of the image of f is given by V pKer ϕq.
In particular, if A is reduced, then f : X Ñ Spec A is dominant if and only if ϕ : A Ñ
OX pXq is injective. △

The previous example generalizes as follows:

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Proposition 7.18. Let f : X Ñ Y be a morphism of integral schemes. Then the follow-

ing are equivalent:
(i) f is dominant
(ii) For all affine open subsets U Ă X and V Ă Y with f pU q Ă V , the induced
ring map OY pV q Ñ OX pU q is injective.
(iii) f maps the generic point of X to the generic point of Y .

Proof (i) ñ (ii): U and V are dense, so also the restriction f |U : U Ñ V is dominant.
Then the injectivity of OY pV q Ñ OX pU q follows from Example 7.17.
(ii)ñ (iii): Write ηX and ηY for the two generic points. Note that for any subset S Ă X ,
we have f pSq Ă f pSq. Applying this to S “ tηX u, we see that f pXq “ f pηX q Ă f pηX q.
Therefore, if f pXq is dense, then so is f pηX q, and hence f pηX q “ ηY (there is only one
point which is dense in Y ).
(iii) ñ (i): If f pηX q “ ηY , then f pXq Ą f pηX q “ ηY “ Y , so f is dominant.

A key property of dominant morphisms is that they induce pullback maps between the
corresponding function fields. More precisely, if X and Y are integral schemes with generic
points ηX and ηY respectively, and f : X Ñ Y is dominant, then by Proposition 7.18, f 7
induces a map between the local rings OY,ηY Ñ OX,ηX , hence between the function fields

f 7 : KpY q ÝÝÑ KpXq. (7.3)

Example 7.19. If k is a field, and X Ă Ank and Y Ă Ank are integral closed subschemes
over k , then the pullback f 7 pgq of a rational function g P KpY q coincides with the pullback
f 7 pgq “ g ˝ f , as defined in Section 1.6. △

If the image f pXq is not dense in Y , there is no hope of defining a pullback map as in
(7.3). For instance, if f : A1k Ñ A2k is the closed embedding of the ‘x-axis’, i.e., induced by
the ring map krx, ys Ñ krx, ys{y , then there is no reasonable way to pull back the element
y ´1 P kpx, yq to KpA1k q “ kpxq. In fact there are no maps of fields kpx, yq Ñ kpxq at all.
Typically, for a dominant morphism f : X Ñ Y , one expects X to be ‘as large as’
or ‘larger than’ Y . For instance, in many good cases X will have dimension at least the
dimension of Y . On the other hand, there are also dominant morphisms with non-intuitive
behaviour, as in the following examples:

Example 7.20. The morphism Spec Q Ñ Spec Z is dominant. More generally, if A is an

integral domain with fraction field K , the canonical map Spec K Ñ Spec A is dominant. △

Example 7.21. Let X be the disjoint union of an infinite set of closed points of the affine
line A1C . Then the embedding X Ñ A1C is dominant. △

7.5 Noetherian schemes

Recall that a ring A is Noetherian if every ideal is finitely generated. This is a strong
requirement, which has many important consequences for the ring.

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Definition 7.22. A scheme X is Noetherian if it has a finite open cover tUi uri“1 where
Ui “ Spec Ai where each Ai is a Noetherian ring.

It is important to note that the cover is required to be finite. This in particular implies that
a Noetherian scheme is quasi-compact, as it can be covered by finitely many affine schemes,
each of which is quasi-compact.

Proposition 7.23. Let X be a scheme. Then the following are equivalent:

(i) X is Noetherian.
(ii) X is quasi-compact, and for every affine U “ Spec A, the ring A is Noethe-
rian.

Proof (ii)ñ(i) is immediate, so we focus on the implication (i)ñ(ii).

Let tUi uri“1 be an affine cover of X , where Ui “ Spec Ai and each Ai is a Noetherian
ring. Since X is covered by finitely many affine open sets, it is quasi-compact.
Now let U “ Spec A be any affine open in X . We need to show that A is a Noetherian
ring. For each i, the intersection U X Ui is an open set in Ui “ Spec Ai , and since Ui is
quasi-compact, it can be covered by distinguished open sets Dpfij q Ă Ui , where fij P Ai .
The Dpfij q’s form a cover of U , so since U is quasi-compact, we may reduce the cover to
a finite one. Since Ai is Noetherian, the localization pAi qfij is also Noetherian. Therefore,
U can be covered by finitely many spectra of Noetherian rings, and hence it is a Noetherian
scheme. Therefore, we reduce to considering X “ U “ Spec A, and proceed to show that if
Spec A is a Noetherian scheme then A is Noetherian.
We may refine the covering Ui to finitely many distinguished open sets, so that Spec A is
covered by finitely many distinguished opens Dpf1 q, . . . , Dpfr q, where f1 , . . . , fr generate
the unit ideal in A.
To conclude, we need to show that every ideal a in A is finitely generated. By assumption,
the ideals aAfi are all finitely generated, and since fi is invertible in Afi , we can find finitely
many elements aij P A which map to generators of aAfi in Afi . Consider the A-linear map
à
ϕ: AÑa
i,j

which sends the standard basis vector eij to aij . Since the Dpfi q’s cover Spec A, the local-
ization ϕp is surjective for every p P Spec A, and from this it follows that ϕ is surjective as
well. Consequently, a is finitely generated and hence A is Noetherian.

Corollary 7.24. The spectrum Spec A is a Noetherian scheme if and only if A is a

Noetherian ring.

Proposition 7.25. Let X be a Noetherian scheme. Then any open or closed subscheme
of X is also Noetherian.

Proof Let tUi uiPI be a finite affine cover with Ui “ Spec Ai for Ai Noetherian. It suffices
to prove that if Y Ă X is a closed or open subscheme, then Y X Ui is Noetherian. In

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particular, since Y X Ui is closed or open subscheme of an affine scheme, it suffices to

consider the case where X “ Spec A and A is Noetherian.
When YŤis an open subscheme: then there are elements g1 , . . . , gn P A such that we
n
have Y “ i“1 Spec Agi . If A is Noetherian, then so is each of the localizations Agi , and
consequently Y is Noetherian.
When Y is a closed subscheme, Y “ SpecpA{aq for some ideal a Ă A. If A is Noetherian,
then so is A{a, and hence Y “ SpecpA{aq is also Noetherian.

Another nice property of Noetherian schemes is that the underlying topological space is
Noetherian (that is, descending chains of closed subsets stabilize, see page 9).

Proposition 7.26. Let X be a Noetherian scheme. Then

(i) The topological space of X is Noetherian.
(ii) Any closed subset Y Ă X admits a unique decomposition into irreducible
components
Y “ Y1 Y ¨ ¨ ¨ Y Yr .

Proof (i): By definition, X may be covered by a finite number of open affine subsets. A
descending chain stabilizes if the intersection with each of those open sets stabilizes, so we
reduce the proof to showing the proposition for X “ Spec A where A is a Noetherian ring,
in which case the statement is clear.
(ii): See the proof of Proposition 1.15.

Examples
š8
Example 7.27. For a field k , the disjoint union X “ i“1 Spec k is not Noetherian (it is
not even quasi-compact). △
`ś8 ˘
Example 7.28. The scheme X “ Spec i“1 k is affine, hence quasi-compact. However
it is not Noetherian, because the ring is not Noetherian. In fact, the set of prime ideals in
infinite products of fields is remarkably complicated: it is described by the set of so-called
‘ultrafilters’ on N. (See also Exercise 10.9.10 for a related example.) △

The topological space Spec A can be Noetherian even without A being Noetherian: the
condition is equivalent to the weaker condition that ascending chains of radical ideals
eventually stabilize, and there are many rings which satisfy this without being Noetherian.
Here are two examples:

Example 7.29. Consider the polynomial ring krt1 , t2 , t3 , . . . s and the maximal ideal m “
pt1 , t2 , . . . q. The ring
A “ krt1 , t2 , t3 , . . . s{m2

has only one prime ideal, the maximal ideal m. Therefore, Spec A consists of a single point,
and is therefore Noetherian as a topological space. The ring A however is not Noetherian, as
m requires infinitely many generators, namely all the ti ’s. △

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Example 7.30. Let A “ Crrtss be the ring of formal power series over C, with fraction field
K “ Cpptqq. Let K denote the algebraic closure of K , that is,
ď
K“ Cpptqqpt1{n q.
ně1

The integral closure of A in K is given by

ď
A“ Crrtsspt1{n q.
ně1

Note that A is not Noetherian, as we have an infinite ascending chain of ideals:

ptq Ă pt1{2 q Ă pt1{4 q Ă ¨ ¨ ¨ .
However, the spectrum Spec A is Noetherian as a topological space. In fact, it consists of
only two points. To see this, observe that each Crrt1{n ss is a discrete valuation ring, hence
its spectrum consists of two points: the generic point and the closed point. Moreover, the
n n`1
inclusions Crrt1{2 ss Ă Crrt1{2 ss induce a directed system
¨ ¨ ¨ Ñ Spec Crrt1{4 ss Ñ Spec Crrt1{2 ss Ñ Spec Crrtss
and Spec A is the inverse limit of this system (see Exercise 9.9.36). As all the maps involved
take the generic point to the generic point and the closed point to the closed point, they are
are homeomorphisms, and hence Spec A also consists of two points. △
While the first example may seem a bit artificial, the second one is one that could appear ’in
nature’. That being said, most of classical algebraic geometry primarily deals with Noetherian
rings. For instance, all of the examples from Chapter 5 are Noetherian, as they are all
constructed by explicitly gluing together finitely many schemes of the form Spec A where A
is a Noetherian ring. Here is another example on the ‘exotic side’.
Example 7.31. In Example 4.52 on page 91, we glued together the affine schemes Xp “
Spec Zppq with p from a finite set P of prime numbers. However, in the gluing conditions for
schemes, there are no restrictions on the number of schemes to be glued together, and we are
free to glue together P infinite. For example, we can let P be the set of all primes.
The resulting scheme XP is rather peculiar: it is neither affine nor Noetherian, but it is
locally Noetherian. As a scheme over Z, the canonical map π : XP Ñ Spec Z is bijective
and continuous, but it is not a homeomorphism. Moreover, for all open subsets U Ă Spec Z
the map induced on sections π 7 : ΓpU, OSpec Z q Ñ Γpπ ´1 U, OXP q is an isomorphism. In
other words, π 7 : OSpec Z Ñ π˚ pOXP q is an isomorphism of sheaves!
As in Example 4.52 the scheme XP is constructed by gluing the different Spec Zppq ’s
together along the generic points. However, when computing the global sections, we see
things changing. As in Example 4.52 the global sections are computed with the help of the
sheaf sequence
ś ś
0 ÝÝÑ ΓpX, OX q pPP ΓpXp , OX q p,qPP ΓpXp X Xq , OX q
“

ś ś
pPP Zppq ρ p,qPP Q,

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p7q p11q
p5q p13q

p3q p17q
..
p2q η .

Figure 7.2 Gluing together all the Spec Zppq ’s

Ş
and the kernel of ρ is still pPP Zppq , but now this intersection equals Z. Indeed, a rational
number α “ a{b lies in Zppq precisely when the denominator b does not have p as factor, so
lying in all Zppq , means that b has no non-trivial prime-factor. That is, b “ ˘1, and hence
α P Z.
One can understand the canonical map π : XP Ñ Spec Z as follows. Each of the schemes
Spec Zppq maps in a natural way into Spec Z, by the map induced by the inclusion Z Ă Zppq .
Here the generic point of Spec Zp map to generic point of Spec Z, and the closed point
maps to ppq P Spec Z. As the maps agree on the generic points, they glue to the canonical
map π : XP Ñ Spec Z. This is a continuous bijection by construction, but it is not a
homeomorphism. Indeed, the subsets Spec Zppq are open in XP by the gluing construction,
but they are not open in Spec Z, as their complements are infinite.
The underlying topological space of XP is not Noetherian, as the subschemes Spec Zppq
form an open cover that obviously cannot be reduced to a finite cover. However, it is locally
Noetherian as the open subschemes Spec Zppq are Noetherian. The sets Up “ XP ´ tppqu
map bijectively to Dppq Ă Spec Z and ΓpUp , OXP q “ Zp , but Up and Dppq are not
isomorphic. △

7.6 The dimension of a scheme

Recall from Chapter 1 that the dimension of a topological space is the supremum of all
integers n such that there exists a chain
Z0 Ă Z1 Ă ¨ ¨ ¨ Ă Zn (7.4)
of distinct irreducible closed subsets of X . If no such finite upper bound exists, we say that
dim X “ 8.
For a point x P X , we will also consider the dimension of X at x, denoted dimx X , which
is defined as the suprenum of lengths of chains as in (7.4) with x P Z0 .

Lemma 7.32. Let X be a topological space.

(i) If Y Ă X is any subset, then dim Y ď dim X .
(ii) Assume that dim X ă 8. If Y Ă X is a closed and irreducible subset and
dim Y “ dim X , then Y is an irreduible component of X .
(iii) If tUi uiPI is an open cover of X , then dim X “ supiPI dim Ui .

Proof (i): as the closure of an irreducible subset is again irreducible, and since any closed

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Z Ă Y satisfies Z X Y “ Z , a chain tZi u of distinct irreducible closed subsets of Y will

yield a chain tZi u of distinct irreducible closed subsets of X .
(ii): Assume dim X ă 8 and let Y Ă X be a closed irreducible subset with dim Y “
dim X . If Y is not an irreducible component, there would exist a strictly larger irreducible
closed subset Z containing Y. But this would contradict the maximality of the dimension of
Y.
(iii): First of all, dim X ě supiPI dim Ui by part (i). To prove the reverse inequality, we
first remark that if Z Ă X is closed and irreducible and Z X U ‰ H, then Z X U “ Z .
Indeed, Z X U Ă Z holds because Z is closed, and if the inclusion is strict, Z would be the
union of two proper closed subsets, namely Z X U and Z ´ U .
Therefore, if Z0 Ă . . . Ă Zn is a chain in X and U an element of the open cover such that
U X Z0 ‰ H, then tZi X U u is a chain of distinct closed subsets in U , and consequently
n ď dim U . This shows that when dim X “ 8, the supremum supiPI dim Ui will be
infinite as well. When dim X is finite, taking the chain to be maximal (that is, not extendable
to a longer chain), we see that dim X “ n ď dim U , which proves the desired reverse
inequality.
If X is a scheme, we define the dimension of X as the dimension of the underlying
topological space. In particular, dim X “ dim Xred (see Exercise 4.14.9).
When X “ Spec A is affine, the closed irreducible subsets are of the form V ppq where p
is a prime ideal of A. Using this observation we find:

Proposition 7.33. The dimension of Spec A equals the Krull dimension of A.

Having finite dimension does not guarantee that a scheme is Noetherian. For instance, Ex-
ample 7.29 gives a counterexample. More surprisingly, there are also Noetherian rings whose
Krull dimension is infinite. Although each maximal chain of prime ideals in a Noetherian
ring will be of finite length (prime ideals satisfy the descending chain condition) there can be
arbitrary long ones. See Exercise 10.9.22 for an explicit example, due to Nagata.
Example 7.34. The spectrum of the integers, Spec Z has dimension 1. The maximal chains
of prime ideals in Z have the form p0q Ă ppq where p is a prime number. △
Example 7.35. The affine line A1Z has dimension 2. The maximal chains of prime ideals
in Zrxs are of the form p0q Ă ppq Ă pf pxq, pq, where p is a prime number and f pxq a
polynomial which is irreducible mod p.
△
More generally, we have dim AnZ “ n ` 1, by the following proposition:

Proposition 7.36. If A is a Noetherian ring, then

dim Spec Arx1 , . . . , xn s “ n ` dim A.

In particular, when A “ k is a field, Ank has dimension n. A maximal chain of irreducible

closed subsets is given by
V px1 , . . . , xn q Ă . . . Ă V px1 , x2 q Ă V px1 q Ă Ank .

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If A is not Noetherian, then the Krull dimension of Arx1 , . . . , xn s is harder to control; it can
be any integer between dim A ` n and dim A ` 2n, inclusive.

Example 7.37 (Zero-dimensional schemes). The schemes

Spec Z{pZ, Spec Crxs{pxn q, Spec Crx, ys{px2 , xy, y 3 q,

have dimension zero.

More generally, the spectrum of an Artinian ring has dimension zero, and for Noetherian
rings A, Spec A śhas dimension zero if and only if A is Artinian.
8
The ring A “ i“1 Z{2Z is not Noetherian, but it has dimension zero, and X “ Spec A
has infinitely many points. △

Codimension
For a closed subset Y Ă X the dimensions dim Y and dim X are defined in terms of closed
irreducible subsets contained in Y and X respectively. If we fix the subset Y Ă X , there is
also a relative notion, the codimension of Y in X , denoted by codimpY, Xq, which is defined
in terms of closed irreducible subsets of X containing Y . These three numbers will in some
important cases be related by the equality dim Y ` codimpY, Xq “ dim X (which justifies
the name ‘codimension’). However this formula does not hold in general (see Example 7.40
below).

Definition 7.38 (Codimension). Let Y Ă X be an irreducible closed subset of X . The

codimension of Y is the supremum of all integers n such that there exists a chain
Y “ Y0 Ă Y1 Ă ¨ ¨ ¨ Ă Yn
of distinct irreducible closed subsets of X .

When X “ Spec A, the bijective correspondence between irreducible closed subsets of

Spec A and prime ideals in A, shows that the codimension of the subset V ppq will be equal
to the height of the prime ideal p, that is, the maximal length of a chain of distinct prime
ideals p0 Ă p1 Ă ¨ ¨ ¨ Ă pr “ p. Equivalently,

codim V ppq “ htppq “ dimpAp q. (7.5)

More generally, we have:

Proposition 7.39. Let X be a scheme and x P X be a point. Set Y “ txu. Then

codimpY, Xq “ dim OX,x . (7.6)

Proof Given a chain Y Ă Y1 Ă ¨ ¨ ¨ Ă Yn of distinct irreducible closed subsets, the generic

points η1 , . . . , ηn of the Yi ’s will be contained in any affine open neighbourhood of x. We
may therefore assume that X “ Spec A is affine, in which the claim follows by (7.5).

A chain of distinct irreducible closed subsets of Y may be extended to a chain in X by

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appending a chain betwen Y and X . Taking the suprenum over all such chains, we find the
inequality
dim Y ` codimpY, Xq ď dim X.
As mentioned above, equality does not hold in general. In fact, there are quite simple examples
where the inequality is strict.
Example 7.40. Consider the discrete valuation ring R “ Crxspxq with maximal ideal
m “ pxq. Consider the principal ideal n “ ptx ´ 1q in the polynomial ring Rrts. It is
a maximal ideal, because Crxspxq {ptx ´ 1q » Cpxq, and one easily checks that it does
not properly contain any non-zero prime ideal, so it is of height 1. Letting Y “ V pnq and
X “ Spec Rrts, we find dim Y “ 0 and codim Y “ 1, but it holds that dim X “ 2. △

Krull’s Principal ideal theorem

TODO.

7.7 Exercises
Exercise 7.7.1. Show that the sections of OSpec A over an open set U Ă X “ Spec A, are
given by the inverse limit of the localizations
OX pU q “ lim
ÐÝ OpDpf qq “ lim
ÐÝ Af . (7.7)
Dpf qĂU Dpf qĂU

Exercise 7.7.2. Let A “ krx, y, zs{pxyzq and X “ Spec A. Compute OX,p where x
corresponds to the prime ideal p “ px ´ 1, y, zq. Show that yz ‰ 0 in OX,p , but takes the
value 0 for all points in a neighbourhood of p.
Exercise 7.7.3. Show that if f : X Ñ Y is a morphism of locally ringed spaces, the stalk
maps fx7 : OY,f pxq Ñ OX,x induce maps between the residue fields κpf pxqq and κpxq. What
happens when X and Y are affine varieties?
Exercise 7.7.4. Let X “ Spec Z. Compute XpFp q, XpQq and XpCq.
Exercise 7.7.5. Show that Spec Qrxs and Spec Z are homeomorphic, but not isomorphic as
schemes.
Exercise 7.7.6. Is Spec Q Ñ Spec Z a closed embedding?
Exercise 7.7.7. Verify the claim about XpQq in Example 4.40. H INT: Compute the second
intersection point a general line trough p0, 1q has with the unit circle.
Exercise 7.7.8. With reference to Example 4.40, show that one may interpret XpQq as the
set of Pythagorean triples:
XpQq “ t pa, b, cq P Z3 | a2 ` b2 “ c2 and a, b, c relatively prime u.
Exercise 7.7.9. With reference to Example 4.40, let p be a prime such that p fl 1 mod 4.
Show that the description in Example 4.40 also is valid for XpFp q.

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Exercise 7.7.10. With reference to Example ??, consider the natural inclusion
A “ Rru, vs{pu2 ` v 2 ` 1q Ă Cru, vs{pu2 ` v 2 ` 1q “ AC .
For each point z “ pa, bq P XpCq consider the ideal nz “ mz X A. Show that nx is maximal
and that nz “ nw if and only if w “ pā, b̄q with z “ pa, bq. Conclude that A has infinitely
many maximal ideals.
Exercise 7.7.11. Let X be a scheme and let x P X be a point.
a) Show that there is a canonical morphism
f : Spec OX,x ÝÝÑ X
b) Show that f induces a homeomorphism between SpecpOX,x q and the subset
W Ă X of points w P X such that w P x.
c) Show that the map ιx : Spec κpxq Ñ X defined in the text factors via f .
d) Show that on the level of topological spaces, the image of f is the intersection
of all open neighbourhoods containing x.
e) Compute the image of f when:
(i) x is the generic point of an irreducible scheme.
(ii) x is a closed point of A2C .
Exercise 7.7.12. Deduce Theorem 4.32 from Theorem 4.21.
n
Exercise 7.7.13. An inclusion Q Ă Q induces a morphism AQ Ñ AnQ . Compute the images
of the following points under the morphism A2Q Ñ A2Q :
? ?
a) px ´ 2, y ´ 2q
2
?´ ω, y?´ ω q where ω is a cube root of unity.
b) px
c) p 2x ` 3yq
Exercise 7.7.14. Let pf, f 7 q : X Ñ Y be a morphism of locally ringed spaces. Show that
pf, f 7 q is an isomorphism if and only if f is a homeomorphism and the sheaf map f 7 is an
isomorphism (that is, fU7 is an isomorphism for every open set U Ă Y ).
Exercise 7.7.15. Show that being a closed embedding is a property which is ‘local on the
target’. In clear text: given a morphism f : Z Ñ X and an open cover tUi u of X . Let
Vi “ f ´1 Ui and assume that each restriction f |Vi : Vi Ñ Ui is a closed embedding. Prove
that then also f is a closed embedding.
Exercise 7.7.16. Show that being a locally closed embedding is ‘local on the image’. Assume
that f : Z Ñ X is a morphism and that tUi u is a collection of open subsets of X covering
the image f pZq. Assume further that each restriction f |f ´1 Ui : f ´1 Ui Ñ Ui is a closed
embedding, then f is a locally closed embedding.
Exercise 7.7.17. Let f : X Ñ Y and g : Y Ñ Z be two morphisms of schemes. Prove that
if both f and g are closed embeddings, then g ˝ f is one as well.
Exercise 7.7.18. Let f : X Ñ Y be a morphism which is both an open embedding and a
closed embedding. Show that f is an isomorphism.
Exercise 7.7.19. Consider the ring R “ Zrts and let X “ Spec R.

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a) For a prime number p, show that m “ pt, pq is a maximal ideal of R.

b) Let U “ X ´ tmu. Show that U “ Dppq Y Dptq and that
OX pU q “ Zrts
c) Deduce that U is not affine.
Exercise 7.7.20. Let X “ ta, b, cu be a set with three elements. Let X have the coarsest
topology so that the two subsets U “ ta, bu, and V “ ta, cu are open. Define a presheaf
OX by OX pU q “ OX pV q “ Crtsptq and OX pU X V q “ Cptq with the restriction map
given by the inclusion Crtsptq Ă Cptq.
a) Show that OX is a sheaf.
b) Show that pX, OX q is a scheme.
c) Show that pX, OX q is not affine.
Exercise 7.7.21. Let X be a scheme.
a) Show that any irreducible and closed subset Z Ă X has a unique generic point.
H INT: Reduce to the affine case.
b) Show that in general schemes are not Hausdorff. What are the possible underly-
ing topologies of affine schemes that are Hausdorff?
c) Show that X satisfies the zeroth separation axiom (they are T0 ); that is, given
two points x and y in X , there is an open subset of X containing one of them
but not the other.
Exercise 7.7.22. a) Show that HomRings pZ, Qq has only one element.
b) Define continuous maps Spec Q Ñ Spec Z, sending p0q to ppq. Is it possible
to make these into morphisms of schemes?
Exercise 7.7.23 (The sheaf of units). Let X be a scheme with structure sheaf OX . We say
that s P OX pU q is a unit if there exists a multiplicative inverse s´1 P OX pU q.
a) Show that s P OX pU q is a unit if and only if for all x P U , the germ sx is a
unit in the ring OX,x ; that is, if and only if sx does not lie in the maximal ideal
of OX,x .
ˆ ˆ
b) We let OX pU q denote the subgroup of units in OX pU q. Show that OX pU q is a
subsheaf of OX .
Exercise 7.7.24. In the same vein as Example 2.33, show that a ring A is a Q-algebra (that is,
it contains a copy of Q) if and only if the canonical map Spec A Ñ Spec Z factors through
the generic point Spec Q Ñ Spec Z.
Exercise 7.7.25. For every ring A, there is a canonical map Z Ñ A which sends 1 to 1.
Hence there is a canonical map Spec A Ñ Spec Z. Show that map factors through the
canonical map Spec Fp Ñ Spec Z if and only if A is of characteristic p.
Exercise 7.7.26 (The Frobenius morphism). Let p be a prime number and let A be a ring of
characteristic p. The ring map FA : A Ñ A given by a ÞÑ ap is called the Frobenius map on
A.
a) Show that FA induces the identity map on Spec A.
b) Show that if A is local, then FA is a map of local rings.

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c) For a scheme X over Fp , define the Frobenius morphism FX : X Ñ X by the

identity on the underlying topological space and with FX7 : OX Ñ OX given
by g ÞÑ g p . Show that FX is a morphism of schemes.
d) Show that FX is natural in the sense that if f : X Ñ Y is a morphism of
schemes over Fp , we have f ˝ FX “ FY ˝ f .
In particular, this exercise shows that for a morphism of schemes f : X Ñ Y , in order to
check that f is an isomorphism, is not enough to check that f is a homeomorphism; also the
map f 7 must be an isomorphism.

Exercise 7.7.27. Let X be an integral scheme over a ring A. Let f P KpXq and let Uf Ă X
be the open set of points x P X such that f P OX,x . Show that there is a morphism
ϕ : Uf Ñ A1A such that ϕ7 : Arts Ñ ΓpUf , OX q is given by t ÞÑ f .
Exercise 7.7.28. Prove Proposition 7.5. That is, prove that a scheme X is integral if and only
if OX pU q is an integral domain for each open U Ă X .

Exercise 7.7.29. Let X “ Spec krx, y, z, ws{pxw ´ yzq and consider the open set U “
X ´ V px, yq. Use the above strategy as in Example 4.26 to compute OX pU q. Conclude that
U is not affine.
Exercise 7.7.30. Prove that a composition of two closed embeddings is a closed embedding.

Exercise 7.7.31. Let X be a scheme so that the underlying topological space is finite and
discrete. Show that X is an affine scheme. H INT: Consider the case where X has one point
first.

Exercise 7.7.32. Prove that a morphism f : X Ñ Y is a closed embedding if and only if f

induces a homeomorphism from X onto a closed subset of Y , and for each x P X , the map
of local rings OY,f pxq Ñ OX,x is surjective. H INT: Consider the stalks of f˚ OX . You need
to use the fact that f is a homeomorphism onto f pXq.

Exercise 7.7.33. Describe the following schemes and the structure sheaf on them.
a) Spec Crts{pt2 ` 1q
b) Spec Rrts{pt3 ´ t2 q
c) Spec F3 rts{pt3 ´ 1q.

Exercise 7.7.34. Let K be a finite field extension of Q and let X “ Spec K . Show that
HomSch pX, Xq can be identified with the Galois group GalpK{Qq.
Exercise 7.7.35. Let A be a ring and consider a morphism g : Am n
A Ñ AA given by poly-
nomials g1 , . . . , gn P Arx1 , . . . , xm s. Consider the morphism f : AA Ñ Am`n
m
A given by
px1 , . . . , xn , g1 , . . . , gn q. Show that f is a closed embedding.
Exercise 7.7.36. Show that the Spec-functor preserves inverse limits of rings. That is, if
tRi uiPI is a directed system of rings, then
Specplim
ÝÑ Ri q “ limÐÝ Spec Ri .
H INT: The Spec-functor is a right adjoint to the global sections functor.

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Exercise 7.7.37. Let A be a ring and let M be an A-module. Show that for each f P A,
there is a natural isomorphism of sheaves on Dpf q “ SpecpAf q

M
Ăf “ M
Ă|Dpf q .

Exercise 7.7.38. Describe the schemes Spec A for

a) A “ Zrxs{p2x ´ 1q
b) A “ Zrxs{p3x ´ 1q
c) A “ Zrxs{p9x ´ 3q
d) A “ Zrxs{p2x2 ´ 1q
Which of these are isomorphic/homeomorphic?
Exercise 7.7.39. Check that Spec ϕ ˝ Spec ψ “ Specpψ ˝ ϕq, whenever ϕ and ψ are
composable ring maps.
Exercise 7.7.40. Prove that Ank ´ tp0, . . . , 0qu is not affine for any n ě 2.
Exercise 7.7.41. Let f : SpecpBq Ñ SpecpAq be the morphism associated to a ring map
ϕ : A Ñ B . Show that the sheaf map f 7 : OSpec A Ñ f˚ OSpec B is given by ϕr : A
rÑB
r,
where we regard B as an A-module via ϕ.
Exercise 7.7.42. Let X be a scheme and let Xred be its reduction. Show that if k is a field,
then there is a natural bijection Xpkq “ Xred pkq. Generalize the statement to reduced rings.
Exercise 7.7.43. Show that the morphism
Spec Crx, b, cs{px2 ` bx ` cq ÝÝÑ Spec Crb, cs
is finite and describe its fibers.
What about the morphism Spec Crx, a, b, cs{pax2 ` bx ` cq Ñ Spec Cra, b, cs?

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Fiber products

8.1 Introduction
Given two scheme morphisms X Ñ S and Y Ñ S , the fiber product X ˆS Y is a scheme
equipped with projection morphisms pX : X ˆS Y Ñ X and pY : X ˆS Y Ñ Y . This
construction generalizes the product of two affine varieties as discussed in Example 1.36.
However, the fiber product is far more than a mere generalization; it is an indispensable tool
in algebraic geometry and it takes on remarkably versatile roles.
The main result of the chapter is the construction of the fiber product X ˆS Y as a scheme.
We first carry out this construction when X , Y and S are affine schemes, and then extend it
to general schemes using gluing techniques.
An important observation is that the fiber product of two varieties is not, in general, a
variety, but rather a scheme. In fact, the fact that fiber products exist is one of the most
important properties of the category of schemes, and one can argue that it is the definitive
reason for transitioning from varieties to schemes.
Towards the end, we will treat the main applications and study a series of examples. We
also explain some of the various contexts where fiber products appear, including base change
and scheme-theoretic fibers.

Fiber products
The notion of a fiber product is meaningful in any category C. Although our main concern
will be the category of schemes, we give the definition in a general setting.
A fiber product of two arrows fX : X Ñ S and fY : Y Ñ S in a category C, is an object
X ˆS Y together with two arrows pX : X ˆS Y Ñ X and pY : X ˆS Y Ñ Y such that
pY
X ˆS Y Y
pX fY (8.1)

X fX
S

commutes, and satisfying the following universal property: for any two arrows gX : Z Ñ X
and gY : Z Ñ Y in C such that fX ˝gX “ fY ˝gY , there is a unique arrow g : Z Ñ X ˆS Y
such that pX ˝ g “ gX and pY ˝ g “ gY .
We call a commutative diagram (8.1) a Cartesian diagram or a Cartesian square. The
universal property can be visualized in the following diagram:

158

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Z gY
g

pY
X ˆS Y Y (8.2)
gX
pX fY

X fX
S

When the fiber product exists, it is unique up to a unique isomorphism. This follows directly
from the universal property: given two candidates for the fiber product, W and W 1 , with
projections pX , pY and p1X , p1Y respectively, then there exists a unique isomorphism θ : W Ñ
W 1 so that pX “ p1X ˝ θ and pY “ p1Y ˝ θ. The uniqueness of θ follows because W 1 is a
fiber product: it is the unique morphism W Ñ W 1 arising from the universal property of W 1 .
For this reason, we allow ourselves to speak about the fiber product.
It is not so hard to come up with examples of categories where fiber products do not
exist. For instance, the fiber product does not exist in the familiar category of differentiable
manifolds, and also not in the category of affine varieties, as we will see later. This is yet
another reason why we need to make the transition from varieties to schemes.
Given two morphisms ϕ : Z Ñ X and ψ : W Ñ Y over S , we can also form their fiber
product morphism
ϕ ˆ ψ : Z ˆS W ÝÝÑ X ˆS Y. (8.3)
To define (8.3), compose ϕ with fX : X Ñ S to obtain a morphism fX ˝ ϕ : Z Ñ S , and
likewise fY ˝ ψ : W Ñ S . Then we can consider the fiber product Z ˆS W , which comes
with projection morphisms pZ : Z ˆS W Ñ Z and pW : Z ˆS W Ñ W . Now composing
pZ with ϕ and pW with ψ , we obtain two morphisms Z ˆS W Ñ X and Z ˆS W Ñ Y . By
the universal property, there is a unique morphism from Z ˆS W to X ˆS Y , and we define
ϕ ˆ ψ to be this morphism. By construction, pX ˝ pϕ ˆ ψq “ ϕ and pY ˝ pϕ ˆ ψq “ ψ .

Fiber products of sets.

In the category of sets, the fiber product has a concrete description. Given two maps of sets
fX : X Ñ S and fY : Y Ñ S , their fiber product X ˆS Y is the subset of X ˆ Y consisting
of pairs px, yq with equal images in S :
X ˆS Y “ t px, yq P X ˆ Y | fX pxq “ fY pyq u.
The projections pX and pY are the usual projection maps pX px, yq “ x and pY px, yq “ y .
We can check directly that X ˆS Y satisfies the universal property: given any two maps
gX : Z Ñ X and gY : Z Ñ Y such that fX ˝ gX “ fY ˝ gY , then the required unique map
g : Z Ñ X ˆS Y is simply given by gpzq “ pgX pzq, gY pzqq.
Note that the fiber product can be written as a disjoint union
ğ
X ˆS Y “ fX´1 psq ˆ fY´1 psq.
sPS

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This is the reason for the name ‘fiber product’; the fibers of the map X ˆS Y Ñ S are the
products of the fibers of the two maps fX and fY .
Example 8.1. If X and Y are subsets of S and fX : X Ñ S and fY : Y Ñ S are the
inclusions, then X ˆS Y “ X X Y . △
Example 8.2. If f : X Ñ S is a map and s P S , then taking Y “ tsu gives X ˆS tsu “
f ´1 psq. △

8.2 Fiber products of schemes

The following existence theorem is the main result of this chapter.

Theorem 8.3 (Existence of fiber products). Let X Ñ S and Y Ñ S be schemes over

a scheme S . Then there is a scheme X ˆS Y satisfying the universal property of the fiber
product.

When the base scheme S is affine, say S “ Spec A, the fiber product X ˆS Y will usually
be denoted by X ˆA Y .
The proof of the theorem consists of a series of reductions to the affine case; for the affine
case the product is defined using the tensor product of rings. The reductions rely heavily on
the gluing techniques developed in Chapter ??.
A remark before we begin: one cannot construct the fiber product X ˆS Y by defining
a structure sheaf on the fiber product of the sets. In fact, the underlying set of a product of
schemes can be very different from the product of the underlying sets of X and Y . This may
seem counterintuitive at first, but is in fact a typical feature of the fiber products of schemes
(see the examples in Section 8.3). It is important to keep in mind that we are taking the fiber
product of the two morphisms X Ñ S and Y Ñ S , not merely of the schemes X and Y
themselves
The points of the fiber product X ˆS Y become easier to understand if we consider the
R-valued points. For a ring R, we have
pX ˆS Y qpRq “ XpRq ˆSpRq Y pRq (8.4)
where the right-hand side is the fiber product of sets. This is just a rephrasing of the universal
property of the fiber product: to give a morphism Spec R Ñ X ˆS Y is equivalent to giving
two morphisms, Spec R Ñ X and Spec R Ñ Y , so that the compositions Spec R Ñ
X Ñ S and Spec R Ñ Y Ñ S are equal.

Products of affine schemes

We being by the constructing fiber products of affine schemes. The main observation is that
the category of affine schemes is equivalent to the opposite category of rings Ringsop , and
that the tensor product of algebras enjoys a universal property dual to the one of the fiber
product.
Consider A-algebras B1 and B2 with structure maps αi : A Ñ Bi for i “ 1, 2. Then the

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tensor product B1 bA B2 is also an A-algebra, via pb1 b b2 qpc1 b c2 q “ pb1 c1 q b pb2 c2 q.

Moreover, there are maps of A-algebras
β1 : B1 Ñ B1 bA B2 , b1 ÞÑ b1 b 1
β2 : B2 Ñ B1 bA B2 , b2 ÑÞ 1 b b2
They fit into the commutative diagram
β2
B1 bA B2 B2
β1 α2 (8.5)
B1 α1 A
where commutativity holds because α1 paq b 1 “ 1 b α2 paq by definition of the tensor
product B1 bA B2 (this is the significance of the tensor product being taken over A; one can
move elements from A between the components of the tensor).
Moreover, the tensor product is universal among diagrams such as (8.5). More presisely,
suppose that γi : Bi Ñ C are maps of A-algebras, i.e. γ1 ˝ α1 “ γ2 ˝ α2 (i.e., they fit
into a commutative diagram analogous to (8.5), but with the βi ’s replaced by the γi ’s).
The assignment b1 b b2 Ñ γ1 pb1 qγpb2 q is A-bilinear and hence extends uniquely to an A-
algebra homomorphism γ : B1 bA B2 Ñ C , which obviously has the property γ ˝ βi “ γi ,
as expressed in the following commutative diagram:
γ2
C
γ
β2
B1 bA B2 B2 (8.6)
γ1
β1 α2

B1 α1 A.
Applying Spec to (8.5), we arrive at the diagram
p2
SpecpB1 bA B2 q Spec B2
p1 (8.7)

Spec B1 Spec A,
and SpecpB1 bA B2 q is universal among affine schemes sitting in a diagram like (8.7).
Hence SpecpB1 bA B2 q serves as the fiber product in the category AffSch of affine schemes.
In fact, it is the fiber product in the larger category Sch of schemes:

Proposition 8.4. For morphisms fi : Spec Bi Ñ Spec A for i “ 1, 2, the scheme

SpecpB1 bA B2 q with the two projection p1 and p2 is the fiber product of the Spec Bi ’s
in the category of schemes Sch.

Unravelled, this means: if Z is any scheme and gi : Z Ñ Spec Bi are morphisms with
f1 ˝ g1 “ f2 ˝ g2 , there exists a unique morphism g : Z Ñ SpecpB1 bA B2 q such that
pi ˝ g “ gi for i “ 1, 2.

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Proof To check the universal property, we apply Theorem 4.17 about maps into affine
schemes. The morphisms gi give maps of A-algebras Bi Ñ OZ pZq. By the universal
property of the tensor product, these induce a unique map of A-algebras B1 bA B2 Ñ OZ pZq,
which in turn gives the desired map g : Z Ñ SpecpB1 bA B2 q of schemes over Spec A
by Theorem 4.17. By construction, this map satisfies pi ˝ g “ gi for i “ 1, 2. Finally, g
is unique by the uniqueness part of Theorem 4.17 and the universal property of the tensor
product.

Products of general schemes

Recall that any open subset U of a scheme X has a canonically defined scheme structure as an
open subscheme with the structure sheaf equal to the restriction OX |U . Hence, if f : X Ñ Y
is any morphism and V Ă Y is an open subscheme, the inverse image f ´1 V is an open
subscheme of X , and any morphism g : Z Ñ X such that f ˝ g factors through V , will
factor through f ´1 V .

Lemma 8.5. If X ˆS Y exists and U Ă X is an open subscheme, then U ˆS Y exists

and is canonically isomorphic to the open subscheme p´1
X U with the two restrictions
pX |p´1
X U
and p | ´1
Y p U
X
as projections.

Proof Write ι : U Ñ X for the open embedding. We need to verify that p´1 X U together
with the restriction of the two projections satisfies the universal property. Suppose Z is a
scheme and gU : Z Ñ U and gY : Z Ñ Y are two morphisms over S . The situation is
displayed in the diagram below

Z ḡ
p´1
X U X ˆS Y pY Y
pX
gU
ι
U X S
The composition gX “ ι ˝ gU is a map into X , and gX and gY induce a unique map of
schemes g : Z Ñ X ˆS Y with gX “ pX ˝ g and gY “ pY ˝ g . Clearly pX ˝ g “ ι ˝ gU
takes values in U . Therefore g takes values in p´1 X U , and we get an induced morphism
g : Z Ñ p´1 X U . Using Exercise 8.7.2, we see that g is unique because g is, and hence p´1
X U
satisfies the universal property of the fiber product.

The following lemma will allow us to construct fiber products in general by gluing.

Lemma 8.6. Let fX : X Ñ S and fY : Y Ñ S be two morphisms and assume that

there is an open cover tUi uiPI of X such that Ui ˆS Y exists for all i P I . Then X ˆS Y
exists. The products Ui ˆS Y form an open cover of X ˆS Y and the projections restrict
to projections.

Proof The proof involves gluing together the different schemes Ui ˆS Y and verifying that

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the result indeed is a product X ˆS Y . Write Uij “ Ui X Uj and Uijk “ Ui X Uj X Uk for

the intersections and let pi : Ui ˆS Y Ñ Ui denote the projections.
By Lemma 8.5 the inverse images p´1 i pUij q serve as fiber products Uij ˆS Y with the re-
strictions of pi and pY as projections. Hence there are unique isomorphisms θji : p´1
i pUij q Ñ
p´1
j pU ij q making the following triangles commute

θji
p´1
i pUij q » p´1
j pUij q
(8.8)
pi pj

Uij

To be able to glue together the p´1

i pUi q’s using the θij ’s, we need to verify the three conditions
of Theorem 4.49 on page 88. The two items (i) and (ii) follow readily. For (iii), note that by
Lemma 8.5 the preimages p´1 i pUijk q serve as products Uijk ˆS Y with the restrictions of pi
and pY as projections. Moreover, the restrictions of the θij ’s fit into a commutative diagram

θji θkj
p´1
i pUijk q » p´1
j pUijk q » p´1
k pUijk q

pj
pi pk

Uijk .

The two small triangles commute, so the big one commutes as well, and it follows by
uniqueness of the isomorphisms that θki “ θkj ˝ θji . The third gluing condition is therefore
fulfilled, and we can glue the p´1
i pUi q’s together to a scheme X ˆS Y . Moreover, in view of
the commutative diagram (8.8) and Proposition 4.50 on page 89, the pi ’s patch together to a
map pX : X ˆS Y Ñ X . The projections Ui ˆS Y Ñ Y are basically unaffected by the
gluing process and glue together to a morphism pY : X ˆS Y Ñ Y .
Finally, we check that X ˆS Y together with pX and pY satisfy the universal property.
Let Z be any scheme with morphisms ϕX : Z Ñ X and ϕY : Z Ñ Y such that fX ˝ ϕX “
fY ˝ ϕY . For each i P I , the restrictions ϕ´1 X pUi q Ñ Ui and ϕY : Z Ñ Y induce a
´1
unique morphism ϕi : ϕX pUi q Ñ Ui ˆS Y by the universal property of Ui ˆS Y . On the
overlaps ϕ´1X pUij q, the morphisms ϕi and ϕj agree by the uniqueness in the universal property.
Therefore, the morphisms ϕi glue together to a (unique) morphism ϕ : Z Ñ X ˆS Y such
that pX ˝ ϕ “ ϕX and pY ˝ ϕ “ ϕY .

An immediate consequence is that fiber products exist when the base S is affine.

Lemma 8.7. Assume that S is affine, then X ˆS Y exists.

Proof First, if Y is affine as well, we are done: cover X by open affine subschemes Ui ;
then each Ui ˆS Y exists by the affine case, and we may apply Lemma 8.6 above. In general,
cover Y by affine open subschemes Vi . As we just verified, the products X ˆS Vi all exist,
and applying Lemma 8.6 again, we conclude that X ˆS Y exists.

For the final reduction, we will need the following lemma.

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Lemma 8.8. If T Ă S is an open subscheme and the fiber product X ˆT Y exists, then
the fiber product X ˆS Y exists and is isomorphic to X ˆT Y .

Proof Let ι : T Ñ S be the inclusion. Let pX and pY be the projection morphisms from
X ˆT Y . For any scheme Z with morphisms ϕX : Z Ñ X and ϕY : Z Ñ Y such that
ι ˝ f ˝ ϕX “ ι ˝ g ˝ ϕY , the condition f ˝ ϕX “ g ˝ ϕY holds because ι is an inclusion
(Exercise 8.7.2). By the universal property of X ˆT Y , there exists a unique morphism
Z Ñ X ˆT Y making the diagram commute. Hence, X ˆT Y satisfies the universal
property of X ˆS Y .
Finally, with Lemmas 8.7 and Lemma 8.8, we can finish the proof of Theorem 8.3.
´1
Proof of Theorem 8.3 Let tSi uiPI be an open affine cover of S and let Ui “ fX pSi q and
´1
Vi “ fY pSi q. By Lemma 8.7 the products Ui ˆSi Vi all exist. By Lemma 8.8 these serve as
products Ui ˆS Vi . Then applying Lemma 8.6 again, we see that the schemes Ui ˆS Vi glue
to the fiber product X ˆS Y and the proof is complete.
Here are some of the basic properties of the fiber product. It is possible to deduce them
directly using gluing arguments, but with the so-called ‘functor of points’, which we will
introduce in Section 12.1, the proofs will become simple and natural.

Proposition 8.9 (Basic formulas). Let X , Y , Z and T be schemes over S . There are
unique canonical isomorphisms over S , all compatible with projections:
(i) (Reflectivity) X ˆS S » X .
(ii) (Symmetry) X ˆS Y » Y ˆS X .
(iii) (Associativity) pX ˆS Y q ˆS Z » X ˆS pY ˆS Zq.
(iv) (Transitivity) pX ˆS T q ˆT Y » X ˆS Y .
In the last claim Y is supposed to be a scheme over T , and X ˆS T is considered a
scheme over T via the projection onto T .

8.3 Examples
As noted in the introduction, the fiber product of schemes can exihibt unexpected behaviour
in some situations, differing from what we are used to in set theory or topology. The main
difference is that the underlying set is almost never the product of the underlying sets of the
factors. The next few examples illustrate this.
Example 8.10. For a ring R and non-negative integers m, n, we have
Rrx1 , . . . , xm s bR Rry1 , . . . , yn s » Rrx1 , . . . , xm , y1 , . . . , yn s,
and so
m`n
Am n
R ˆR A R » A R .
Even when R “ C, the affine space Am`nC has an underlying set which is different from
m n
the Cartesian product AC ˆ AC , and the topology is not equal to the product topology (see
Example 2.17). △

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Example 8.11. Let X Ă Am m

R “ Spec Rrx1 , . . . , xm s and Y Ă AR “ Spec Rry1 , . . . , yn s
be two closed subschemes, defined by the ideals a “ pf1 , . . . , fr q Ă krx1 , . . . , xm s and
b “ pg1 , . . . , gs q respectively. Then X ˆR Y is isomorphic to the closed subscheme of
Am`n
R given by
Spec Rrx1 , . . . , xm , y1 , . . . , yn s{pf1 , . . . , fr , g1 , . . . , gs q
This is the scheme version of the product in Example 1.36. △
Here are three examples of fiber products of spectra of fields:
Example 8.12. A simple but illustrative example is the product Spec C ˆR Spec C. This
scheme has two distinct closed points, even if both factors are singletons. Note also that
the product is not integral, not even connected. So the product of integral schemes is not
necessarily integral.
The tensor product C bR C is in fact isomorphic to the direct product C ˆ C of two copies
of the complex field C. One sees this using that C “ Rrts{pt2 ` 1q, which gives
C bR C “ Rrts{pt2 ` 1q bR C “ Crts{pt2 ` 1q “ Crts{pt ´ iqpt ` iq “ C ˆ C,
where the last equality follows from the Chinese Remainder Theorem and that the rings
Crts{pt ˘ iq both are isomorphic to C. △
Example 8.13. The fiber product Spec F2 ˆZ Spec F3 is empty. Indeed, it is the spectrum
of the ring
Z{2 bZ Z{3 » Z{pgcdp2, 3qq “ 0
See Exercise 8.7.3 for a generalization. △
Example 8.14. The fiber product Spec Cpxq ˆC Spec Cpyq is even more extreme: it has
infinitely many points! This is because the tensor product A “ Cpxq bC Cpyq is a ring of
Krull dimension 1, and it contains infinitely many maximal ideals.
To prove this, we note that the ring A can be written as the localization S ´1 Crx, ys of the
polynomial ring Crx, ys in the multiplicative set
S “ t ppxqqpyq | ppxqqpyq ‰ 0 u.
Therefore, if p Ă A is a prime ideal, it is of the form p “ S ´1 q for some prime ideal
q Ă Crx, ys that does not intersect S . Bearing in mind that the maximal ideals in Crx, ys
are of the form px ´ a, y ´ bq with a, b P C, we find that q is not maximal, and hence of
height of at most 1. We must also have that q X Crxs “ 0 and q X Crys “ 0, and we find
that either q “ p0q or q “ pf px, yqq, where f is an irreducible polynomial neither lying in
Crxs nor in Crys.
In conclusion, all non-zero primes in A are therefore maximal, and so A has dimension 1.
Moreover, A has infinitely many maximal ideals, in fact, uncountably many. △

8.4 Base change

The fiber product construction leads us to one of the most powerful techniques in algebraic
geometry - the ability to change base schemes. This process generalizes the familiar idea of

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considering polynomial equations, such as x2 ` 1, over different rings, but in the geometric
framework of schemes.
To make this more precise, let us consider a scheme X over S with structure morphism
p : X Ñ S . If T Ñ S is a morphism, we can form the fiber product X ˆS T , which
is then naturally a scheme over T . One frequently writes XT for X ˆS T and says that
XT is obtained from X by base change. The scheme XT and the new structure morphism
pT : XT Ñ T fits into the Cartesian diagram

XT X
pT p

T S.
The square in the diagram indicates that the map pT is created by the base change process.
While the ’base change’ is essentially a fiber product, the novelty lies in the terminology
rather than the construction. We start with a scheme over S , and obtain new schemes XT by
changing the base schemes T . This philosophy turns out to be very powerful for studying
schemes and proving properties about schemes and morphisms, as we will see shortly.
Example 8.15 (Field extensions). If X is a scheme over k and k Ă K is a field extension,
then the fiber product XK “ X ˆk K is a scheme over K . XK is the scheme defined by the
same equations as X , but viewed over K instead of k .
For instance, if X “ Spec Rrx, ys{px2 ` y 2 ` 1q, then the base change XC via the field
extension R Ă C is given by
XC “ Spec Rrx, ys{px2 ` y 2 ` 1q
There is a significant change in the geometry: X has no R-points, but XC has infinitely many
C-points. △
Example 8.16. Suppose that X is a scheme over a ? field k and that σ : k Ñ k is a field
?
automorphism.
? For instance, we can consider k “ Qp 2q and σ the map sending a ` b 2
to a ´ b 2. Then σ induces a morphism Spec k Ñ Spec k , and by the fiber product, a new
k -scheme denoted by σX .
σX X

ι
Spec k Spec k

Note that X and σX are isomorphic as abstract schemes, but they need not be isomorphic as
k -schemes. ? ? 2 ?
2
? 2For instance, if X “ Spec Q p 2qrx, ys{px
? ` 2y `1q, then σX “ Spec Q p 2qrx, ys{px2 ´
2y ` 1q. The first scheme has no Qp 2q-valued points, wheras the latter has infinitely
many. △
This is a functorial construction: If f : X Ñ Y is a morphism over S , there is induced a
morphism fT “ f ˆ idT from XT to YT over T , and one checks that pT ˝ fT coincides with
the natural projection map XT Ñ T (or in other words, the outer rectangle in the diagram

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below is Cartesian).
XT fT
YT pT T
pY
f
X Y S.

Properties stable under base change

When studying morphisms of schemes, it is natural to ask how their properties behave
under base change. If P is a property of morphisms, one says that P is stable under base
change if for any T over S , the map fT has the property P whenever the original morphism
: X Ñ Y f does. The same convention applies to properties of schemes.
Examples 8.18 and 8.19 below show that neither being irreducible nor being reduced are
properties stable under base change.
On the other hand, many imporant geometric properties are preserved by base change. For
instance, being a closed or open embedding:

Proposition 8.17 (Embeddings and base change). Let f : Z Ñ X be a morphism and

consider its base change along Y Ñ X :

ZY Z
fY f

Y X.
If the morphism f is a closed, open or locally closed embedding, then the morphism
fY : ZY Ñ Y is as well.

Proof The case of an open embedding is a consequence of Lemma 8.5 on page 162. The
case of locally closed embeddings follow directly from the two others, so it suffices to show
that closed embeddings are stable under base change.
Assume first that X “ Spec A and Y “ Spec B are affine. Then a closed subscheme
f : Z Ñ Spec A is of the form Z “ SpecpA{aq for some ideal a (Proposition 4.30 on
page 76), and therefore

ZY “ Z ˆX Y “ SpecpA{a bA Bq “ SpecpB{aBq.
and fY corresponds to the closed subscheme SpecpB{aBq Ñ Spec B .
For the general case, the statement is local on Y (Exercise 9.9.15 on page 195). For any
open affine U Ă Y mapping into an open affine V Ă X , Lemma 8.8 on page 164, implies
that f ´1 V ˆX Y “ f ´1 V ˆV U , and by the affine case this is a closed subscheme of U .
Finally, one may cover Y by such U ’s by first covering X by affine opens and subsequently
cover each of their inverse images in Y by affine opens.

Example 8.18 (Being irreducible is not stable under base change). Consider the R-algebra
A “ Rrx, ys{px2 ` y 2 q. Over R, the polynomial x2 ` y 2 is irreducible, so X “ Spec A is

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an irreducible R-scheme. The base change to C however is not irreducible, because

A bC C » Crx, ys{px ´ iyq ˆ Crx, ys{px ` iyq
and so XC is the union of two conjugate lines in Spec Crx, ys △
Example 8.19 (Being reduced is not stable under base change). An example was already
given in Exercise 8.7.5. Consider the scheme X “ Spec Zrx, ys{px2 ´ 2y 2 q, viewed as a
scheme over Spec Z. Clearly X is integral, as x2 ´ 2y 2 is irreducible. However, if we take
the base change via the morphism Spec F2 Ñ Spec Z, the resulting scheme is non-reduced:
XF2 “ SpecpF2 rx, ys{px2 qq
△
Example 8.20. For a related ? example,
? consider the polynomial T 4 ´ 10T 2 ` 1, which is
the minimal polynomial of 2 ` 3. This polynomial has the interesting property that it is
irreducible over Q, but its reduction modulo p factors for every prime p. This means that for
the morphism
Spec ZrT s{pT 4 ´ 10T 2 ` 1q ÝÝÑ Spec Z,
the fiber over the generic point is irreducible, but all of the closed fibers are reducible. △

8.5 Scheme-theoretic fibers

Suppose that f : X Ñ Y is a morphism of schemes and that y P Y is a point. One of the
first applications of the fiber product is to define a scheme structure on the preimage f ´1 pyq.
Having the fiber product at our disposal, inspired by part a) of Exercise 8.7.1, nothing is more
natural than defining the fiber to be the fiber product Xy “ Spec κpyq ˆY X . It appears in
the diagram
Xy “ X ˆY Spec κpyq X
f

Spec κpyq Y,
where Spec κpyq Ñ Y is the map corresponding to the point y . Recall that the field κpyq
is given as κpyq “ OY,y {my , and that the ‘point-map’ Spec κpyq Ñ Y is the composition
Spec κpyq Ñ Spec OY,y Ñ Y of the two canonical maps.
Note that the fiber Xy satisfies the following universal property: a morphism g : Z Ñ X
factors through Xy if and only if f ˝ g factors through Spec κpyq Ñ Y (topologically this
means it maps Z to y P Y ).
It is common to write Xy for the scheme-theoretic fiber and reserve the notation f ´1 pyq
for the preimage as a topological space. In any case, the next proposition shows that the
underlying topological space of Xy is equal to f ´1 pyq.

Proposition 8.21. Let X and Y be schemes and f : X Ñ Y a morphism. Let y P Y be

a point. Then the inclusion Xy Ñ X of the scheme-theoretic fiber is a homeomorphism
onto the topological fiber f ´1 pyq.

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Proof We may assume that Y is affine, say Y “ Spec A.

We first treat the case where X is also affine, say X “ Spec B and f : X Ñ Y is induced
by a ring map ϕ : A Ñ B . In this situation Proposition 2.36 states that the fiber f ´1 ppq over
a point p P Spec A is homeomorphic to the spectrum of the ring pB{pBqp . On the other
hand, standard formulas for tensor products give the equality
pB{pBqp “ B bA Ap {pAp “ B bA κppq,
and the Zariski topology on the spectrum SpecpB{pBqp (i.e. the induced topology on
f ´1 ppq) coincides with the Zariski topology on SpecpB bA κppqq (i.e. the topology on the
scheme Xy ), and hence the proposition holds when X is affine.
In the general case, let U be open and affine in X . Denote by ι the inclusion ι : Xy Ñ X ;
that is, the projection X ˆY Spec κpyq Ñ X . According to Lemma 8.5 on page 162, it
holds that U ˆY Spec κpyq “ ι´1 U (equipped with the unique open scheme structure on
the open set ι´1 U ), and clearly ι´1 U “ U X Xy . By the affine case, the two topologies we
examine agree on Xy X U , and as U can be any open affine, the two topologies share a basis
and must be equal.
Example 8.22 (The fiber product is the fiber product). Let f : X Ñ S and g : Y Ñ S be
two morphisms of schemes and let s P S be a point with ‘point map’ ι : Spec κpsq Ñ S .
Denote by h : X ˆS Y Ñ S the structure map, i.e. h “ f ˝ pX “ g ˝ pY . Then the
scheme-theoretic fiber of h is the fiber product of the scheme-theoretic fibers of f and g :
pX ˆS Y qs “ Xs ˆκpsq Ys .
This is immediate, applying associativity and transitivity of the fiber product (formulas (iii)
and (iv) of Proposition 8.9 on page 164):
pX ˆS Spec κpsqq ˆκpsq pY ˆS Spec κpsqq “ X ˆS pY ˆS Spec κpsqq
“ pX ˆS Y q ˆS Spec κpsq.
△
Example 8.23. In Example ??, we showed that any scheme X admits a unique morphism
f : X Ñ Spec Z. This means that X can be viewed as a sort of ‘fibered object’ over Spec Z;
X is the union of the schemes X ˆZ SpecpFp q where p is a prime number and X ˆZ SpecpQq.
Compare this with Figure 2.42. △
Example 8.24. More generally, if f : Y Ñ X is a morphism and Z Ă X is a closed
subscheme, the base change scheme ZY “ Z ˆX Y is called the the scheme-theoretic
inverse image of Z . If k is a field, the k -points in ZY are exactly the k -points that map into
Z. △

8.6 The Segre embedding

In this section, we describe the Segre embedding of a product of two projective spaces Pn pkqˆ
Pm pkq. At the level of k -points, the map defined by taking the products of homogeneous
coordinates:
px0 : ¨ ¨ ¨ : xn q ˆ py0 : ¨ ¨ ¨ : ym q ÞÑ px0 y0 : x1 y0 : ¨ ¨ ¨ : xi yj : ¨ ¨ ¨ : xn ym q.

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Here, there are pn ` 1qpm ` 1q distinct products wij “ xi yj . Scaling the xi ’s or the yj ’s will
scale the products wij accordingly. Moreover, if at least one of the xi ’s and one of the yj ’s are
non-zero, one of the products will be non-zero as well. Therefore we obtain, a well-defined
map of sets

σ : Pn pkq ˆ Pm pkq Pnm`n`m pkq.

Note that twrs ‰ 0u “ txr ‰ 0u ˆ tys ‰ 0u. Indeed, xr ys ‰ 0 precisely when both
xr ‰ 0 and ys ‰ 0.
The map σ is injective, because if say wpq “ xp yq ‰ 0, we recover the coordinates of the
points in Pn pkq and Pm pkq with xp ‰ 0 and yq ‰ 0 respectively, via xi {xp “ wiq {wpq and
yi {yq “ wqi {wpq .
This construction has a natural generalization to schemes.
For i “ 0, . . . , m and j “ 0, . . . , n, consider the rings PmA , let
„ ȷ „ ȷ
x0 xm y0 yn
Ri “ A ,..., and Sj “ A ,...,
xi xi yj yj
Then the fiber product Pm n
A ˆA PA is the scheme obtained by gluing together the affine
schemes Uij “ SpecpRi bA Sj q for all i, j .
The upshot is that the scheme Pm n
A ˆA PA admits a closed embedding into the projective
pm`1qpn`1q´1
space PA .
To see this, let wij be a set of pm ` 1qpn ` 1q variables, where i “ 0, . . . , m and
j “ 0, . . . , n. The projective space is obtained from the rings
„ ȷ
w00 wmn
Tpq “ A ,...,
wpq wpq
Consider the map of A-algebras
„ ȷ „ ȷ „ ȷ
w00 wmn x0 xm y0 yn
A ,..., ÝÝÑ A ,..., bA A ,...,
wpq wpq xp xp yq yq
wij xi yi
ÞÝÑ b .
wpq x p yq
This is surjective, because all the monomials on the right-hand are in the image of a monomial
on the left. Therefore, we get a closed embedding

σpq : SpecpRp q ˆA SpecpSq q ÝÝÑ Spec Tpq .

It is not hard to see that these closed embeddings glue to a closed embedding
pm`1qpn`1q´1
σ : Pm n
A ˆA PA ÝÝÑ PA .

Proposition 8.25. The scheme Pm n

A ˆA PA is projective over A.

pm`1qpn`1q´1
The ideal of the image of σ in PA can be described explicitly as the ideal

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generated by the 2 ˆ 2-minors of the matrix of variables

¨ ˛
w00 w01 ¨ ¨ ¨ w0n
˚ w10 w11 ¨ ¨ ¨ w1n ‹
˚ ‹
˚ .. .. . . .. ‹
˝ . . . . ‚
wm0 wm1 ¨ ¨ ¨ wmn

For a proof of this fact, see ???.

Example 8.26. In the special case R “ Arx0 , x1 s and R1 “ Ary0 , y1 s, the map wij ÞÑ
xi b yj yields an isomorphism

S “ Arw00 , w01 , w10 , w11 s{pw00 w11 ´ w01 w10 q.

In particular, we recover the classical Segre embedding of P1 pkq ˆ P1 pkq as a quadric surface
in P3 pkq. △

P1 ˆ P1 Q Ă P3

Example 8.27. The product P1A ˆA P2A embeds into P5A as the closed subscheme defined by
the 2 ˆ 2-minors of the matrix
ˆ ˙
w0 w1 w2
w3 w4 w5

Just like homogeneous ideals in Arx0 , . . . , xm s give rise to closed subchemes of Pm A , it is

bihomogeneous polynomials in Arx0 , . . . , xm , y0 , . . . , yn s which give rise to subschemes of
Pm n
A ˆA PA . More precisely, a polynomial F P Arx0 , . . . , xm , y0 , . . . , yn s is bihomogeneous
of bidegree pd, eq if all monomials in F have degree d in the xi ’s and degree e in the yj ’s.
If a is an ideal generated by bihomogeneous polynomials, then we can consider the
‘dehomogenization’
app,qq “ t x´d ´e
p yq F | F P a u.

The bigraded ring Arx0 , . . . , xm , y0 , . . . , yn s has the corresponding ‘irrelevant ideal’ defined
by
px0 , . . . , xm q X py0 , . . . , yn q.

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8.7 Exercises
Exercise 8.7.1. Working with the fiber product of sets, show that
a) If Y is a subset of S and fY is the inclusion, then X ˆS Y equals the preimage
fX´1 pY q.
b) X and Y are both subsets of S , the fiber product X ˆS Y will be equal to the
intersection X X Y .
c) When S has one element, X ˆS Y is the usual Cartesian product X ˆ Y .
Exercise 8.7.2. Assume that U Ă X is an open subscheme and let ι : U Ñ X be the
inclusion map. Let f and g be two maps from a scheme Z to U and assume that ι ˝ f “ ι ˝ g .
Show that f “ g .
Exercise 8.7.3. Let p and q be two different prime numbers. Show the following identities:
a) Spec Fp ˆZ Spec Fq “ H.
b) Spec Zppq ˆZ Spec Zppq “ Spec Zppq .
c) Spec Zppq ˆZ Spec Zpqq “ Spec Q.
Exercise 8.7.4. Example 8.12 can be generalized as follows. Let K{k be a finite Galois
extension of fields with Galois group G. Show that the map x b y ÞÑ pxgpyqqgPG defines an
isomorphism
ź
K bk K Ñ K.
gPG

Hint: Write K “ krxs{pf pxqq for a minimal polynomial f pxq and compute K bk K using
the Chinese Remainder Theorem and the fact that f factors in K .
Deduce that Spec K ˆk Spec K has an underlying set with |G| points.
Exercise 8.7.5. This exercise goes along the same lines as Exercise 8.7.4 and gives an
example that a fiber product X ˆk Spec L may not be reduced even if X is.
Let k “ Fp paq for a prime number p and let L “ krxs{pxp ´ aq. Show that
L ˆk L » Lrts{ptp ´ aq » Lrts{pt ´ xqp .
Conclude that Spec L ˆSpec k Spec L is not reduced.
Exercise 8.7.6. Let X and Y be schemes over S with open affine covers tUi u and tVj u.
Show that Ui ˆS Vj is an open cover of X ˆS Y .
Exercise 8.7.7. Let X “ Spec Rrx, ys{px2 ` y 2 q.
a) Show that X is irreducible.
b) Show that X ˆR C is not irreducible, and describe the irreducible components.
c) Compute XpRq and XpCq.
Exercise 8.7.8. a) Show that the base change of a surjective morphism is surjec-
tive.
b) Show that the base change of an injective morphism need not be injective.
H INT: Consider Spec C Ñ Spec R.
Exercise 8.7.9. Let X “ SpecpQrx, ys{xyq and Y “ SpecpQrx, ys{px2 ` y 2 qq.

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a) Compute HomSch pSpec Q, Xq and HomSch pSpec Q, Y q.

b) Show that X and Y are not isomorphic.
c) Let X 1 “ SpecpQpiqrx, ys{xyq and Y 1 “ SpecpQpiqrx, ys{px2 `y 2 qq. Show
that X 1 and Y 1 are isomorphic.
Exercise 8.7.10. a) Show that the map sending σ P GalpQ{Qq to p “ KerpQ bQ
Q Ñ Qq defines a bijection
` ˘
GalpQ{Qq ÝÝÑ Spec Q bQ Q
b) Describe the topological space X “ SpecpQ bQ Qq. Is it connected? Irre-
ducible? Reduced? Noetherian?
Exercise 8.7.11. In the setting above, show that the scheme-theoretic intersection is naturally
a closed subscheme of X , with underlying topological space equal to the intersection ipY q X
jpZq in X .
Exercise 8.7.12. Prove statements (i) and (iv) in Proposition 8.9.
Exercise 8.7.13. Let A “ Rrx, ys{px2 ` y 2 ` 1q and let X “ Spec A. Show that the
base-change XC is isomorphic to A1C ´ V ptq, but X is not isomorphic to A1R ´ V ptq.
Exercise 8.7.14. Show that if B is an A-algebra, then AnB » AnA ˆA Spec B and that
PnB » PnA ˆA Spec B .
Exercise 8.7.15. Let Lm Ñ P1k be the line bundle constructed in Section 6.3 on page 129,
and let fn : P1 Ñ P1 be the map u ÞÑ un . Show that Lm ˆP1k P1k “ Lnm .
Exercise 8.7.16 (Finite type and base change).
a) Show that the properties ‘finite’ and ‘finite type’ are stable under base change.
b) Show that the product of two morphisms of finite type is of finite type. Similarly
for two finite morphisms.
Exercise 8.7.17. Let m “ px1 , . . . , xn q the origin in Ank .
À
(i) Show that the graded k -algebra iě0 mi {mi`1 is isomorphic to the polynomial
ring krt1 , . . . , tn s, where ti denotes the class of xi in m{m2 .
(ii) Let π : X Ñ An be the blow up of the origin 0 in Ank (that is, of m). Show that
X0 “ Pn´1
k .
Exercise 8.7.18 (Flat base change). Let A be a ring and let B be a flat A-algebra. Let X be
a Noetherian scheme over A and let XB denote the base change to B . Show that there is a
natural isomorphism
ΓpXB , OXB q “ ΓpX, OX q bA B (8.9)
Find an example where (8.9) fails for a non-flat A-algebra.
Exercise 8.7.19. Let X be a scheme over a field k and let k Ñ K be a field extension. If
XK denotes the base change to K , show that there is a natural bijection XpKq “ XK pKq.
Exercise 8.7.20. Show that P1C ˆC P1C fi P2C .

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Exercise 8.7.21. Consider the projective schemes X “ V` px2 ` y 2 ` z 2 q and Y “

V` px2 ` y 2 ` 3z 2 q in P2 .
a) Show that XpQq? “ Y pQq “ H. ?
b) Show that XpQp 3qq “ H, but Y pQp 3qq ‰ H. Deduce that X and Y are
not isomorphic.
c) Show that X ˆQ C and Y ˆQ C are isomorphic.
Exercise 8.7.22. Let P and Q be two lattice polytopes. Show that the product polytope
P ˆ Q is a lattice polytope. Show that for a field k , there is a natural isomorphism of toric
varieties
XP ˆQ » XP ˆk XQ
Exercise 8.7.23. Let k be a field with char k ‰ 2.
a) Show that X “ Spec krx, ys{py 2 ´ x2 ´ x3 q is an integral scheme.
b) Show that the base change to krrx, yss, Spec krrx, yss{py 2 ´ x2 ´ x3 q is
reducible. What is the geometric intuition here?
Exercise 8.7.24 (Scheme-theoretic intersections). If X is a scheme and Y, Z are two closed
subschemes we define their scheme-theoretic intersection as the fiber product Y ˆX Z of the
closed embeddings i : Y Ñ X and j : Z Ñ X .
a) Show that when X “ Spec A and Y and Z are closed subschemes given by
ideals I and J respectively, then Y ˆX Z is the closed subscheme associated
to the ideal I ` J .
b) Show that in general, Y ˆX Z is a closed subscheme of X with underlying
topological space is homeomorphic to ipY q X jpZq in X

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Morphisms of schemes

9.1 Properties of affine open subsets

A common theme in scheme theory is to prove that if a property holds for the open sets in a
given affine cover tUi uiPI of a scheme X , then it holds for every affine open set U Ă X . For
instance, we encountered such a situation in our discussion of closed subschemes in section
4.7. In this section, we introduce the notion of distinguished properties, which will allow us
to systematically prove such statements.
The idea is to work with properties that behave well with respect to both localization and
gluing. More precisely, a property P of open affine subschemes of a scheme X is said to be
distinguished if the following two conditions hold:
(D1) If U is an open affine subscheme with property P and g P OX pU q, then Dpgq
also has property P .
(D2) If U is an open affine subscheme covered by a finite collection of distinguished
open sets tDpgi qu, each of which has property P , then U itself has P .
Note that the two conditions (D1) and (D2) mirror the two sheaf axioms. (D1) is analogous
to the Locality axiom, as it ensures that a property holds locally if it holds in U . (D2) is
analogous to the Gluing axiom, as it allows us to patch properties together from a finite cover.

Proposition 9.1. Let P be a property of open affine subschemes of X .

(i) If P is distinguished, and there is one open affine cover tUi uiPI of X so that
each Ui has P , then every open affine in X has P .
(ii) If P satisfies (D1), then P is distinguished provided that (D2) holds for all
covers consisting of two distinguished opens

We first prove a basic lemma, saying that we can cover the intersection of any two affine
open sets U and V by affine open subsets which are distinguished in both U and V .

Lemma 9.2. Let X be a scheme and let U “ Spec A and V “ Spec B be two open
affine subschemes. For a point x P U XV , there exist an open set W Ă U XV containing
x, which is distinguished in both U and V .

Proof As the distinguished open sets form a basis for V , we may find a f P B such that
Dpf q Ă U X V containing x. As a distinguished open set in a distinguished open set is a
distinguished open (Exercise 2.7.37), we may replace V by Dpf q, and hence assume that
V Ă U.
175

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Now the inclusion map ι : V Ñ U is given by a ring map ϕ : A Ñ B . Let a P A

be such that Dpaq Ă V and set W “ Dpaq Ă U . Then by Proposition 2.29, we have
W “ ι´1 pDpaqq “ Dpϕpaqq and so W is distinguished in both U and V .
Proof of Proposition 9.1 Let V Ă X be any affine open set. Then the open sets Ui X V
form an open cover of V . By the previous lemma, we may for each i cover Ui X V by affine
open sets Vij which are distinguished in both V and Ui . Then the collection tVij u forms an
affine cover of V . As V is quasi-compact, we may reduce this to a finite subcover. Now each
of the Ui have property P and hence the Vij have P , by (D1). Therefore, V has property P
by (D2).
For the second statement in the lemma, assume (D2) is fulfilled for covers with two
elements. We prove that (D2) holds in general by induction of the number r of opens in the
given cover tDpgi qu of V .
Because the Dpgi q’s cover V , there is a relation a1 g1 ` ¨ ¨ ¨ ` ar gr “ 1 in OV pV q. Let
g “ a2 g2 ` ¨ ¨ ¨ ` ar gr . Each Dpgi gq with i ě 2 is distinguished in Dpgi q, and hence has
property P by (D1). On the other hand, they are also distinguished in Dpgq and cover Dpgq,
hence Dpgq has P by induction. Now, V is the union of Dpg1 q and Dpgq and hence has P
by the r “ 2 case.
To check that a property P holds for every affine subset of a scheme X , it therefore suffices
to show that P satisfies the two conditions (D1) and (D2). This is often a simpler task, as
the open sets SpecpAf q Ă Spec A are more concrete than general open sets. Moreover, it
allows us to translate geometric questions to algebraic questions about localizations of rings
and modules. We will see several examples of this in the next few sections.
Example 9.3. The property of being reduced is a distinguished property. More generally,
any property that can be checked on stalks is distinguished.
On the other hand, the property of being integral is not a distinguished property (if X is a
disjoint union of two affine schemes, then (D1) holds, but (D2) does not). △

9.2 Morphisms of finite type

In this section we work over a base ring R. The main example to keep in mind is when
R “ k is a field.
Recall that an R-algebra A is called finitely generated, or of finite type over R if there exists
a finite set of elements x1 , . . . , xr P A such that every element f P A can be expressed as a
polynomial in the xi ’s with coefficients in R. Equivalently, A is a quotient of a polynomial
ring Rrx1 , . . . , xn s.
If X is an R-scheme, we say that X is of finite type over R if it admits a finite open cover
tUi u where each Ui “ Spec Ai and each Ai is a finitely generated R-algebra.
The prototypical example of an R-scheme of finite type is X “ Spec A where A is a
finitely generated R-algebra. In fact, the next result implies that every affine R-scheme of
finite type is of this form:

Proposition 9.4. Let X be a scheme of finite type over R. Then for any affine open
U Ă X , the ring OX pU q is a finitely generated R-algebra.

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Proof Consider the following property P of affine open subsets U Ă X : PpU q holds if
OX pU q is a finitely generated R-algebra. We need to check that P is a distinguished property.
The property (D1) is easy: If A “ OX pU q is finitely generated over R, then we may
pick generators x1 , . . . , xn that generate A as an R-algebra, i.e., A “ Rrx1 , . . . , xn s. Then
Af » Rrx1 , . . . , xn , f1 s is also finitely generated as an R-algebra.
For (D2), suppose that f1 , . . . , fr P A is a finite set of elements generating the unit ideal,
and assume ř that Afi is finitely generated over R for each i. Let a1 , . . . , ar P A be elements
such that ai fi “ 1 and pick x1 , . . . , xn P A such that Afi is generated by the xi and f1i
for every i. We claim that A is generated by a1 , . . . , ar , x1 , . . . , xn as an R-algebra.
Given any a P A, we may write a{1 “ bi {fiN P Afi where bi PřRrx1 , . . . , xn s Ă A for
some N P N. By the lemma below, there is a relation of the form i ci fiN “ 1 where each
ci is a polynomial in the ai and xj ’s with integer coefficients. Therefore,
ÿ ÿ
a“a¨1“ ci ¨ afiN “ ci bi .
i i

which shows that a is a polynomial in a1 , . . . , ar , x1 , . . . , xn .

Lemma
řn 9.5. Let S be a ring and a1 , . . . , an , f1 , . . . , fn PřS be elements such that
n N
i“1 ai fi “ 1. Then for each N P N, there is also a relation i“1 ci fi “ 1 where the
ci P S are polynomials with integer coefficients in a1 , . . . , an , f1 , . . . , fn .
ř
Proof Expand p i ai fi q2nN and observe that each term contains some power fim with
m ě N . Regrouping the appropriate terms gives the claim.
Example 9.6. The R-schemes AnR and PnR are of finite type, for any ring R. △
Example 9.7. If X{R is of finite type, then any locally closed subscheme Z Ă X is also of
finite type. In particular, any closed subscheme of PnR is of finite type over R. △
Example 9.8. If R is Noetherian and X is of finite type over R, then X is Noetherian. The
converse here is not true: Spec Q is not of finite type over Z, but it is Noetherian. △
š8 1
Example 9.9. The morphism i“1 Ak Ñ A1k which is the identity on each component, is
not of finite type (even though it is ‘locally of finite type’). △
More generally, a morphism of scheme f : X Ñ S is said to be of finite type if for every
open affine V “ Spec R Ă S , the inverse image f ´1 pV q is of finite type over R.
At first glance, this condition seems difficult to check in practice, as it is required to hold
for every affine V Ă Y . Luckily, we have the following result:

Proposition 9.10. Let f : X Ñ S be a morphism. If there exists an affine cover Si “

Spec Ri of S such that for each i, the scheme f ´1 pSi q is of finite type over Ri , then f is
of finite type.

Proof We consider the property P for an open affine subscheme Spec R Ă S : for ev-
ery open affine Spec A Ă f ´1 Spec R, the algebra A is finitely generated over R. By
assumption, there exist one affine cover of S whose open affines all satisfy P . To prove the
proposition, we need to check that P is a distinguished property.

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(D1): If A is finitely generated over R, then for any g P R, the localization Ag is finitely
generated over Rg .
(D2): Suppose tDpgi qu is a cover of Spec R such that P holds for each Spec Rgi . Let
Spec A be an open affine subscheme of f ´1 Spec R. Then Spec Agi is an open subscheme
of f ´1 Spec Agi . By assumption, each Agi is finitely generated over Rgi . But then it will be
finitely generated over R as well, and we may apply Proposition 9.4 to conclude that A is
finitely generated over R.

Example 9.11. A closed embedding ι : X Ñ Y is of finite type. Indeed, by definition there

is an open affine cover tSpec Ai u of Y so that ι´1 Ui » Spec Ai {ai , and Ai {ai is of finite
type. △

Example 9.12. An open embedding ι : U Ñ X is not of finite type in general. For instance,
the open embedding
ď
Dpti q ÝÝÑ Spec krt1 , t2 , . . . s
tPN

is not of finite type, because the scheme on the left is not quasi-compact (and so cannot be
covered by finitely many affine subschemes).
However, if U is quasi-compact, then an open embedding is of finite type. In that case, for
any open affine Spec A in X , U X Spec A is open in Spec A, and can be covered by finitely
many distinguished open sets Dpgi q “ SpecpAgi q, and each Agi is finitely generated over A
(being generated by gi´1 ). In particular, if X is Noetherian, then any subset is quasi-compact,
so any open embedding ι : U Ñ X is of finite type. △

9.3 Affine morphisms

A morphism f : X Ñ Y is said to be affine if f ´1 pU q is affine for every open affine U Ă Y .
As for finite type morphisms, one can verify that a morphism is affine if the condition holds
over an affine covering:

Proposition 9.13. Let f : X Ñ Y be a morphism and assume that there is an open affine
covering Vi “ Spec Bi of Y such that f ´1 pVi q is affine for every i. Then f is affine.

Proof Consider the following property P of affine subsets V Ă Y : PpV q holds if f ´1 pV q

is affine. We show that P is a distinguished property.
(D1): Let V “ Spec A be an affine open set in Y and assume that f ´1 pV q “ Spec B is
affine. Then Spec B Ñ Spec A is induced by a ring map ϕ : A Ñ B . Then for each f P A,
we have f ´1 pDpf qq “ Dpϕpf qq “ SpecpBϕpf q q is also affine, so (D1) follows.
(D2): Let V “ Spec A be an affine subset of Y satisfying P , so that the preimage
U “ f ´1 pV q is affine. Assume that the distinguished open subsets Dpg1 q and Dpg2 q form a
cover of V with each inverse image Ui “ f ´1 Dpgi q being affine, say f ´1 Dpgi q “ Spec Bi .
As Spec Bi maps into Spec A, we may regard Bi as A-algebras.
We claim that U is affine, so that U “ Spec R, where R “ OX pU q.
We begin with establishing that Bi » OX pU qgi . To this end, consider the sheaf exact

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sequence
α β
0 OX pU q OX pU1 q ˆ OX pU2 q OX pU1 X U2 q. (9.1)

Note that U1 X U2 is a distinguished subset in both U1 and U2 ; it is the spectrum of

pB1 qg2 “ ΓpU12 , OX q “ pB2 qg1 . (9.2)
With the identification (9.2), the above sequence takes the following form:
α β
0 OX pU q B1 ˆ B2 pB1 qg2 “ pB2 qg1 . (9.3)

As usual, the components of the map α are the restriction maps, and the map β sends pa, bq
to the difference a{1 ´ b{1.
Now we localize (9.3) with respect to g1 . Note that both B1 , and B12 are already Ag1 -
modules and so do not change when localized. Hence we obtain the sequence
β
0 OX pU qg1 B1 ˆ pB2 qg1 pB2 qg1

where βp0, bq “ b{1. This is actually a split exact sequence: the map β is surjective, and
b ÞÑ p0, bq defines a splitting of β . Therefore, we get OX pU qg1 » B1 , and by symmetry,
OX pU qg2 » B2 .
Next, consider the canonical morphism θU : U Ñ Spec OX pU q from Corollary 4.19 on
page 73. There is a commutative diagram

U1 U U2
θ U1 θU θ U2

Spec B1 Spec OX pU q Spec B2

where each inclusion is the inclusion of a distinguished open set. As the two outer vertical
morphisms are isomorphisms, and U1 and U2 cover U , we see that θU is an isomorphism.
Example 9.14. Any morphism Spec B Ñ Spec A between affine schemes is affine. △

9.4 Closed embeddings

Closed embeddings are important examples of affine morphisms. Using Proposition 9.13, we
can give a proof of Proposition 4.30 on page 76.

Proposition 9.15. Let X be a scheme and let ι : Y Ñ X be a closed embedding. If

U “ Spec A Ă X is an affine open subscheme, then ι´1 pU q “ Spec B is also affine,
and ι|ι´1 U : ι´1 U Ñ U is induced by a surjective ring map ϕ : A Ñ B .

Proof By definition, ι : Y Ñ X is a closed embedding if there is a covering Ui of X so

that for each i, the preimage ι´1 pUi q Ă Y is affine and ι7Ui : OX pUi q Ñ OY pι´1 pUi qq
is surjective. By Proposition 9.13, we deduce that in fact ι´1 pU q is affine for any affine
U Ă X . Writing U “ Spec A and ι´1 pU q “ Spec B , it remains to check that the ring map
ϕ “ ι7U : A Ñ B is surjective. Note that ϕ is surjective if and only if ϕp is surjective for

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every p P Spec A. But the ϕp ’s are simply the stalk maps ι7y : OX,ιpyq Ñ OY,y . These in turn
must be surjective because they are localizations of the ring maps OX pUi q Ñ OY pι´1 pUi qq,
which are surjective by assumption.

Corollary 9.16. Any closed subscheme of an affine scheme Spec A is isomorphic to one
of the form SpecpA{Iq Ñ Spec A for some ideal I Ă A.

Example 9.17. Open embeddings need not be affine morphisms. For instance, the inclusion
A2k ´ tp0, 0qu Ñ A2k is not affine (Example 4.26). △

9.5 Finite morphisms

Recall that an A-module B is said to be finite over A if it is finitely generated as an A-
module. In other words, there ř is a finite set of elements b1 , . . . , br so that each b is an
A-linear combination b “ ai bi with ai P A.
Even though the names ‘finite type’ and ‘finite’ are similar, the two notions are very
different. To say that B is of finite type is to say that B is a ring quotient of a polynomial ring
Art1 , . . . , tr s, where as B being finite means that B is a quotient module of a free module
Ar of finite rank. So for instance, Crts is of finite type over C, but it is not a finite C-module.

Definition 9.18. A morphism f : X Ñ Y is said to be finite if for every open affine

U “ Spec A Ă Y , the preimage f ´1 pU q is affine, say, f ´1 pU q » Spec B and the
induced ring map A Ñ B makes B a finite A-module.

If Y “ Spec A, we say that an A-scheme X is finite over A if the structure morphism

f : X Ñ Spec A is finite.

Proposition 9.19. Let f : X Ñ Y be a morphism and assume that there is an open affine
covering Vi “ Spec Bi of Y such that f ´1 pVi q is finite over Vi for every i. Then f is
finite.

Proof From Proposition 9.13 we know that f ´1 V “ Spec B for some ring B , and it only
remains to prove that B is a finite A-module. As before, we consider the property P of affine
subsets V “ Spec A Ă Y , that f ´1 pV q is the spectrum of a finite A-module. We check that
P is a distinguished property:
(D1): Clearly f ´1 Spec Ag “ Spec Bg so the first requirement is fulfilled.
(D2): Assume that finitely many Dpgi q’s cover V and that f ´1 Dpgi q “ Spec Bgi with
each Bi a finite modules over Agi . Let tij be generators of Bgi over Agi , which we may
choose to be images of elements bij in B . We contend that the bij ’s generate B over A.
ř
Given an element b P B , it holds that gin b “ j aij bij for some n P N independent of i
and with aij P A. Since the Dpgi q’s cover V , there is relation

1 “ c1 g1n ` ¨ ¨ ¨ ` cr grn ,

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which yields
ÿ ÿ
b“ cj gjn b “ cj aij bij .
j

Example 9.20. Let f : Spec krxs Ñ Spec krys be the map induced by a ring map
ϕ : krys Ñ krxs sending y ÞÑ ppxq, where p is a nonconstant polynomial. Then f is
a finite morphism. Indeed, if ppxq “ xn ` an´1 xn´1 ` ¨ ¨ ¨ ` a0 , then as a krxs-module,

krys » krxsrys{py ´ ppxqq » krxs ‘ krxsx ‘ ¨ ¨ ¨ ‘ krxsxn´1 .

△
Example 9.21. The morphism Spec C Ñ Spec R is finite, while Spec C Ñ Spec Q is not.
△
Example 9.22. Consider the ‘blow-up’ morphism

f : Spec krx, y, zs{px ´ yzq Ñ Spec krx, ys

Then f is not finite, as the ring on the left is not finitely generated as a krx, ys-module. △

To underline the huge difference between the two finiteness conditions of this section,
we observe the following: X is of finite type over a field k simply means it can be covered
by open affine subschemes of the form Spec krt1 , . . . , tr s{a. On the other hand, for X is
to be finite over a field k means that X “ Spec A is affine, and A is a k -algebra of finite
dimension over k . Such a ring A is Artinian and has only finitely many prime ideals all being
maximal. Hence the spectrum Spec A is a finite set, and the underlying topology is discrete.

Example 9.23. For n ě 1, the structure morphisms Ank Ñ Spec k and P1k Ñ Spec k are of
finite type, but not finite. △
Example 9.24. The embedding Spec Ag ãÑ Spec A of a distinguished open subscheme is of
finite type, but typically not finite. For instance, the morphism Spec krx, x´1 s Ñ Spec krxs
is not finite, as krx, x´1 s is not a finite krxs-module. △
Example 9.25. Consider ‘the hyperbola’ X “ V pxy ´ 1q Ă A2k “ Spec krx, ys and
the projection X Ñ A1k “ Spec krxs onto the x-axis, which is induced by the inclusion
krxs Ă krx, x´1 s. The algebra krx, x´1 s is not finite over krxs, as it requires all the
negative powers x´n as generators. However, for any elements a, b of k with ab ‰ 0, the ring
krx, x´1 s is finite over krax`bx´1 s. Indeed, krx, x´1 s is generated by x over krax`bx´1 s,
and x satisfies the monic equation

T 2 ´ a´1 pax ` bx´1 qT ` ba´1 “ 0.

△
Example 9.26 (Noether’s Normalization). Let X “ Spec A be an affine scheme of finite type
over a field k . Then by Noether’s Normalization Theorem (A.33), we may find algebraically

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independent elements x1 , . . . , xn P A such that krx1 , . . . , xn s Ă A is a finite extension. In

particular, there is a finite dominant morphism
f : X ÝÝÑ Ank . (9.4)
△

Properties of finite morphisms

An important fact about finite morphisms is that they have finite fibers. Thus there is an
elegant connection between the algebraic condition of ‘finite module’ and the geometric
condition ‘finite fibers’.

Proposition 9.27. Let f : X Ñ Y be a finite morphism. Then:

(i) f is a closed map.
(ii) If f is dominant, then it is surjective.
(iii) Each scheme-theoretic fiber Xy has an underlying topological space which
is finite and discrete.

Proof (i): As f is affine, and ‘closedness’ of a subset can be checked over an open cover, we
reduce to the case X “ Spec B , Y “ Spec A and f is induced by a ring map ϕ : A Ñ B .
By Proposition 2.29 on page 36, the closure f pV pbqq of the image of a closed subset
V pbq Ă Spec B equals V pϕ´1 pbqq. On the other hand, note that the restriction of f to V pbq
can be identified with the map on spectra induced by A{ϕ´1 pbq Ñ B{b. As this ring map is
both injective and finite (since ϕ is finite), it is integral, and hence the Lying-Over Theorem
(Theorem 9.30 (i) below) implies that f is closed.
(ii): Again we may reduce to the affine case. Note that if B is an A-algebra which is
finite as an A-module, then Bred is an Ared -algebra which is finite as a Ared -module. Hence
replacing ϕred : A Ñ B with ϕred : Ared Ñ Bred , we do not change the induced map on
spectra, and we may assume to the case where A and B are reduced. In that case, f is
dominant if and only if ϕ is injective. But an injective and finite ring map is integral, so we
conclude by the Going-Up theorem (Theorem 9.30 (ii) below)
(iii): If y P Y is a point, choose an affine U “ Spec A containing it. As f is finite,
´1
f pU q “ Spec B is also affine, so we reduce to the case where X and Y are affine, and f
is induced by a ring map A Ñ B , making B into a finite A-module.
In this situation, y corresponds to a prime ideal p Ă A, and it follows that Bp {pBp “
B bA Ap {pAp is a finite vector space over κppq “ Ap {pAp (images of generators persist
being generators). In other words, Bb {pBp is an Artinian ring, and hence its spectrum
Xy “ SpecpBp {pBp q is finite and discrete.
Example 9.28. The converse of Proposition 9.27 does not hold. The open embedding
A1k ´ t0u ãÑ A1k has at most one point in each fiber, but it is not a finite morphism. △
Example 9.29. The morphism f : Spec C Ñ Spec R is finite of degree 2, but of course
there is only a single point in the preimage. △
In order to complete the proof of Proposition 9.27 (ii), we will prove a slightly more
general result, for affine morphisms which are locally of the form Spec B Ñ Spec A where

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A Ă B is an integral ring extension. Recall that a ring extension A Ă B is said to be integral

if any element y P B satisfies some monic polynomial

y n ` an´1 y n´1 ` ¨ ¨ ¨ ` a0 “ 0 (9.5)

with coefficients in A. (See Section A.10 for more background).

Theorem 9.30. Let A Ă B be an integral ring extension and let f : Spec B Ñ Spec A
be the induced map. Then:
(i) (Lying-Over) f is surjective with discrete fibers.
(ii) (Going-Up) f is closed

In concrete terms, f being surjective means that for every prime ideal p Ă A there is a
prime ideal q Ă B such that p “ q X A.

Proof (i): Let p P Spec A be a point. If A Ă B is integral, then so is Ap Ă Bp . We want to

show that the fiber f ´1 ppq “ SpecpBp {pBp q is non-empty. In other words, we need to show
that Bp {pBp is not the zero ring, or equivalently, that pBp ‰ Bp . Assuming the opposite, let
x P p be an element so that x{1 is invertible in Bp . Then there is an element y P Bp such
that xy “ 1 in Bp . As Ap Ă Bp is integral, y satisfies an integral equation of the form (9.5)
with coeffients in Ap . Multiplying this relation by xn´1 shows that y P Ap , contradicting the
fact that x is not invertible in Ap .
The fibers being discrete means that each point q P f ´1 ppq is closed in Spec B . In
other words, if q, q1 P f ´1 ppq and q1 P tqu “ V pqq, then q “ q1 . Replacing A and B by
A{pq X Aq Ă B{q, we reduce to the case where A and B are integral domains and p “ p0q.
If there is a nonzero prime ideal q1 such that q1 X A “ p0q, take any y P q1 . Then y satisfies
an integral equation of the form (9.5) and we may choose n minimal. Note that the equation
(9.5) implies that a0 P q1 X A “ p “ p0q. Therefore, ypy n´1 ` ¨ ¨ ¨ ` a1 q “ 0 P B . As n is
chosen minimal, y n´1 ` ¨ ¨ ¨ ` a1 ‰ 0, contradicting the fact that B is an integral domain.
(ii): Let b Ă B be an ideal. We claim that f pV pbqq “ V pb X Aq. The containment ‘Ă’ is
clear. Conversely, applying (i) to the extension A{pb X Aq Ñ B{b (which is again integral),
we see that the restriction V pbq Ñ V pb X Aq is surjective, as we want.

Example 9.31. Let k be an algebraically closed field and consider the closed subscheme
X “ V py 2 ` P pxqq in A2k “ Spec krx, ys, where P pxq is a polynomial in krxs. Let
π : A2k Ñ A1k denote the projection onto the x-axis (induced by the inclusion krxs Ă krx, ys).
Then the restriction π|X will be finite. Indeed, its algebraic counterpart is the ring extension
krxs Ă krx, ys{py 2 ` P pxqq, and the latter ring has a basis as module over krxs consisting
of 1 and y .
On the other hand, if Y “ V pxy 2 ` P pxqq and P p0q ‰ 0, then π|Y is not a finite
morphism. Indeed, the origin 0 P A1 pkq does not belong to its image, and so π|Y is not
surjective. △

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Proposition 9.32. Let f : X Ñ Y be a finite morphism. Then

dim X ď dim Y
with equality if and only if f is dominant.

Proof We first prove the statement when X and Y are affine, say X “ Spec B , Y “
Spec A, and f is induced by a ring map ϕ : A Ñ B .
We have dim f pXq Ă dim Y (Lemma 7.32). Therefore, we may as well replace Y by
f pXq and assume f is dominant. On the level of rings, this corresponds to replacing A by
A{ Ker ϕ, and ϕ : A Ñ B injective. Then B is an integral extension of A, and f is surjective
by Theorem 9.30. Moreover, dim A “ dim B by Proposition A.19.
For the general case, cover Y by affines tUi uiPI . Then the collection f ´1 Ui form an affine
cover of X . By the affine case and (iii) of Lemma 7.32 we find
dim X “ sup dim f ´1 pUi q ď sup Ui “ dim Y.
i i

Proposition 9.33. Let f : X Ñ Y be a finite surjective morphism of Noetherian integral

schemes. Then:
(i) KpXq is a finite field extension of KpY q.
(ii) There is a nonempty open set V Ă Y such that the fiber f ´1 pyq consists of
exactly d points, where d “ rKpXq : KpY qs.

Proof As f is affine, we may reduce to the affine case where f : Spec B Ñ Spec A
induced by a ring map ϕ : A Ñ B , making B into a finite A-module. By the Generic
Freeness Theorem (Theorem A.54) there is g P A such that Bg is free as an Ag -module, i.e.,
Bg » Adg for some integer d. This implies that KpBg q is a finite extension of KpAg q of
degree d.
Over the open set V “ Dpgq, we have for each p P Spec A,
d
ź
B bA κppq » κppq.
i“1

This implies that f ´1 ppq “ SpecpB bA κppqq consists of exactly d points, as required.

Definition 9.34 (The degree of a finite morphism). Let f : X Ñ Y be a finite morphism

of integral schemes. The degree of the field exension KpY q Ă KpXq is called the degree
of f and is denoted deg f .

Note that if f : X Ñ Y and g : Y Ñ Z are finite morphisms of integral schemes, the

composition g ˝ f is finite and degpg ˝ f q “ deg f ¨ deg g . This follows because rM : Ks “
rM : LsrL : Ks for a tower of field extensions K Ă L Ă M .
Example 9.35. The ’n-th power map’ f : P1k Ñ P1k of Example 14.34 has degree n. Indeed,
locally it is given by the ring map krus Ñ krxs sending u to xn , and kpxq “ kpuqrxs{pu ´
xn q has dimension n as a kpuq-vector space. △

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Example 9.36. If X is a hyperelliptic curve, as discussed in Section 6.4, then the morphism
X Ñ P1k has degree 2. In this case, KpXq is obtained from kpxq by adjoining a square root
y of a polynomial f pxq P krxs, so it is spanned by 1 and y as a kpxq-vector space. △

9.6 Separated schemes

We have seen several examples showing that the topology on schemes behaves very differently
from the usual Euclidean topology. In particular, schemes are essentially never Hausdorff
– the open sets in the Zariski topology are simply too large. Still we would like to find an
analogous property that can serve as a satisfactory substitute, so that we have good properties
such as ‘uniqueness of limits’. This leads to the notion of ‘separatedness’.
The route we take to defining separatedness involves the diagonal morphism. The motiva-
tion comes from the following basic fact from basic topology.

Proposition 9.37. A topological space X is Hausdorff if and only if the diagonal ∆ “

t px, xq | x P X u is a closed subset of X ˆ X (in the product topology).

Proof The diagonal ∆ Ă X ˆ X is closed if and only if the complement X ˆ X ´ ∆ is

open, and with the product topology, this is equivalent to any point px, yq P X ˆ X with
x ‰ y being contained in U ˆ V where U, V Ă X are open and U ˆ V Ă X ˆ X ´ ∆.
But this is equivalent to U X V ‰ ∅.
Even for the affine line X “ A1k over a field, the usual Hausdorff condition does not
hold: any open set will contain the generic point p0q (or even in the context of varieties, two
non-open subsets intersect). On the other hand, the Zariski topology on a product is typically
much finer than the product topology on the underlying sets. For instance, for A1k , we have
A1k ˆk A1k “ A2k , and it makes perfect sense to talk about the subset V px ´ yq Ă A2k of
points on the ‘diagonal’, and this is indeed a Zariski closed subset.
It turns out that the ’diagonal perspective’ gives a completely satisfactory notion of
‘Hausdorffness’ for schemes. In fact, it works for relative schemes X{S as well, and we will
speak of a morphism X Ñ S being separated, rather than the scheme itself.
The freedom to glue schemes together leads to many examples of non-separated schemes,
but they are not commonly encountered in practice. For instance, all affine schemes and
all projective schemes are separated. More importantly, some very nice and advantageous
properties hold only for separated schemes, and this legitimates the notion. For instance, in
a separated scheme, the intersection of two affine subsets is again affine (this is a property
which will be important later on).

Separated schemes
Let X{S be a scheme over S . The diagonal morphism is the morphism
∆X{S : X ÝÝÑ X ˆS X
defined by the two identity maps X Ñ X as components. In other words, the defining
property of ∆X{S is that pi ˝ ∆X{S “ idX for i “ 1, 2 where the pi ’s denote the two
projections.

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The following little lemma gives intuition for the diagonal morphism. In particular, it
tells us that if K is a field and x1 , x2 P XpKq are two K -points, the induced K -point
x1 ˆ x2 : Spec K Ñ X ˆS X factors via the diagonal precisely whenever x1 “ x2 .

Lemma 9.38. A morphism f : Z Ñ X ˆS X factors via the diagonal if and only if

p1 ˝ f “ p2 ˝ f .

Proof If f factors, the equality holds by definition of the diagonal. If the equality holds,
we put g “ p1 ˝ f : Z Ñ X , and the uniqueness part of the universal property gives that
∆X{S ˝ g “ f .

In the case that X and S are affine schemes, say X “ Spec B and S “ Spec A, the
diagonal has a simple and natural interpretation in terms of algebras; it corresponds to the
most natural map, namely the multiplication map:

µ : B bA B ÝÝÑ B.

Here µ sends b b b1 to the product bb1 , and we extend it to B bA B by A-linearity. The

projections correspond to the two maps of A-algebra βi : B Ñ B bA B that send B to b b 1
respectively to 1 b b. Clearly it holds that µ ˝ βi “ idB for i “ 1, 2. On the level of schemes
this translates into the defining property of the diagonal map. Moreover, µ is clearly surjective
so it induces a closed embedding. We have shown the following:

Proposition 9.39. If X is an affine scheme over the affine scheme S , then the diagonal
morphism ∆X{S : X Ñ X ˆS X is a closed embedding.

The conclusion here is not generally true for schemes, and we will see simple counterex-
amples shortly. However, from the proposition we just proved, it follows readily that the
image ∆X{S pXq is always locally closed, i.e. the diagonal is locally a closed embedding:

Proposition 9.40. The diagonal ∆X{S is locally a closed embedding.

Proof Begin with covering S by open affine subsets and subsequently cover each of their
inverse images in X by open affines as well. In this way one obtains a cover of X by affine
open subsets Ui whose images in S are contained in affine open subsets Si . The products
Ui ˆSi Ui “ Ui ˆS Ui are open and affine, and their union is an open subset containing the
image of the diagonal. By Proposition 9.39 above the diagonal restricts to a closed embedding
of Ui in Ui ˆSi Ui .

With this in place, we are ready to give the general definition of separatedness:

Definition 9.41. One says that the scheme X{S is separated over S , or that the structure
map X Ñ S is separated, if the diagonal map ∆X{S : X Ñ X ˆS X is a closed
embedding. One says for short that X is separated if it is separated over Spec Z.

Recall that being a closed embedding is a local property on the target. Translating this to

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the case of ∆X{S , a morphism f : X Ñ S is separated if and only if for some open cover
tSi u of S it holds that all the restrictions f ´1 pSi q Ñ Si are separated.
Since ∆X{S is a locally closed embedding, it suffices to check that the image ∆X{S pXq is
a closed subset of X ˆS X . In particular, this means that being separated is a condition that
only involves the underlying topological part of the morphism f : X Ñ S .
Example 9.42. Any morphism Spec B Ñ Spec A of affine schemes is separated, by
Proposition 9.39. This, together with the above paragraph, shows that any affine morphism
f : X Ñ Y is separated (Exercise 9.8.4). △
Example 9.43. Open embeddings are separated: if U Ă X is an open set, then U ˆX U » U
and under this identification, the diagonal map ∆U {X : U Ñ U is simply the identity map.
△
Example 9.44. The affine line X with two origins constructed in Section ?? on page ?? is not
separated over Spec k . Recall that X is constructed by gluing two copies U1 , U2 of the affine
line A1k “ Spec krus along their common open subset Spec kru, u´1 s. Let g1 : U1 Ñ X
and g2 : U2 Ñ X denote the two open embeddings. The scheme X has two ‘origins’, which
are the images 01 and 02 of the origin 0 P A1k under respectively g1 and g2 .
To see that X is not separated, it is instructive to study what happens with the diagonal.
The scheme X ˆk X is glued together by four affine charts Ui ˆk Uj for i, j P t1, 2u, each
isomorphic to A2k . Each A2k contains an origin, so there are four origins in total. These are the
images 0i ˆ0j of p0q P A1k under the four embeddings gij : A1k Ñ X ˆk X with components
gi and gj . Away from the origin, these maps coincide and agree with the diagonal map.
By Lemma 9.38, only 01 ˆ 01 and 02 ˆ 02 lie on the diagonal. However, all four lie
in the closure of the diagonal. Consider 01 ˆ 02 , for instance, which lies in the image of
´1
the map g12 . If V is an open subset containing 01 ˆ 02 , the inverse image g12 V will be a
1 1
non-empty open subset of Ak , and hence must intersect Ak ´ t0u. This means that V must
intersect g12 pA1k ´ t0uq, which is open in the diagonal. Thus, 01 ˆ 02 lies in the closure of
the diagonal.

X X ˆk X

Heuristically, the maps gij agree on A1 ´ t0u, but they differ at the origin, bridging the gap
differently by passing through distinct points 0i ˆ 0j . All four lie in the closure, but the
diagonal itself only passes through 01 ˆ 01 and 02 ˆ 02 . △

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Example 9.45. An even more basic example of a scheme that is not separated is obtained by
gluing the prime spectrum of a discrete valuation ring to itself along the generic point.
To give more details, let R be a DVR with fraction field K . Then Spec R “ tx, ηu where
x is the closed point and η is the generic and open point. By the Gluing Lemma for schemes
(Proposition 4.49 on page 88), we may glue two copies of Spec R together by identifying
the generic points; that is, the open subschemes Spec K in the two copies.
In this manner we construct a scheme ZR together with two open embeddings gi : Spec R Ñ
ZR . They send the generic point η to the same point, which is an open point in ZR , but they
differ on the closed point x. It follows ZR is not separated.
The similar-looking examples of Examples 4.51 and 4.52 are separated however, because
they are affine.

Spec R
x
η x

y η
y
η ZR
Spec R

9.7 Properties of separated schemes

We introduce separatedness mostly because they give good formal properties. In some sense
the schemes category is still a little bit ‘too large’, and separated schemes have properties
that make them closer to varieties. In this section we survey a few of these properties.

Intersection of affine open subsets

Proposition 9.46. Let X be an A-scheme. The following are equivalent:

(i) X is separated over Spec A.
(ii) For any two affine open subschemes U and V , the intersection U X V is
also affine, and the natural multiplication map
OX pU q bA OX pV q ÝÝÑ OX pU X V q
is surjective.
(iii) There is an affine cover tUi uiPI such that all intersections Ui X Uj are affine,
and OX pUi q bA OX pUj q Ñ OX pUi X Uj q is surjective for every i, j P I .

Proof (i)ñ (ii): The product U ˆS V is an affine open subset of X ˆS X , and U X V “

∆X{S pXqXpU ˆS V q. If X is separated, the diagonal is closed, U XV is a closed subscheme
of the affine scheme U ˆS V , hence affine (Proposition 4.30). By the construction of the

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fiber product of affine schemes, we have

ΓpU ˆS V, OU ˆS V q “ ΓpU, OU q bA ΓpV, OV q,
and as U X V is a closed subscheme of U ˆS V , the restriction map
ΓpU ˆS V, OU ˆS V q ÝÝÑ ΓpU X V, OU XV q
is surjective, which gives (ii).
The implication (ii)ñ(iii) is clear, so let us show the last implication (iii)ñ(i). Let
p1 , p2 : X ˆA X Ñ X denote the two projections and let ∆ : X Ñ X ˆA X denote
the diagonal morphism over Spec A. Let Ui “ Spec Bi and Uj “ Spec Bj be two affines
in the cover tUi u. Then we have
∆´1 pp´1 ´1
1 pUi q X p2 pUj qq “ ∆
´1 ´1
pp1 pUi qq X ∆´1 pp´1
2 pUj qq “ Ui X Uj , (9.6)
Also, by the universal property of the fiber product it follows that X p pUj q “ p´1
1 pUi q
´1

Ui ˆA Uj Ă X ˆA X , and from this we deduce that ∆ is a closed embedding if each

restriction
∆ij : Ui X Uj Ñ Ui ˆA Uj
of ∆ is a closed embedding. But this follows from the assumptions: the intersection Ui XUj is
affine, say Ui X Uj “ Spec Cij , and the ring homomorphism Bi bA Bj Ñ Cij is surjective.
This completes the proof.
Example 9.47. The above proposition provides us with a convenient criterion to check that
a scheme is separated. For instance, projective space PnA is separated over A, because it is
covered by the standard affine covering, and the multiplication map
„ ȷ „ ȷ „ ȷ
x0 xn x0 xn x0 xn xi
A ,..., bA A ,..., ÑA ,..., ,
xi xi xj xj xi xi xj
is surjective (all monomials on the right hand side are products of monomials on the left).
In fact, one can argue directly that the diagonal in PnA ˆA PnA is a closed embedding. In the
notation of Section 8.6, the diagonal is the closed subscheme defined by the bihomogeneous
ideal generated by the 2 ˆ 2-minors of the matrix
ˆ ˙
x0 x1 . . . xn
(9.7)
y0 y1 . . . y n
△
Example 9.48. More generally, Proj R is a separated scheme over R0 . The argument is
similar: Proj R is covered by the affine open sets D` pf q where f runs over the homogeneous
elements of R` , and each intersection D` pf q X D` pgq “ D` pf gq is also affine. To prove
that it is separated, it suffices to check that the multiplication map pRf q0 bR0 pRg q0 Ñ
pRf g q0 is surjective for any f, g P R` , which is the case. △
Example 9.49 (The affine plane with two origins). Here is a non-separated scheme where
two affine open subsets have non-affine intersection. We glue two copies of the affine plane
A2k together along the complement U12 “ A2k ´ V px, yq of the origin. If U1 and U2 denote
the two open embeddings of the affine plane, then U1 X U2 “ U12 , but the open set U12 is not

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affine (see the example in Section 4.26 on page 75). In this example, the multiplication map
in the proposition coincides with krx, ys b krx, ys Ñ ΓpU12 , OU12 q, which is surjective. △

Extensions of rational maps

A very useful property of separated schemes is that morphisms into separated schemes are
determined on open dense subschemes, at least when the source is reduced. This can be
viewed as an analogue of the fact that ‘limits are unique’ in Hausdorff spaces.

Proposition 9.50. Let X and Y be two schemes over S and let f, g : X Ñ Y with be
two morphisms over S . Assume that
(i) X is reduced, and
(ii) Y is separated over S .
Then if there is a dense open subscheme U Ă X such that f |U “ g|U , then f “ g .

Proof In order to prove that f “ g , we may assume that X is affine, say X “ Spec A. The
two morphisms f and g induce a morphism pf, gq : X Ñ Y ˆS Y . We want to show that H
factors through the diagonal Y Ñ Y ˆS Y , as this will imply that f “ g by Lemme 9.38.
Consider the pullback of the diagonal ∆Y {S via pf, gq. This fits into the following Cartesian
diagram:

E Y
j ∆Y {S

ι
U X pf,gq
Y ˆS Y

The scheme E is the ’equalizer’ of the two morphisms, and j : E Ñ X represents the sub-
scheme of points in X where the morphisms coincide. Since pullbacks of closed embeddings
are closed embeddings, j : E Ñ X is a closed subscheme of X . Since X is assumed to be
affine, Proposition 4.30 on page 76 implies that E is isomorphic to a subscheme of the form
SpecpA{aq for some ideal a.
The assumption f |U “ g|U implies that there exists a lift U Ñ E of ι. Hence the image
jpEq contains the dense set U and therefore jpEq “ X . By Lemma 2.4, a is contained in
the nilradical of A, which is zero as A is reduced. Consequently, j is an isomorphism, H
factors through the diagonal, and it follows that f “ g .

Here are two examples demonstrating that the hypotheses in the proposition are necessary.

Example 9.51. For the affine line with two origins, X , the two embeddings g1 : A1k Ñ X
and g2 : A1k Ñ X agree over a dense open set, but they are not equal. △

Example 9.52. Consider the non-reduced scheme X “ Spec krx, ys{py 2 , xyq and the two
morphisms f, g : X Ñ Spec krus, defined by u ÞÑ x and u ÞÑ x ` y respectively. These
agree over the distinguished open set Dpxq Ă X , but they are not equal. △

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Formal properties

Proposition 9.53.
(i) (Embeddings) Locally closed embeddings are separated, in particular open
and closed embeddings are.
(ii) (Composition) Let f : X Ñ S and g : Y Ñ X be morphisms. If both f
and g are separated, the composition f ˝ g is separated as well. Moreover, if
Y is separated over S , it is separated over X .
(iii) (Base change) Being separated is a property stable under base change: if
f : X Ñ S is separated and T Ñ S is any morphism, then fT : XT Ñ T
is separated.
(iv) If f : X Ñ Y is a morphism, and tUi uiPI is a cover of Y so that each
f ´1 pUi q Ñ Ui is separated, then f is separated.

Proof (i): Closed embeddings are affine, hence separated. Open embeddings are separated
by Example 9.43. A locally closed embedding is the composition of an open and and closed
embedding, so this case follows from (ii).
(ii): Consider the diagram
∆Y {X
Y Y ˆX Y Y ˆS Y

∆X{S
X X ˆS X

The square in the diagram is Cartesian. As the morphism Y ˆX Y Ñ Y ˆS Y is the base

change of ∆X{S , which is a closed embedding, it is itself a closed embedding. Therefore,
∆X{S , which equals the composition Y Ñ Y ˆX Y Ñ Y ˆS Y , is also a closed embedding.
For the second part of the claim, if the composition Y Ñ Y ˆX Y Ñ Y ˆS Y is a closed
embedding, then also the first map, ∆Y {X , must be a closed embedding, by Exercise 9.8.12.
(iii): There is a natural isomorphism

XT ˆT XT “ pX ˆS T q ˆT pX ˆS T q “ pX ˆS Xq ˆS T
Under this identification, the diagonal morphism ∆XT {T is given by the map X ˆS T Ñ
pX ˆS Xq ˆS T , that is, the pullback of ∆X{S along T Ñ S . As ∆X{S is a closed
embedding, and closed embeddings are stable under base change, we find that ∆XT {T is a
closed embedding as well.
(iv): This was proved in the paragraph following Definition 9.41.

Example 9.54. Combining Proposition 9.53 with Example 9.47 we see that any projective
subscheme of PnA is separated over A. △

9.8 Exercises
Exercise 9.8.1. Let X be the gluing of two copies of Spec Z along the open set U “
Spec Z ´ tp2qu. Show that X is not separated and describe the diagonal map.

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Exercise 9.8.2. Let X{S be a scheme over S and let U Ă X be an open set. Show that
∆X{S |U “ ∆U {S .
Exercise 9.8.3. Let X “ Spec C and S “ Spec R. Recall that the product X ˆS X
consists of two (closed) points. Which one corresponds to the diagonal? Can you find another
R-algebra A so that if Y “ Spec A it holds that Y ˆS Y » X ˆS X and the diagonal is
the other point?
Exercise 9.8.4. Show that affine morphisms are separated.
Exercise 9.8.5. Let X and Y be schemes separated over a scheme S . Show that their product
X ˆS Y is separated over S .
Exercise 9.8.6. Let X be a separated scheme. Show that, for any affine opens U1 , . . . , Um Ď
X , U1 X ¨ ¨ ¨ X Um is affine.
Exercise 9.8.7. Show that if a scheme X is separated (over Z), then for every scheme Y and
every morphism f : X Ñ Y , the morphism f is separated.
Exercise 9.8.8. Let T Ñ S be a morphism and let X and Y be two schemes over T . Show
that there is a Cartesian diagram
ι
X ˆT X X ˆS X
f ˆf
∆T {S
T T ˆS T,
and conclude that the natural map ι : X ˆT Y Ñ X ˆS Y is a locally closed embedding.
Hint: Use the functor of points to reduce to a statement of sets.
Exercise 9.8.9 (Pullback of diagonals). Let X Ñ S and T Ñ S be morphisms between
schemes, and let XT “ X ˆS T . Show that the diagonal ∆X{S pulls back to the diagonal
∆XT {T ; in other words, that there is a canonical Cartesian square
∆XT {T
XT XT ˆT XT

∆X{S
X X ˆS X.
Exercise 9.8.10. Let X{S be a scheme and let ι : W Ñ X be an open subscheme or a
closed subscheme (over S ). Show that the diagram below is Cartesian
W X
∆W {S ∆X{S

W ˆS W X ˆS X
Conclude that W {S is separated if X{S is.
Exercise 9.8.11 (The graph of a morphism). Let S be a scheme and let f : X Ñ Y be a
morphism over S . Assume that Y is separated over S . We define the graph Γf as the map
X Ñ X ˆS Y with components idX and f .

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a) Show that Γf is the pullback of the diagonal ∆Y {S under the morphism idX ˆ
f : X ˆS Y Ñ X ˆS Y .
b) Show that Γf is a closed embedding.
c) Suppose that X and Y are integral. Show that Γf (with the reduced scheme
structure) is isomorphic to X .

Exercise 9.8.12 (Closed embeddings). Let f : X Ñ Y and g : Y Ñ Z be morphisms of

schemes.
a) Assume that g is separated. Show that if the composition g ˝ f is a closed
embedding, then f is a closed embedding. H INT: Consider the diagram
Γf
X X ˆZ Y Y
g

X g˝f
Z

where the square is Cartesian and Γf is the graph of f .

b) Show by an example that in general f is not necessarily a closed embedding
even if g ˝ f is. H INT: Let X be the affine line with two origins and consider
a morphism a morphism X Ñ A1 .

Exercise 9.8.13 (Equalizers). Let X and Y be schemes over S and f1 , f2 : Y Ñ X two

morphisms over S . Define f : Y Ñ X ˆS X be the morphism whose components are the
fi ’s; that is, fi “ πi ˝ f (as usual, the πi ’s are the two projections). The equalizer of the fi is
defined as the pullback of the diagonal ∆X{S along f . In other words, the diagram below is
Cartesian:
η
E Y
f
∆X{S
X X ˆS X.

a) Show that a morphism g : Z Ñ Y satisfies f1 ˝ g “ f2 ˝ g if and only if g

factors via η .
b) Show that X is separated if and only if for every pair of morphisms f1 , f2 : Y Ñ
X , the equalizer η : E Ñ Y is a closed embedding.
c) Describe the equalizer of the two inlusions ι1 , ι2 : A1 Ñ X into the affine line
with two origins.

Exercise 9.8.14 (Monomorphisms). Let f : X Ñ Y be a morphism of schemes. We say that

f is a monomorphism if the composition function
HompT, Xq ÝÝÑ HompT, Y q
is injective defined by α ÞÑ f ˝ α is injective for any scheme T . That is, if g1 : T Ñ X and
g2 : T Ñ X are morphisms such that f ˝ g1 “ f ˝ g2 , then g1 “ g2 .
a) Prove that if f is a closed embedding or an open embedding then it is a monomor-
phism.

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b) Prove that f is a monomorphism if and only if the diagonal ∆ : X Ñ X ˆY X

of the morphism f is an isomorphism.
c) Prove that if f is a monomorphism, then f is separated.
Exercise 9.8.15. Let X be scheme. We say that a scheme X has affine diagonal if for every
affine opens U and V of X the intersection U X V is affine.
a) Prove that X has affine diagonal if and only if the diagonal morphism ∆ : X Ñ
X ˆSpec Z X is an affine morphism. In particular, separated schemes (over
Spec Z) have affine diagonal.
b) Find a scheme with affine diagonal which is not separated.

9.9 Exercises
Exercise 9.9.1. Show that the sections of OSpec A over an open set U Ă X “ Spec A, are
given by the inverse limit of the localizations
OX pU q “ lim
ÐÝ OpDpf qq “ lim
ÐÝ Af . (9.8)
Dpf qĂU Dpf qĂU

Exercise 9.9.2. Let A “ krx, y, zs{pxyzq and X “ Spec A. Compute OX,p where x
corresponds to the prime ideal p “ px ´ 1, y, zq. Show that yz ‰ 0 in OX,p , but takes the
value 0 for all points in a neighbourhood of p.
Exercise 9.9.3. Show that if f : X Ñ Y is a morphism of locally ringed spaces, the stalk
maps fx7 : OY,f pxq Ñ OX,x induce maps between the residue fields κpf pxqq and κpxq. What
happens when X and Y are affine varieties?
Exercise 9.9.4. Let X “ Spec Z. Compute XpFp q, XpQq and XpCq.
Exercise 9.9.5. Show that Spec Qrxs and Spec Z are homeomorphic, but not isomorphic as
schemes.
Exercise 9.9.6. Is Spec Q Ñ Spec Z a closed embedding?
Exercise 9.9.7. Verify the claim about XpQq in Example 4.40. H INT: Compute the second
intersection point a general line trough p0, 1q has with the unit circle.
Exercise 9.9.8. With reference to Example 4.40, show that one may interpret XpQq as the
set of Pythagorean triples:
XpQq “ t pa, b, cq P Z3 | a2 ` b2 “ c2 and a, b, c relatively prime u.
Exercise 9.9.9. With reference to Example 4.40, let p be a prime such that p fl 1 mod 4.
Show that the description in Example 4.40 also is valid for XpFp q.
Exercise 9.9.10. With reference to Example ??, consider the natural inclusion
A “ Rru, vs{pu2 ` v 2 ` 1q Ă Cru, vs{pu2 ` v 2 ` 1q “ AC .
For each point z “ pa, bq P XpCq consider the ideal nz “ mz X A. Show that nx is maximal
and that nz “ nw if and only if w “ pā, b̄q with z “ pa, bq. Conclude that A has infinitely
many maximal ideals.

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Exercise 9.9.11. Let X be a scheme and let x P X be a point.

a) Show that there is a canonical morphism
f : Spec OX,x ÝÝÑ X
b) Show that f induces a homeomorphism between SpecpOX,x q and the subset
W Ă X of points w P X such that w P x.
c) Show that the map ιx : Spec κpxq Ñ X defined in the text factors via f .
d) Show that on the level of topological spaces, the image of f is the intersection
of all open neighbourhoods containing x.
e) Compute the image of f when:
(i) x is the generic point of an irreducible scheme.
(ii) x is a closed point of A2C .
Exercise 9.9.12. Deduce Theorem 4.32 from Theorem 4.21.
n
Exercise 9.9.13. An inclusion Q Ă Q induces a morphism AQ Ñ AnQ . Compute the images
2 2
of the following points under the morphism AQ Ñ AQ :
? ?
a) px ´ 2, y ´ 2q
2
?´ ω, y?´ ω q where ω is a cube root of unity.
b) px
c) p 2x ` 3yq
Exercise 9.9.14. Let pf, f 7 q : X Ñ Y be a morphism of locally ringed spaces. Show that
pf, f 7 q is an isomorphism if and only if f is a homeomorphism and the sheaf map f 7 is an
isomorphism (that is, fU7 is an isomorphism for every open set U Ă Y ).
Exercise 9.9.15. Show that being a closed embedding is a property which is ‘local on the
target’. In clear text: given a morphism f : Z Ñ X and an open cover tUi u of X . Let
Vi “ f ´1 Ui and assume that each restriction f |Vi : Vi Ñ Ui is a closed embedding. Prove
that then also f is a closed embedding.
Exercise 9.9.16. Show that being a locally closed embedding is ‘local on the image’. Assume
that f : Z Ñ X is a morphism and that tUi u is a collection of open subsets of X covering
the image f pZq. Assume further that each restriction f |f ´1 Ui : f ´1 Ui Ñ Ui is a closed
embedding, then f is a locally closed embedding.
Exercise 9.9.17. Let f : X Ñ Y and g : Y Ñ Z be two morphisms of schemes. Prove that
if both f and g are closed embeddings, then g ˝ f is one as well.
Exercise 9.9.18. Let f : X Ñ Y be a morphism which is both an open embedding and a
closed embedding. Show that f is an isomorphism.
Exercise 9.9.19. Consider the ring R “ Zrts and let X “ Spec R.
a) For a prime number p, show that m “ pt, pq is a maximal ideal of R.
b) Let U “ X ´ tmu. Show that U “ Dppq Y Dptq and that
OX pU q “ Zrts
c) Deduce that U is not affine.

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Exercise 9.9.20. Let X “ ta, b, cu be a set with three elements. Let X have the coarsest
topology so that the two subsets U “ ta, bu, and V “ ta, cu are open. Define a presheaf
OX by OX pU q “ OX pV q “ Crtsptq and OX pU X V q “ Cptq with the restriction map
given by the inclusion Crtsptq Ă Cptq.
a) Show that OX is a sheaf.
b) Show that pX, OX q is a scheme.
c) Show that pX, OX q is not affine.
Exercise 9.9.21. Let X be a scheme.
a) Show that any irreducible and closed subset Z Ă X has a unique generic point.
H INT: Reduce to the affine case.
b) Show that in general schemes are not Hausdorff. What are the possible underly-
ing topologies of affine schemes that are Hausdorff?
c) Show that X satisfies the zeroth separation axiom (they are T0 ); that is, given
two points x and y in X , there is an open subset of X containing one of them
but not the other.
Exercise 9.9.22. a) Show that HomRings pZ, Qq has only one element.
b) Define continuous maps Spec Q Ñ Spec Z, sending p0q to ppq. Is it possible
to make these into morphisms of schemes?
Exercise 9.9.23 (The sheaf of units). Let X be a scheme with structure sheaf OX . We say
that s P OX pU q is a unit if there exists a multiplicative inverse s´1 P OX pU q.
a) Show that s P OX pU q is a unit if and only if for all x P U , the germ sx is a
unit in the ring OX,x ; that is, if and only if sx does not lie in the maximal ideal
of OX,x .
ˆ ˆ
b) We let OX pU q denote the subgroup of units in OX pU q. Show that OX pU q is a
subsheaf of OX .
Exercise 9.9.24. In the same vein as Example 2.33, show that a ring A is a Q-algebra (that is,
it contains a copy of Q) if and only if the canonical map Spec A Ñ Spec Z factors through
the generic point Spec Q Ñ Spec Z.
Exercise 9.9.25. For every ring A, there is a canonical map Z Ñ A which sends 1 to 1.
Hence there is a canonical map Spec A Ñ Spec Z. Show that map factors through the
canonical map Spec Fp Ñ Spec Z if and only if A is of characteristic p.
Exercise 9.9.26 (The Frobenius morphism). Let p be a prime number and let A be a ring of
characteristic p. The ring map FA : A Ñ A given by a ÞÑ ap is called the Frobenius map on
A.
a) Show that FA induces the identity map on Spec A.
b) Show that if A is local, then FA is a map of local rings.
c) For a scheme X over Fp , define the Frobenius morphism FX : X Ñ X by the
identity on the underlying topological space and with FX7 : OX Ñ OX given
by g ÞÑ g p . Show that FX is a morphism of schemes.
d) Show that FX is natural in the sense that if f : X Ñ Y is a morphism of
schemes over Fp , we have f ˝ FX “ FY ˝ f .

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In particular, this exercise shows that for a morphism of schemes f : X Ñ Y , in order to

check that f is an isomorphism, is not enough to check that f is a homeomorphism; also the
map f 7 must be an isomorphism.
Exercise 9.9.27. Let X be an integral scheme over a ring A. Let f P KpXq and let Uf Ă X
be the open set of points x P X such that f P OX,x . Show that there is a morphism
ϕ : Uf Ñ A1A such that ϕ7 : Arts Ñ ΓpUf , OX q is given by t ÞÑ f .
Exercise 9.9.28. Prove Proposition 7.5. That is, prove that a scheme X is integral if and only
if OX pU q is an integral domain for each open U Ă X .
Exercise 9.9.29. Let X “ Spec krx, y, z, ws{pxw ´ yzq and consider the open set U “
X ´ V px, yq. Use the above strategy as in Example 4.26 to compute OX pU q. Conclude that
U is not affine.
Exercise 9.9.30. Prove that a composition of two closed embeddings is a closed embedding.
Exercise 9.9.31. Let X be a scheme so that the underlying topological space is finite and
discrete. Show that X is an affine scheme. H INT: Consider the case where X has one point
first.
Exercise 9.9.32. Prove that a morphism f : X Ñ Y is a closed embedding if and only if f
induces a homeomorphism from X onto a closed subset of Y , and for each x P X , the map
of local rings OY,f pxq Ñ OX,x is surjective. H INT: Consider the stalks of f˚ OX . You need
to use the fact that f is a homeomorphism onto f pXq.
Exercise 9.9.33. Describe the following schemes and the structure sheaf on them.
a) Spec Crts{pt2 ` 1q
b) Spec Rrts{pt3 ´ t2 q
c) Spec F3 rts{pt3 ´ 1q.
Exercise 9.9.34. Let K be a finite field extension of Q and let X “ Spec K . Show that
HomSch pX, Xq can be identified with the Galois group GalpK{Qq.
Exercise 9.9.35. Let A be a ring and consider a morphism g : Am n
A Ñ AA given by poly-
nomials g1 , . . . , gn P Arx1 , . . . , xm s. Consider the morphism f : AA Ñ Am`n
m
A given by
px1 , . . . , xn , g1 , . . . , gn q. Show that f is a closed embedding.
Exercise 9.9.36. Show that the Spec-functor preserves inverse limits of rings. That is, if
tRi uiPI is a directed system of rings, then
Specplim
ÝÑ Ri q “ limÐÝ Spec Ri .
H INT: The Spec-functor is a right adjoint to the global sections functor.
Exercise 9.9.37. Let A be a ring and let M be an A-module. Show that for each f P A,
there is a natural isomorphism of sheaves on Dpf q “ SpecpAf q

M
Ăf “ M
Ă|Dpf q .

Exercise 9.9.38. Describe the schemes Spec A for

a) A “ Zrxs{p2x ´ 1q

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b) A “ Zrxs{p3x ´ 1q
c) A “ Zrxs{p9x ´ 3q
d) A “ Zrxs{p2x2 ´ 1q
Which of these are isomorphic/homeomorphic?
Exercise 9.9.39. Check that Spec ϕ ˝ Spec ψ “ Specpψ ˝ ϕq, whenever ϕ and ψ are
composable ring maps.
Exercise 9.9.40. Prove that Ank ´ tp0, . . . , 0qu is not affine for any n ě 2.
Exercise 9.9.41. Let f : SpecpBq Ñ SpecpAq be the morphism associated to a ring map
ϕ : A Ñ B . Show that the sheaf map f 7 : OSpec A Ñ f˚ OSpec B is given by ϕr : A
rÑB
r,
where we regard B as an A-module via ϕ.
Exercise 9.9.42. Let X be a scheme and let Xred be its reduction. Show that if k is a field,
then there is a natural bijection Xpkq “ Xred pkq. Generalize the statement to reduced rings.
Exercise 9.9.43. Show that the morphism
Spec Crx, b, cs{px2 ` bx ` cq ÝÝÑ Spec Crb, cs
is finite and describe its fibers.
What about the morphism Spec Crx, a, b, cs{pax2 ` bx ` cq Ñ Spec Cra, b, cs?

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Schemes of finite type over a field

10.1 The general definition of a variety

In the literature, one can find several definitions of the notion of a ‘variety’. Some texts do
not require varieties to be irreducible (but most texts define them to be reduced), and many
require the base field to be algebraically closed. It is also convenient to include the empty
scheme as a variety (over any field k ). In this book, the term ‘variety’ will be defined as
follows:

Definition 10.1. A variety over a field k is an integral, separated scheme of finite type
over k .

A curve is a variety of dimension 1. A surface is a variety of dimension 2. A threefold is a

variety of dimension 3, and so on.
A morphism or simply map of varieties over k is a morphism f : X Ñ Y of schemes over
k . In this way, the varieties over k form a subcategory Var{k of the category of k -schemes,
Sch{k .
One drawback of the above definition is that the property of ‘being a variety’ is not stable
under base change. For instance, X “ Spec Rrx, ys{px2 ` y 2 q is a variety over R, but its
base change XC is not a variety over C, as it is not irreducible. Similarly, the fiber product
of two varieties needs not be a variety under this definition, unless the field is algebraically
closed (Theorem 10.14).

Proposition 10.2 (Subvarieties). Let X be a variety over a field k .

(i) (Open subvarieties) Every open subscheme U Ă X is a variety.
(ii) (Closed subvarieties) Every closed, integral subscheme Y Ă X is a variety.
(iii) Every closed irreducible subset Y Ă X has a unique structure as closed
subvariety.

Proof (i): Open subschemes of integral schemes are integral by Proposition 7.5 on page 142,
and open embeddings are separated (Proposition ?? on page ??), so U is integral and separated
over k . Finally, U is of finite type over k : it is covered by finitely many open affine subschemes
(because X is a variety, and because every open set in an affine scheme can be written as a
union of finitely many distinguished open sets), and each of these are of finite type over k by
Propoisition 9.10.
(ii): According to Example 9.11 the subscheme Y is locally of finite type, and since X

199

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is quasi-compact, it is of finite type. By hypothesis, it is integral, and it is separated by

Proposition ??.
(iii): Each closed subset carries a unique reduced scheme structure (Proposition 14.54),
which is integral when the subset is irreducible. The rest follows from (ii).

10.2 Schemes of finite type over a field

The most important class of schemes are the schemes of finite type over a field k . These
schemes have similar properties to those of affine varieties, especially when k is assumed to
be algebraically closed.

Proposition 10.3. If X is a scheme of finite type over a field k , then a point x is closed
if and only if the residue field κpxq is a finite extension of k . Moreover, the closed points
form a dense subset of X .

In particular, when k is algebraically closed, it follows that κpxq “ k . In other words, the
closed points of X are exactly the k -points. In particular, for X “ Ank , this is precisely the
content of Theorem 1.8.

Proof The point x is contained in an affine open subscheme Spec A of X with A of finite
type over k , so we reduce to the case where X “ Spec A. Here the statement follows from
Theorem 1.7.
To prove density, it suffices to see that any open subset of X contains a closed point. By
Proposition 9.4, X has a basis consisting of open affines U of finite type over k , and each
of these have closed points. Therefore, any non-empty open subset of X contains a closed
point.

Example 10.4. The assumption that X is of finite type over k is essential. For instance, if X
is the spectrum of the discrete valuation ring A “ krxspxq then there is a single closed point,
corresponding to the maximal ideal pxq. This point is not dense in X .
In contrast, if A “ krxsx “ krx, x´1 s, the algebra is finitely generated, so the closed
points are dense in Spec A. △

Corollary 10.5. Let f : X Ñ Y be a morphism of finite type schemes over a field k . If

x P X is a closed point, then f pxq is closed in Y .

Proof We may assume that X and Y are affine, say X “ Spec B and Y “ Spec A, and
that A and B are of finite type over k . The point x corresponds to a maximal ideal m in B ,
and κpxq “ B{m is a finite extension of k , by Proposition 10.3. Let p Ă A be a prime ideal
corresponding to f pxq, that is, the preimage of m under the ring map ϕ : A Ñ B inducing
f . Note that ϕ induces an injection A{p ãÑ B{m “ κpxq. Now κpxq is integral over k ,
and hence also integral over A{p. Therefore, by the Going–Up theorem (Theorem 9.30), the
quotient A{p is a field, and hence the point is closed.

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10.3 Dimension theory for schemes of finite type over a field

For integral schemes of finite type over fields, the theory of dimension is much simpler, as
we may study the dimension in terms of the function field:

Theorem 10.6 (Dimension and transcendence degree). Let X be an integral scheme

of finite type over a field k .
(i) dim X “ trdegk KpXq.
(ii) For each non-empty open subscheme U Ă X , it holds that dim U “
dim X .
(iii) codimpY, Xq “ dim X ´ dim Y .

Proof In view of Lemma 7.32 on page 150, the general case follows from the affine case,
so we may assume that X is affine, say X “ Spec A. The Normalization Lemma tells us
that there is a finite surjective morphism p : X Ñ Ank where n “ trdegk KpXq. Applying
Proposition 9.32, we deduce that dim X “ dim Ank “ n.
Statement (ii) holds because U has the same function field as X .
Finally, claim (iii): TODO.
Example 10.7. The projective space Pnk contains Ank as a dense open subscheme, and hence
has dimension n. △
Example 10.8 (Affine hypersurfaces). The quadric cone Q “ Spec krx, y, zs{px2 ´ yzq
of Example 7.10 on page 144 has dimension 2. This follows directly from ?? of Lemma ??.
More generally, for any irreducible non-constant polynomial f P krt1 , . . . , tn s, the closed
subvariety V pf q Ă Ank is of dimension n ´ 1. △
Example 10.9 (Projective hypersurfaces). Any irreducible homogeneous polynomial f P
krt0 , . . . , tn s of positive degree defines a closed subscheme Z Ă Pnk , which is a closed sub-
variety of dimension n ´ 1. Indeed, Z must intersect at least one distinguished open set, say
Dpt0 q, in a non-empty open subscheme Ui “ Dpti qXZ , which equals Spec krt1 {t0 , . . . , tn {t0 s{pF q,
where F “ f pt1 {t0 , . . . , tn {t0 q is the dehomogenization of f (see Section ??). From this
we see that dim Ui “ n ´ 1 and so also dim Z “ n ´ 1.
△
Example 10.10. In A “ krx, y, zs{pxy, xzq, the two chains pxq Ă px, yq Ă px, y, zq and
py, zq Ă px, y, zq are maximal chains of different lengths. △
There is a generalization of the notion of ‘hypersurfaces’ which is meaningful for any
scheme X . A subscheme is said to be locally given by one equation if one may find an open
affine cover tUi u of X and non-zerodivisors fi P OX pUi q so that Z X Ui “ V pfi q.

Proposition 10.11. Let X be variety over k and let Z Ă X be a closed subvariety locally
defined by one equation. Then dim Z “ dim X ´ 1.

For schemes which are not integral, but of finite type over k , we still have a good control
over the dimension. First of all, the dimension of X is the same as of the reduction
Ť Xred
(see Exercise 4.14.9), so we may assume that X is reduced. Then, if X “ Xi is the

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decomposition into irreducible components, each Xi is integral, and dim X is the maximum
of all dim Xi .
Example 10.12. Let A3k “ Spec krx, y, zs and consider the subscheme X “ V pxy, xzq.
Then as pxy, xzq “ pxq X py, zq, we see that X consists of two components, V pxq and
V py, zq. The dimension of X is maximum of the dimensions of its irreducible components,
and so we find that dim X “ 2. △
Example 10.13. Consider A3k “ Spec krx, y, zs and X “ V paq where a is the ideal
px2 , xz, yz, z 2 ´ zq. This ideal has a primary decomposition
a “ px2 , zq X px, y, z ´ 1q.
This means that the associated primes of a are p1 “ px, zq and p2 “ px, y, z ´ 1q. Geometri-
cally, X consists of two components: V px, zq, a line; and V px, y, z ´ 1q, the point p0, 0, 1q.
From this, we see that X has dimension 1. △

10.4 Products of varieties

Theorem 10.14. Let X and Y be two schemes of finite type over an algebraically closed
field k . Then
(i) X ˆk Y is of finite type over k .
(ii) pX ˆk Y qpkq “ Xpkq ˆ Y pkq.
(iii) if X and Y are reduced, then so is X ˆk Y .
(iv) if X and Y are integral, then so is X ˆk Y .
(v) dimpX ˆk Y q “ dim X ` dim Y .

Proof (i): We may reduce to the case where X “ Spec A and Y “ Spec B , where A
and B are finitely generated k -algebras. We need to show that the ring A bk B is a finitely
generated k -algebra. If x1 , . . . , xr and y1 , . . . , ys are generators of A and B respectively,
then A bk B is generated by the products xi b yj .
(ii): By the Nullstellensatz, the k -points of X ˆk Y are precisely the closed points, so the
set of closed points in the product equals the Cartesian product of the sets of closed points of
the factors. Of course, the fiber product may also have many non-closed points which do not
come from the closed points in each factor (Example 8.10).
(iii): We need to show that A bk B is reduced. Let m be a maximal ř ideal of B and
ř consider
the ring map ϕ : Abk B Ñ Abk B{m » Abk k “ A, sending p i ai bbi qbp j cj bdj q
to p i ai b¯i q ¨ p j cj d¯j q. Suppose that f “
ř ř ř
ai b bi is a nilpotent element of A bk B .
Without loss of generality, we may assume that the ai are linearly independent over k . Then
the image of f under ϕ must be zero, as B is reduced. Therefore, ai b¯i “ 0, and hence
ř
bi P m for every i because the ai are Ş linearly independent. The conclusion is that bi lies in
every maximal ideal, and hence bi P m m “ p0q, by the Nullstellensatz (Exercise 1.8.23).
This means that f “ 0. ř ř
(iv): Suppose that f “ ai b bi and g “ ci b di are two elements such that f g “ 0.
As in (iii), we may arrange it so that the ai ’s are linearly independent over k , and likewise with
the ci ’s. Then since B is an integral domain, we see that either bi P m for every i or di P m.

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In any case, if b and d denote the ideal generated by the bi and di respectively, then b X d Ă m.
Since this holds for every m, we get b X d “ 0, and hence V pbq Y V pdq “ Spec B . As
Spec B is an integral domain, we have either b “ 0 or d “ 0, which means that either f “ 0
or g “ 0.
(v): Again one reduces to the case where X “ Spec A and Y “ Spec B are affine and
integral. If we choose Noether normalizations krx1 , . . . , xr s Ñ A and kry1 , . . . , ys s Ñ B ,
then a Noether normalization for A bk B is given by krx1 , . . . , xr , y1 , . . . , ys s Ñ A bk B ,
which gives that the Krull dimension of A bk B is equal to r ` s “ dim A ` dim B .

10.5 A structure result for morphisms

Theorem 10.15 (Generic structure theorem for morphisms of finite type). Let X
and Y be integral schemes and f : X Ñ Y a dominant morphism of finite type. Then
there are open affine subsets U Ă Y and V Ă X such that f pV q “ U and such that
f |V factors as
q p
V U ˆ An U
where q is finite; p is the first projection; and n “ trdegKpY q KpXq.
If X and Y are affine, we may take V to be the inverse image of a distinguished open set.

In the theorem, An denotes the affine space over Z, and the product U ˆ An is the product
over Z. If X , Y and f are defined over a ring R, the product can be replaced by the product
U ˆR AnR .
Proof Choose affine open sets Spec A Ă Y and Spec B Ă X such that f pSpec Bq Ă
Spec A. As f is of finite type, the A-algebra B will be of finite type. Moreover, since f is
dominant, the ring map f 7 : A Ñ B is injective, and we may assume that A Ă B is a ring
extension.
Applying Theorem A.34 to A Ă B , we may find algebraically independent elements
x1 , . . . , xn and a g P A such that Bg is finite over Ag rx1 , . . . , xn s.
Let U “ Dpgq Ă Spec A and V “ Dpgq Ă Spec B , and note that Spec Ag rx1 , . . . , xn s “
Spec Ag ˆ An . Then f |U is given by the extension Ag Ă Bg which factors as
Ag Ă Ag rx1 , . . . , xn s Ă Bg .
Here the first inclusion corresponds to p, and the second to q .
By construction f pV q Ă U . Now g is surjective by Proposition ?? on page ??, and p
is surjective because pU ˆ An qpLq “ U pLq ˆ An pLq and An pLq ‰ H for every field L.
Therefore f is surjective as well, and so f pV q “ U .

10.6 The dimensions of the fibers of a morphism

When studying a morphism f : X Ñ Y it is useful to know how the dimensions of a fibers
Xy vary with the closed point y . As a good first estimate, at least when f is dominant, one
could hope to relate dim Xy to the ‘relative dimension’ r “ dim X ´ dim Y . In general,

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we have an inequality dim Xy ě dim X ´ dim Y , as the next proposition shows. The proof
is based on Krull’s Principal Ideal Theorem combined with the fact that all maximal ideals in
krt1 , . . . , tn s are generated by n elements.

Proposition 10.16. Let f : X Ñ Y be a dominant morphism between varieties over

a field k . Then for each closed point y P Y in the image of f and every irreducible
component Z of the fiber Xy , it holds that
dim Z ě dim X ´ dim Y. (10.1)

Proof Replacing Y by some open affine neighbourhood U of y and X by some open affine
subscheme that meets Z and maps into U , we may assume that X and Y both are affine, say
X “ Spec B and Y “ Spec A.
We first treat the case where Y “ Ank . Let m be the maximal ideal in krt1 , . . . , tn s that
corresponds to y . It is generated by n elements g1 , . . . , gn . Consequently, the fiber Xy is
given as
Xy “ Spec B{mB “ Spec B{pg1 , . . . , gn q,
and the actual component Z of the fiber Xy equals V ppq for a minimal prime ideal p of
pg1 , . . . , gn q. By the Principal Ideal Theorem, we find that codimpZ, Xq “ dim Bp ď n.
Hence by (iii) of Theorem 10.6 we conclude that dim X ´ dim Z ď n “ dim Ank ; or on
other words, that dim Z ě dim X ´ dim Ank .
For the general case, we use the Normalization Lemma to find a finite and dominant
morphism p : Y Ñ Ank , and consider the composition h “ p ˝ f : X Ñ Ank . The point is
that z “ ppyq is closed in Ank , and that Z is a component of the fiber h´1 pzq the fiber p´1 pzq
is finite and discrete.
While this inequality can certainly be strict (e.g., when f is a blow-up), Theorem 10.15
combined with Going–Up shows that it is an equality for most fibers:

Proposition 10.17 (Dimension of generic fibers). Let X and Y be varieties over k

and let f : X Ñ Y be a dominant morphism. There is an open dense subset U Ă Y so
that for all closed points y P Y and all irreducible components Z of Xy , it holds that
dim Z “ dim X ´ dim Y .

Proof We may clearly assume that Y is affine, and we cover X by finitely many open affine
subschemes tWi u.
For each Wi we choose open affines Vi Ă Wi and Ui Ă Y such that fi “ f |Vi factors as
in Theorem 10.15; that is, as the composition of two maps
gi pi
Vi Ui ˆ Ar Ui
with gi finite and pi the projection and r “ dim Vi Ş
´dim Ui . Note that r “ dim X ´dim Y
by (ii) of Theorem 10.6. We claim that the set U “ i Ui will be as required. Indeed, consider
a closed point y P U and a component Z of the fiber Xy . At least one of the Wi meets the
given component Z in an open dense set, and hence the corresponding Vi meets Z as well.
Then Zi “ Z X Wi is open and dense in Z , and dim Z “ dim Zi by (ii) of Theorem 10.6.

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Proposition 10.18 (Semicontinuity of the fiber dimension). Let X and Y be varieties

over k and let f : X Ñ Y be a surjective morphism. Then for all integers s the set
Fs pf q “ t y P Y | dim Xy ě s u is closed in Y .

Proof The proof goes by induction on dim Y . The case dim Y “ 0 is trivial, so assume
that dim Y ą 0. If s ď r “ dim X ´ dim Y , it holds that Fs pf q “ X by Proposition 10.16
(remember that f is surjective). Suppose then that s ą r, and let U Ă Y be an open set as
in Proposition 10.17. Let Zi be the components of Y ´ U and let Wij be the components
of f ´1 Zi . Then dim Zi ă dim Y , and by induction each Fs pf |Wij q is closed in Zi . We
contend that
ď
Fs pf q “ Fs pf |Wij q, (10.2)
ij

and this will imply that Fs pf q is closed since Zi is closed in Y .

in y P U , each component W of Xy has dim W “ r ă
As to (10.2), note that for all pointsŤ
s, and hence the inclusion Fs pf q Ă ij Fs pf |Wij q holds. Then pick a point y P Fs pf |Wij q.
Each component of f |´1 Wij pyq is contained in a component of f
´1
pyq, so we infer that
´1 ´1
dim f pyq ě dim f |Wij pyq ě s.

10.7 Applications to intersections

The Principal Ideal Theorem has important consequences for the intersections of subvarieties
in both affine spaces An and projective spaces Pn . It provides upper bounds for the dimension
of the intersection of two closed subvarieties in terms of their dimensions.
A direct application of the Principal Ideal Theorem is often insufficient because varieties are
generally not complete intersections and may require more equations than their codimension
suggests. However, there is a useful trick, which uses the diagonal which provides a way
forward.
The main observation is that if X and Y are subvarieties of An , their product X ˆ Y is a
closed subvariety of A2n , and X X Y is isomorphic to ∆ X pX ˆ Y q, where ∆ is the diagonal
in A2n . The diagonal is defined by the simple equations xi ´ yi “ 0 for i “ 1, . . . , n. By
Krull’s Principal Ideal Theorem, each component Z of X X Y therefore satisfies

dim Z ě dim X ` dim Y ´ n.

This leads to the following proposition:

Proposition 10.19. Let X and Y be subvarieties of An . Then each (non-empty) compo-

nent Z of X X Y satisfies
codim Z ď codim X ` codim Y.

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Example 10.20. The inequality in Proposition 10.19 does not hold for subvarieties of general
varieties. For instance, the two planes Z1 “ Zpx, yq and Z2 “ Zpz, wq in A4 intersect only
at the origin, and both are contained in the quadratic cone X “ Zpxz ´ ywq, which is three-
dimensional. As subvarieties of X , the planes have codimension one, but their intersection
has codimension three. △

In the projective case, we get a stronger statement. The following theorem is a fundamental
result in projective geometry:

Proposition 10.21. Let X and Y be projective varieties in Pnk . If dim X ` dim Y ě n,

then X X Y is non-empty. Moreover, each component Z of X X Y satisfies
codim Z ď codim X ` codim Y.

Proof If dim X ` dim Y ě n, then dim CpXq ` dim CpY q ě n ` 2. Since the cones
CpXq and CpY q both contain the origin, their intersection is non-empty. Any component W
of CpXq X CpY q satisfies dim W ě dim X ` dim Y ´ n ` 1 ě 1, so CpXq X CpY q is
not reduced to the origin. The result now follows from Proposition 10.19.

Exercise 10.7.1. Any two curves in P2k intersect.

Exercise 10.7.2 (The Resolution of a Determinantal Variety). Let M pxq “ pxij qij be a
generic 3ˆ2-matrix, where the xij ’s are variables. Thus, M has coefficients in the polynomial
ring krx11 , . . . , x23 s. The matrix M pxq is given by
¨ ˛
x11 x12
M pxq “ ˝x21 x22 ‚.
x31 x32

Endow the space of matrices A6k “ Homk pk 2 , k 3 q with coordinates xij , and let W Ă A6 be
the locus of matrices of rank less than one, defined by the vanishing of the three 2 ˆ 2-minors
of M . Show that W is an integral scheme.
Now, introduce a copy of P1 with homogeneous coordinates v “ pt1 : t2 q. Inside the
product A6 ˆ P1 , consider the subvariety W Ă of pairs pM, rvsq such that M ¨ v “ 0; that is,
the locus where t1 x1j ´ t2 x2j “ 0 for j “ 1, 2, 3.
Ă Ñ P1 are hyperplanes in A6 .
a) Show that all fibers of the projection p2 : W
3
b) For i “ 1, 2, let Ui “ p´1
2 pD` pti qq. Exhibit isomorphisms Ui » A ˆ D` pti q
compatible with the projection. Deduce that W Ă is 4-dimensional.
c) Show that the projection p1 : W Ñ W is birational and describe all its fibers.
Ă

Exercise 10.7.3. a) Give examples of projective varieties X and Y that do not

satisfy the inequality codim X `codim Y ď n and have an empty intersection.
b) Give examples of two closed subvarieties in affine space An satisfying codim X`
codim Y ď n but having an empty intersection.

Exercise 10.7.4. Find examples of two irreducible quadratic curves in A2 with empty
intersection. Do the same for two cubic curves.

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Exercise 10.7.5. View P3C with homogeneous coordinates a, b, c, d as the projective space
parameterizing complex cubic polynomials aX 3 ` bX 2 ` cX ` d.
a) Find equations parameterizing the locus of cubics having (i) having a double
root and (ii) cubics having a triple root. Describe the corresponding projective
schemes.
b) Using a), show that there exist s, t P C such that X 3 `sX 2 `tX `ps5 `t5 `1q
has a triple root.

10.8 Rational maps

A rational map between integral schemes X and Y is a morphism U Ñ Y defined on some
open subset U of X . To avoid the ambiguity in the domain of definition, we say that two
morphisms f : U Ñ Y and g : V Ñ Y define the same rational map if f |W “ g|W for
some non-empty open subset W Ă U X V (this is an equivalence relation on pairs pU, f q).
It follows immediately from the Gluing Lemma for morphisms (Proposition 4.50) that there
is a maximal open set U for which the rational map is defined, and this is another way of
resolving the ambiguity. A rational map is denoted with a dashed arrow f : X 99K Y .
One says that a rational map f : X 99K Y is dominant if f pU q is dense in Y where U is
some open set where f is defined (if this holds for one U , it holds for all).
Dominant rational maps can be composed: if f : X 99K Y and g : Y 99K Z are rational
maps which are defined over U Ă X and V Ă Y , then there is a (non-empty!) open set
W Ă V such that f ´1 pW q Ă U and we define g ˝ f : X 99K Z to be the rational map
corresponding to the morphism f ´1 pW q Ñ Z . Hence the k -varieties together with the
dominant rational maps form a category Ratk .
One says that a dominant rational map f : X 99K Y is birational if there exists a dominant
rational map g : Y 99K X such that f ˝ g and g ˝ f are the identity maps as rational maps,
that is, they equal the identity map on some open set.
Two varieties are said to be birationally equivalent if they have isomorphic open subsets.
This is a much weaker relation than being isomorphic: for instance, blowing up a point in P2k
yields a variety which is birationally equivalent with but not isomorphic to P2k .

Example 10.22. The Cremona transformation

ϕ : P2k ÝÝÑ P2k

pu0 : u1 : u2 q ÞÑ pu1 u2 : u0 u2 : u0 u1 q
is a rational map defined away from the three coordinate points p0 : 1 : 0q, p1 : 0 : 1q and
p1 : 1 : 0q. It is a birational map, which satisfies ϕ ˝ ϕ “ id. △
Example 10.23. The map

ϕ : P1k ˆ P1k ÝÝÑ P2k

pu0 : u1 q ˆ pv0 : v1 q ÞÑ pu0 v0 : u1 v0 : u1 v1 q
is a birational map defined away from p1 : 0q ˆ p0 : 1q. The inverse is given by pt0 : t1 :
t2 q ÞÑ pt0 : t1 q ˆ pt1 : t2 q, and this is defined away from p0 : 0 : 1q and p1 : 0 : 0q. △

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Birational maps and function fields

A fundamental fact in the theory of varieties is that the study of dominant rational maps is
basically reduced to the study of extensions of function fields. This is because a dominant
rational map f : X 99K Y induces a map between the function fields

f 7 : KpY q ÝÝÑ KpXq (10.3)

Indeed, if U Ă X, V Ă X are open sets where f is defined, they contain the generic points
of X and Y , ηX and ηY . The induced map on stalks OV,ηY Ñ OU,ηX is exactly the map
(10.3). If f is dominant, then f 7 is injective.

Theorem 10.24. Let X and Y be two varieties over k . Then there is a one-to-one corre-
spondence between rational dominant maps X 99K Y and k -algebra homomorphisms
KpY q Ă KpXq. In particular, two varieties are birationally equivalent if and only if
their function fields are isomorphic as k -algebras.

Proof We have already showed that any rational map f : X 99K Y induces a map of
k -algebras KpXq Ñ KpY q. The inverse assignment is constructed as follows.
Let U “ Spec B Ă X and V “ Spec A Ă Y be open affine subsets. Then KpXq is the
fraction field of B and KpY q that of A, and we have A Ă KpXq and B Ă KpY q.
Let ϕ : KpY q Ñ KpXq be a given map of k -algebras. If a1 , . . . , ar generate A over
k , then each ϕpai q is of the form ϕpai q “ bi {ci with bi , ci P A. Therefore, if we set d “
c1 ¨ ¨ ¨ cr we have ϕpAq Ă Bd . This means that ϕ induces a map of k -algebras ϕ̄ : A Ñ Bd ,
and hence a morphism Spec Bd Ñ Spec A “ V Ă Y hence a rational map f : X 99K Y .
Evidently, A maps injectively into Bd so the morphism is dominant. By construction, ϕ̄
localizes to the map ϕ, so f induces the map ϕ.

Associating X to the function field KpXq defines a functor from the category of varieties
over k and dominant rational maps to the category of finitely generated field extensions of
k . The next result shows that these categories are ‘essentially equivalent’: X ÞÑ KpXq is
fully faithful, but there is no natural functor that serves as the inverse functor, as there is no
systematic way to pick out one particular variety X for each field K .

Theorem 10.25. The assignment X ÞÑ KpXq is defines an equivalence between the

following categories:
(i) The category of affine varieties and dominant rational maps.
(ii) The category of projective varieties and dominant rational maps.
(iii) The category of finitely generated field extensions of k and k -algebra homo-
morphisms.

Theorem 10.24 tells us that the functor is fully faithful; that is, rational maps X 99K Y are
in bijection with maps of function fields KpY q Ñ KpXq. To conclude, we need to show
that every finitely generated field extension K of k is of the form KpXq for some projective
variety X .

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Lemma 10.26. Let k be a field and let K be a finitely generated field extension of k .
Then there exists a projective variety X over k with KpXq » K .

Proof Let t1 , . . . , tn be a set of generators for k as a field extension of k and let A “

krt1 , . . . , tn s be the subalgebra of K generated by them. Then A is an integral domain
with fraction field equal to K . The scheme V “ Spec A is an affine variety over k with
KpV q “ K . Pick a closed embedding V Ñ Ark and embed Ark as a distinguished open set
in Prk . Then if we define X to be the Zariski closure of V inside Prk , then X is a projective
variety with KpXq “ KpV q “ K .
Recall that a field k is said to be perfect if every algebraic extension k Ñ L is separable.
Any field of characteristic 0 is perfect, as is every algebraically closed field, and every finite
field. The field k “ Fp ptq is not perfect, as L “ Fp pt1{p q is a non-separable extension.
The next result says that any variety over a perfect field is birational to a hypersurface. To
prove this we need the following fact from Algebra.

Theorem 10.27 (Primitive Element Theorem). If K{k is a finite separable field exen-
sion, then there exists an element ξ P K such that K “ kpξq.

Proposition 10.28. Let X be an integral scheme of finite type over a perfect field k .
Then X is birational to a hypersurface in Ank .

Proof Let K “ KpXq be the function field of X . As k is perfect, K is a separable

extension of k . Let y1 , . . . , yn be a transcendence basis of K over k , so that K is a finite
separable extension of L “ kpy1 , . . . , yn q Ă K . By Theorem 10.27, we have K “ Lpξq
for some ξ P K . The element ξ is a root of a minimal polynomial equation
P ptq “ tr ` ar´1 tr´1 ` ¨ ¨ ¨ ` a0 “ 0
where the ai P L. Multiplying α with a suitable polynomial in the yi , we may assume that
ai P kry1 , . . . , yn s for each i. But this means that X has the same function field as the
hypersurface
Y “ Spec pkry1 , . . . , yn , ts{P ptqq Ă An`1
k .

Rationality
One says that a variety X over a field k is rational if it is birational to Pnk for some n.
Equivalently, the function field of X satisfies
KpXq » kpt1 , . . . , tn q.
We have seen many examples of rational varieties already. For instance, Ank , Pnk , Pm n
k ˆk P k ,
the nodal cubic, and all toric varieties are rational.
A central question in algebraic geometry is to determine whether a given variety is rational.

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Here the challenge is usually to find a method to disprove rationality, i.e., find a birational
invariant which is non-trivial of the given example, but trivial for Pnk .

Example 10.29. If X Ă An`1k is a hypersurface defined by an equation which is linear in

one of the variables, say

Apx1 , . . . , xn q ` xn`1 Bpx1 , . . . , xn q “ 0

then X is rational. Indeed, we may use the relation to eliminate xn`1 :

KpXq “ kpx1 , . . . , xn`1 q{pApx1 , . . . , xn q ` xn`1 Bpx1 , . . . , xn qq » kpx1 , . . . , xn q.

If X is defined by a quadratic polynomial, then one can often perform a change of variables
over k to bring it into this form. For instance, the quadric curve y02 ` y12 ´ y22 “ 0 in A2R is
rational, by the variable change x1 “ y1 ` y2 and x2 “ y1 ´ y2 .
On the other hand, if X is the curve defined by x2 ` y 2 ` 1 “ 0 in A2R , then X cannot be
rational, as rational varieties have a dense set of R-points. On the other hand, XC , which is
defined by the same equation in A2C , is rational, as we may bring it to the form uv ` 1 by the
coordinate change u “ x ` iy , v “ x ´ iv . △

Exercises 10.8.1, 10.8.2 and 10.8.3 discuss some basic techniques to disprove rationality.
We will see more sophisticated examples in Section 20.10 after discussing differentials .

Exercise 10.8.1 (Elliptic curves are irrational). Let k be a field of characteristic not equal
to 2 and let X Ă A2k be the curve defined by the equation y 2 “ xpx ´ 1qpx ` 1q. In this
exercise, we will show that X is not rational.
a) Write K “ kpxq and L “ Kpyq “ KpXq. Show that for any valuation
ν : Lˆ Ñ Z, the number νpxq is even. H INT: Consider the cases νpxq ă 0
and νpxq ě 0 separately.
b) Show that x is not a square in L. H INT: If x “ pa ` byq2 , then expand using
the equation y 2 “ xpx ´ 1qpx ` 1q.
c) Show that kptq has the following property: if g P kptq is an element such that
νpgq is even for every valuation ν : kptqˆ Ñ Z, then g is a square.
d) Conclude that L fi kptq and that X is not rational.

Exercise 10.8.2 (Cubic surfaces over R). a) * Let X and Y be a nonsingular pro-
jective varieties over R and suppose that X and Y are birational. Show that
XpRq and Y pRq have the same number of connected components.
b) Consider the cubic surface X Ă A3R defined by the equation

x2 ` y 2 “ z 3 ´ z. (10.4)

Show that XpRq has 2 connected components. Show that X is not rational over
R.
c) Show that XC Ă A3C is rational over C.

Exercise 10.8.3. Let k be a field and let X be a variety over k . Show that if X is rational,
then the k -points Xpkq are dense in X .

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10.9 Exercises
Exercise 10.9.1. Find an example of a connected scheme X with a disconnected open subset
U Ă X.
Exercise 10.9.2. Let A be a Noetherian ring such that Spec A is of dimension 0. Show that
A is Artinian.
Exercise 10.9.3. Show that if A is a Noetherian ring, then each local ring Ap is also
Noetherian.

Exercise 10.9.4. Describe X “ Spec Zrxs{p5x ´ 15q. Is X irreducible? Reduced? What

are the fibers of the canonical map X Ñ Spec Z?

Exercise 10.9.5. Let X be an integral scheme and U Ă X an open subset. Show that x P U
if and only if OX pU q Ă OX,x inside KpXq.

Exercise 10.9.6. Let X be a topological space. Show that the following two conditions are
equivalent.
(i) X is Noetherian.
(ii) Every open subset of X is quasi-compact.

Exercise 10.9.7. a) Find an open subset of an affine scheme which is not quasi-
compact.
b) Let U, V be quasi-compact open subsets of Spec A for some ring A. Show that
U X V is also quasi-compact.
Exercise 10.9.8. For a Noetherian scheme X , the ring OX pXq may fail to be Noetherian in
general. Consider P3k with homogeneous coordinates x0 , x1 , x2 , x3 . Let H0 “ V` px0 q and
H1 “ V` px1 q and let l “ V` px0 , x2 q Ă D be a line on H0 different from H0 X H1 . Let
Y “ H0 Y H1 and X “ Y ´ l.
a) Show that X is Noetherian.
b) Show that U “ H0 ´ l is an open set of X and that OX pU q “ krx, ys where
x “ xx12 and y “ xx23 .
c) Show that V “ H1 ´ H1 X L is an open set of X and that OX pV q “ k
d) Show that OX pXq “ k ` xkrx, ys Ă krx, ys.
e) Show that the ring in (d)) is not Noetherian. H INT: pxq Ă px, xyq Ă
px, xy, xy 2 q Ă . . . .
Exercise 10.9.9 (A 1-dimensional non-Noetherian domain). Let R be the subring of Cpx, yq
consisting of rational functions f px, yq that are defined and constant along the y -axis. The
elements of R, when written in lowest terms, have a denominator not divisible by x, and
f p0, yq, which is then meaningful, is constant.
a) Show that the ideals ar “ px, xy ´1 , . . . , xy ´r q with r P N form an ascending
chain that does not stabilize. Conclude that R is not Noetherian.
b) Show that R is local with the set m of elements f P R that vanish along the
y -axis as the maximal ideal.
c) Prove that there are no other primes than m and p0q in R. H INT: Show first

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that any element in R is of the form xi y j α where i ě 0, j P Z and α is a unit

in R.
Exercise
ś8 10.9.10 (The ring of eventually constant sequences). Consider the subring A of
i“1 Z {2 Z consisting of sequences pei qiě1 which are eventually constant, that is, sequences
with ei “ ei`1 for i " 0.
a) Show that all elements of A are idempotents and conclude that every prime
ideal is maximal. H INT: the only idempotents in a domain are 0 and 1.
b) Let mn denote the ideal generated by 1 ´ an where an “ p0, . . . , 0, 1, 0, . . . q
with a ‘1’ in the n-th factor. Show that mn is a maximal ideal.
c) Show that Dpan q “ tmn u and conclude that the one-point set tmn u is both
open and closed in Spec A.
d) Let m8 denote the ideal consisting of sequences which are eventually zero, i.e.,
ei “ 0 for all i " 0. Show that m8 is a maximal ideal. H INT: Consider the
‘limit map’ A Ñ Z{2.
e) Show that A is not Noetherian. H INT: Show that m8 is not finitely generated.
f) Show that these are all the prime ideals of A, i.e., that Spec A “ t mi | i P
N u Y tm8 u. H INT: Consider the cases ai R m for some i and ai P m for all i
separately. Use the identity ai p1 ´ ai q “ 0.
g) Show that Spec A is homeomorphic to the set t n1 | n P N u Y t0u (with the
standard topology).
Exercise 10.9.11. For each of the following rings A, decide whether the corresponding
morphism Spec A Ñ Spec Z is finite or finite type.
Zris, Zr1{ps, Zppq , Z ˆ Z, Zrxs.
Exercise 10.9.12. Let f : X Ñ Y be a morphism of finite type. Show that if X is Noetherian,
then so is Y .
Exercise 10.9.13. Show that the composition of two morphisms (locally) of finite type is
(locally) of finite type. Show that if S is quasi-compact and f : X Ñ S is of finite type, then
X will be quasi-compact.
Exercise 10.9.14. Assume that ι : Spec B ãÑ Spec A is an open embedding. Show that B
is of finite type over A.
Exercise 10.9.15. Assume that S is a Noetherian scheme and that f : X Ñ S is of finite
type. Prove that X is Noetherian. H INT: Hilbert’s Basis Theorem.
Exercise 10.9.16 (The ”Lying Over Theorem”). Let A Ă B an integral extension of integral
domains.
a) Show that A is a field if and only if B is a field.
b) If p is a prime in A, show that p lies in the image of Spec B Ñ Spec A if and
only if pAp lies in the image of Spec Bp Ñ Spec Ap .
c) Conclude that Spec B Ñ Spec A is surjective.
Exercise 10.9.17. a) Show that there exist surjective morphisms X Ñ Y with Y
integral and such that all fibers Xy are irreducible, without X being irreducible.

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b) Let k be an algebraically closed field. Show that there exist morphisms X Ñ A1k
with X integral, the generic fiber Xη non-empty and integral, but no closed
fiber integral.
c) Show that there exists morphisms X Ñ A1Q with X integral and infinitely many
irreducible and inifintiely many reducible closed fibers. What happens for the
geometric closed fibers?
d) Show that there exists morphisms f : X Ñ Y with X and Y integral whose
geometric generic fiber is not reduced.
Exercise 10.9.18. Consider the ring R “ Qrx, x´1 s ˆ Q. Show that the induced map
Spec R Ñ Spec Qrxs is surjective, of finite type, and has finite fibers, but not finite.
Exercise 10.9.19. Consider the map
ϕ : A1k ÝÝÑ P1k
u ÞÑ pu : 1 ´ u2 q.
Show that ϕ is finite and surjective.
Exercise 10.9.20. Let f : X Ñ Y be an affine morphism and let V Ă Y be an open set.
Show that f ´1 pV q Ñ V is affine.
More generally, if V Ñ Y is any morphism, show that the base change morphism
X ˆY V Ñ V is affine. Thus affine morphisms are stable under base change.
Exercise 10.9.21. Show that the composition of two finite morphisms is finite.
Exercise 10.9.22 (Nagata’s example). Let R “ krx1 , Ť x2 , . . . s be the polynomial ring in
countably many variables, and decompose N as N “ i Ji be a disjoint union of finite
sets Ji whose cardinality tends to infinity with i. For instance, we could let Ji be the set of
integers with i digits in their binary expansion.
Let ni Ă R be the ideal generated
Ş by the xj ’s for which j P Ji , and let S Ă R be the
multiplicatively closed subset i pR ´ ni q, that is, the set of elements in R not lying in any
of the ni ’s. Nagata’s example is the localized ring A “ S ´1 R. The aim of the exercise is to
prove that A is Noetherian, but of infinite Krull dimension. We let mi denote ni A; the ideal
in A generated by the xj ’s with j P Ji .
Consider the rational function field Ki “ kpxj |j R Ji q in the variables xj whose index
does not lie in Ji , and the polynomial ring Ki rxj |j P Ji s over Ki in the remaining variables.
Also, define ai “ pxj |j P Ji q in Ki rxj |j P Ii s.
a) Show that Bni » Ki rxj |j P Ii sai .
b) Show that Ami “ Bni and conclude that each local ring Ami is Noetherian
with dim Ami “ |Ji | and hence that dim A “ 8.
c) * Show that A is Noetherian. H INT: Any ideal is contained in finitely many of
the of the mi ’s, and is therefore finitely generated.
Exercise 10.9.23. Let X be a quasi-compact scheme. Show that for any x P X , there exists
a closed point in txu Ă X .
Exercise 10.9.24. Let X be a quasi-compact scheme. Show that X is reduced if and only if
OX,p is reduced for each closed point p. (See Exercise 10.9.23.)

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214 Schemes of finite type over a field
š
Exercise 10.9.25. Let X denote the disjoint union of the schemes ně1 pSpec Qrxs{xn q.
Compute Xred . Show that the formula (7.1) does not hold in general for non-affine open sets
in X .
Exercise 10.9.26 (Criterion for affineness). a) Let f : X Ñ Y be a morphism
and let Vi be an open cover of Y such that for each i, the restriction f ´1 pVi q Ñ
Vi is an isomorphism. Show that f is an isomorphism.
b) Show that a scheme is affine if and only if there is a finite set of elements
f1 , . . . , fn P OX pXq such that each of the open sets Xfi “ tx P X|fi pxq ‰
0u are affine, and f1 , . . . , fn generate the unit ideal. H INT: Show that X “
Ť
Xfi . Glue the morphisms Xfi Ñ Spec OX pXq to a morphism X Ñ
Spec OX pXq.
Exercise 10.9.27. Show that the Frobenius morphism is finite.
?
Exercise 10.9.28. Let C Ă R2 denote the cone generated by p1, 0q and p1, 2q. Show that
the toric algebra krCs is not a Noetherian ring.
Exercise 10.9.29. Let X be a scheme and let x P X be a point. Show that x is a closed point
if and only if the corresponding morphism Spec κpxq Ñ X is finite.
Exercise 10.9.30 (Noetherian Induction for Schemes). Let T be a noetherian scheme. Let
P p¨q be a property of closed subschemes of T . Suppose that:
(i) P pempty schemeq holds, and
(ii) For all closed subschemes C of T , if P pC 1 q holds for all proper closed sub-
schemes C 1 Ĺ C , then P pCq holds.
Prove that P pT q holds.
Exercise 10.9.31. Let X be a Noetherian scheme. Show that Xred is affine if and only if X
is affine.
Exercise 10.9.32. Let X be a quasi-compact scheme. Prove that X has a closed point.
Exercise 10.9.33 (Properties of Noetherian spaces).
a) Show that any Noetherian topological space T is quasi-compact. H INT: If
tUi uiPI is a cover, start with any Ui1 , and pick Ui2 so that Ui1 Ĺ Ui1 Y Ui2 ,
and so on.
b) Show that if T is Noetherian, then every subset S Ă T is also Noetherian.
H INT: The closed subsets of S are of the form W X S where W Ă T is closed.
Exercise 10.9.34. Show that if k is algebraically closed, then the product of two projective
k -varieties X and Y is a projective variety.
Exercise 10.9.35. Let X be a scheme of finite type over a field k and let x P X be a point.
Show that the following are equivalent:
a) x is closed
b) There exists an affine open neighbourhood U “ Spec A of x P X with A a
finitely generated k -algebra, with x P U corresponding to a maximal ideal in
A.

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c) The residue field κpxq is a finite extension of k .

Exercise 10.9.36. Let A be a ring and m Ă Arts a maximal ideal. Let m0 “ A X m and
k “ A{m0 .
a) Show that Arts{m0 Arts » krts;
b) Show that if m0 is maximal and generated by r elements, then m is generated
by r ` 1 elements. H INT: krts is a principal ideal domain.
c) Show by induction on the number of variables that each maximal ideal in a
polynomial ring krt1 , . . . , tr s over a field k is generated by r elements.
d) (Alternative proof that dim Ank “ n) Show that if A is an algebra of finite type
over a field k , then dim Arts “ dim A ` 1. H INT: Claim ?? of Corollary ??
is useful.
e) If X is a variety over k , show that dim X ˆk Ank “ dim X ` n.

Exercise 10.9.37. Show that the constructible sets in a topological space form the small-
est Boolean algebra containing the open (or the closed) sets. Show inverse images under
continuous maps of contructible sets are constructible.

Exercise 10.9.38. Let X be a scheme and x P X a point. One says that a point y P X is a
specialization of x if y P x̄, and that y is a generalization of x if x P ȳ .
One says that a subset E Ă X is closed under specialization if specializations of points in
E belong to E . Likewise, E is said to be closed under generalization if generalizations of
points in E belong to E .
a) Show that E is closed under specializations if and only if the complement
X ´ E is closed under generalizations.
b) Show that E is closed under specialization if and only if it has the following
property: if x P E and Z Ă X is a closed irreducible set with x P E then
ZĂE
c) Show that closed sets are closed under generalization and that open sets are
closed under generalization.
d) Show that if E set closed under specialization and x R E , then each irreducible
component Z of X containing x is disjoint from E .
e) Show that in a Noetherian scheme, a constructible subset E is closed if it is
closed under specialization and that it is open if it is closed under generalization.
f) Give example that the Noetherian hypothesis is necessary. H INT: Consider the
spectrum in Exercise 10.9.10.

Exercise 10.9.39. Let Qpx0 , . . . , xn q be a homogeneous quadratic polynomial. Show that

the subvariety of Pn`2
k given by xn`1 xn`2 ` Qpx2 , . . . , xn q is birational to Pn`1
k .

Exercise 10.9.40. Let f : X Ñ Y be a morphism of varieties over k and suppose that

f induces a homeomorphism onto a closed subset of Y and f 7 induces an isomorphism
OY,f pxq Ñ OX,x for every closed point x P X . Show that f is a closed embedding.
H INT: Consider the kernel and cokernel of f 7 locally.

Exercise 10.9.41 (The Jounolou trick). Let k be an algebraically closed field. Consider the

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closed subset Γ Ă Pn ˆk Pn given by the equation

x0 y0 ` ¨ ¨ ¨ ` xn yn “ 0 (10.5)
a) Show that Γ is irreducible.
b) Show that Pn ˆ Pn ´ Γ is affine.
c) Show that the fibers of the first projection π : Pnk ˆ Pnk ´ Γ Ñ Pnk are affine
spaces Ank .
d) Show that π is Zariski locally trivial, i.e., there is an affine covering Ui of Pnk
such that π ´1 pUi q » Ui ˆ Ank .
e) Deduce that for any projective variety Y , there is an affine variety X and a
morphism X Ñ Y which is an Zariski locally trivial Ank -bundle.
Exercise 10.9.42 (The Jounolou trick II). Let k be an algebraically closed field. Consider the
set V pkq Ă Mn`1,n`1 pkq of matrices A such that A2 “ A.
a) Show that V pkq naturally forms the k -points of an affine variety V .
b) Show that there is a morphism π : V Ñ Pnk given on k -points by A ÞÑ Im pAq.
c) Show that the fibers of π over k -points are isomorphic to affine spaces Ank .
d) Show that π is Zariski locally trivial, i.e., there is an affine covering Ui of Pnk
such that π ´1 pUi q » Ui ˆ Ank .
e) Deduce that for any projective variety Y , there is an affine variety X and a
morphism X Ñ Y which is an Zariski locally trivial Ank -bundle.
Exercise 10.9.43 (Graph of a morphism II). Let X and Y be varieties over a field k . Let
f : X 99K Y be a rational map, defined over a maximal open set U Ă X . Define Γf to be
the closure of the graph Γf Ă U ˆk Y in X ˆk Y , and give it the reduced scheme structure.
a) Show that Γf is integral.
b) Show that the first projection Γf Ñ X is a birational map.
c) Show that f extends to a morphism f : X Ñ Y if and only if Γf Ñ X is an
isomorphism.

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Local properties of schemes

This chapter explores the local properties of schemes. These are properties of schemes which
take place at and around specific points.
The main definition is that of a nonsingular scheme. The intuition behind a nonsingular
scheme is that it locally looks like an affine space. Taking the definition of a manifolds as a
guide, one could naively try to define a scheme over a field k to be nonsingular of dimension n
if it is covered by open subsets which are isomorphic to an open subscheme of Ank . However,
this would be incorrect, as only very special schemes satisfy this condition. The issue is that
the Zariski open subsets are too large: over C, the curve x3 ` y 3 ´ 1 “ 0 should by all
accounts be considered ‘nonsingular’, but there is no open set which is isomorphic to affine
space, as X is not rational.
There is however another idea from differential geometry which generalizes to algebraic
geometry. A subset X Ă Rn of dimension d is a manifold if and only if locally around each
point p P X , X can be defined by ´ r equations
¯ f1 “ ¨ ¨ ¨ “ fr , where r “ n ´ d, and so that
Bfi
for each the Jacobian matrix J “ Bx j
has rank r at p. Indeed, if this condition holds, the
Implicit Function Theorem tells us that f1 , . . . , fr can be extended to a coordinate system in
a neighbourhood of p, and one proves that X is locally diffeomorphic to Rd there. We will
see that this generalizes well to schemes over a field in Section 11.3.
We begin by discussing tangent spaces and cotangent spaces of schemes. We then introduce
ingular points of schemes. Nonsingular schemes have local rings which are regular, i.e., where
the maximal ideal can be generated by n elements, where n is the dimension of X . Thus,
they admit “systems of parameters” that resemble local coordinates on manifolds.
Towards the end of the chapter, we discuss normal schemes and the process of normaliza-
tion. Like nonsingular schemes, normal schemes also have algebraic properties that make
them easier to study than most schemes. For instance, their singular loci are of codimension
at least 2. This property will prove particularly useful in Chapter 17, were we study divisors.

11.1 Tangent spaces

In this section, we define the tangent space of a scheme at a point. As a warm-up example,
let us consider a plane curve C Ă R2 defined by a real polynomial f px, yq “ 0 at the point
p “ p0, 0q. From basic calculus, the tangent space of the curve f px, yq “ 0 at p is defined
by the linear equation
Bf Bf
ppqx ` ppqy “ 0 (11.1)
Bx By
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This linear form is the first-order Taylor expansion of f px, yq at p. It is the linear form which
best approximates f near p.
The cotangent space at p is defined as the dual of this vector space: it consists of the linear
forms in x and y modulo the linear form (11.1). Equivalently, the cotangent space can be
viewed as the R-vector space of smooth functions vanishing at p, up to second order terms.
If X is a scheme and p P X , we would like to define the tangent space Tp X of X at p.
While we want to keep the above geometric picture, it turns out to be more convenient to
define things by defining the cotangent space first and then define the tangent space as its
dual. The definitions also work best when X is a scheme over a field k , and p is a k -point.
The definition of the cotangent space uses the local ring OX,p and the maximal ideal
mp . This is a natural choice, as OX,p captures the main geometric properties of the scheme
locally near p. If X is integral, the ring OX,p consists of the rational functions f P KpXq
which are defined at p and mp consists of the functions that vanish at p. The quotient
mp {m2p intuitively captures the space of functions vanishing at p up to ‘second order terms’.
Note that mp {m2p “ mp bOX,p OX,p {mp is naturally an OX,p {mp -module; in other words
it is a κppq-vector space. For example, if p is the origin in Ank “ Spec krx1 , . . . , xn s
then mp “ px1 , . . . , xn q and mp {m2p is the k -vector space generated by the linear forms
x1 , . . . , x n .
With this in mind, we make the following definition.

Definition 11.1. Let X be a scheme and let p P X be a point.

(i) The cotangent space of X at p is defined the κppq-vector space mp {m2p ,
where mp is the maximal ideal in the local ring OX,p .
(ii) The tangent space of X at p is defined as the dual κppq-vector space
Tp X “ Homκppq pmp {m2p , κppqq

For a morphism ϕ : X Ñ Y , there is an induced map of cotangent spaces. Let x P X , and

let y “ ϕpxq. Then the map of local rings ϕ7x : OY,y Ñ OX,x maps the maximal ideal into
the maximal ideal, and being a ring map, it sends m2y into m2x . Therefore, it induces a map of
κpyq-vector spaces:
ϕ˚x : my {m2y ÝÝÑ mx {m2x . (11.2)
˚
For a second morphism ψ : Y Ñ Z , we have pψ ˝ ϕq˚x “ ϕ˚x ˝ ψϕpxq .
The tangent space, however, does not satisfy the same functorial property in general. The
issue is that the required duals will be with respect to different fields. More precisely, ϕ˚x
is only a map of κpyq-vector spaces, and there is no natural way to make my {m2y into a
κpxq-vector space.
An exception occurs when X and Y are schemes of finite type over a field k , and both
x P X and y P Y are k -points, so that κpxq “ κpyq “ k . In this case, each quotient m{m2
is a finite-dimensional k -vector space. We can then take duals to obtain a map

dϕx : Tx X ÝÝÑ Ty Y. (11.3)

In this setting, the tangent maps satisfy dpψ ˝ ϕqx “ dψϕpxq ˝ dϕx for any morphisms of
k -schemes ϕ : X Ñ Y and ψ : Y Ñ Z .

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Classical tangent spaces

To understand how the previous definition relates to the classical situation, let us consider
an affine variety X in Ank . Before proceeding, we note that one does not have to go all the
way to the local ring OX,p to compute the cotangent space. If A is a ring, and m is a maximal
ideal, then there is an isomorphism of A{m-vector spaces
m{m2 ÝÝÑ mAm {m2 Am .
This follows from the canonical isomorphism of A-modules A{m “ Am {mAm , and
m{m2 “ m bA A{m “ m bA Am {mAm “ mAm {m2 Am .
Let us apply this to compute the tangent space of affine space Ank “ Spec krx1 , . . . , xn s at
a k -point p. For simplicity, we assume that p “ p0, . . . , 0q is the origin, corresponding to
the maximal ideal m “ px1 , . . . , xn q. (We may always achieve this by a linear change of
coordinates.) The cotangent space m{m2 is then a k -vector space of dimension n, with basis
x1 , . . . , x n .
More generally, consider an affine scheme X “ Spec A, where A “ krx1 , . . . , xn s{a for
an ideal a Ă krx1 , . . . , xn s, and assume that X contains p. Writing n Ă A for the maximal
ideal of the origin in A, the cotangent space at p is isomorphic to the k -vector space
m
n{n2 » 2 . (11.4)
m `a
Example 11.2. Say a “ pf q is generated by f “ y 2 ` x3 ` 3x ` 4y in A2C . Then
m px, yq px, yq
“ 2 “ 2 “ Cx.
m2 ` pf q px , xy, y 2 , y 2 ` x3 ` 3x ` 4yq px , xy, y 2 , 3x ` 4yq
If f “ y 2 ` x3 ` x2 , then
m px, yq px, yq
“ 2 “ 2 “ Cx ‘ Cy.
m2 ` pf q px , xy, y 2 , y 2 ` x3 ` x2 q px , xy, y 2 q
△
It is clear that the quotient (11.4) only depends on the ‘linear part’ of the Taylor expansion
of the elements of a at p. More precisely, let us define, for f P krx1 , . . . , xn s,
Bf Bf
dp f “ ppq ¨ x1 ` ¨ ¨ ¨ ` ppq ¨ xn (11.5)
Bx1 Bxn
Bf
Here the Bx i
denote the formal derivatives of f . They are defined using the usual formulas for
differenting monomials, that is, if f “ xe11 . . . xenn , then Bx
Bf
1
“ e1 x1e1 ´1 . . . xenn , and so on.
All the usual rules for derivatives still hold, such as the chain rule, quotient rule etc. However,
one should keep in mind that there may be unusual behaviour when k has characteristic p.
For instance, if f “ xp then Bf Bx
“ 0, even though it is not constant.
In terms of (11.5), the cotangent space of X at p is given by
m{pm2 ` dp paqq,
where dp paq Ă m{m2 is the subspace spanned by all the linear parts dp f for f P a. In

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other words, the cotangent space is obtained from m{m2 by modding out by the linearized
equations defining X at p.
To compute Tp X , we are interested in the dual of this vector space. For this, note that m{m2
is generated by x1 , . . . , xn , so we may identify m{m2 “ pk n q_ . Under this identification,
dp f corresponds to a linear functional on k n . Taking duals, we get an isomorphism
Tp Ank » k n .
where the i-th basis vector of k n corresponds to the functional m{m2 Ñ k which sends xi to
1 and the other xj ’s to 0.
To compute Tp X , we recall the following fact from linear algebra. Let V be a finite
dimensional k -vector space and let S Ă V _ be a linear subspace of the dual space of V .
Consider the subspace of V defined by
␣ ˇ (
W “ v P V ˇ f pvq “ 0 for all f P S
Then restrictions of linear functionals induces an isomorphism W _ » V _ {S . In particular,
since W is isomorphic to pW _ q_ , there is an isomorphism W “ pV _ {Sq_ .
Applying this to V “ k n and S “ dp paq Ă m{m2 “ pk n q_ , we may identify the tangent
space Tp X with
" *
n Bf Bf
Tp X “ pv1 , . . . , vn q P k | v1 ppq ` ¨ ¨ ¨ ` vn ppq “ 0 @f P a . (11.6)
Bx1 Bxn
This is the ‘classical tangent space’ of X at p. The description (11.6) is valid for any point
p “ pa1 , . . . , an q, not just the origin.
If a “ pf1 , . . . , fr q, then Tp X is the subspace of k n defined by the r linear equations in
kn :
dp pf1 q “ ¨ ¨ ¨ “ dp pfr q “ 0
One sometimes also considers the affine tangent space of X at the point p. This is the
subvariety of Ank defined by the r linear equations
n
ÿ Bfj
ppq ¨ pxi ´ ai q “ 0 for j “ 1, . . . , r. (11.7)
i“1
Bxi

One can view this as the tangent space Tp X , but translated so that p is the origin. Another

Tp X

p
X

Figure 11.1 The affine tangent space

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way to view the tangent space is in terms of the Jacobian matrix

¨ Bf1 Bf1 ˛
Bx1
¨ ¨ ¨ Bx n

J “ ˝ ... .. ‹ . (11.8)
˚
. ‚
Bfr Bfr
Bx1
¨¨¨ Bxn

Then the matrix Jppq “ J b´A κppq is ¯ an r ˆ n-matrix with entries in κppq “ k , which can
Bfi
be identified with the matrix Bxj ppq . From (11.6), we see that Tp X can be identified with
the null space of Jppq. In particular, the dimension of Tp X is given by
dimk Tp X “ n ´ rank Jppq. (11.9)
Example 11.3. Consider the quadric surface X “ V px2 ` yz ´ zq in A3C . Then the Jacobian
at a closed point p “ pa, b, cq is equal to
` ˘
J “ 2a c ´ 1 b .
Therefore, Tp X has dimension 3 at the k -point p “ p0, 0, 1q and dimension 2 otherwise. △

Zariski tangent spaces and the ring of dual numbers

When X is a scheme over a field k , there is an interesting relation between the Zariski tangent
space at k -points and the ring krϵs{ϵ2 . This ring is called the ring of dual numbers over k .
The spectrum of krϵs{ϵ2 is a very simple scheme: its underlying topological space is a single
point. However, the non-reduced structure on Spec krϵs{ϵ2 shows that it is more interesting
than Spec k . It is common to picture it as a point ε with a vector ‘sticking out of it’.
The following proposition generalizes the perspective in differential geometry where
tangent vectors are defined as equivalence classes of curves passing through p.

Proposition 11.4. Let X be a scheme over a field k and let p P X be a k -point. Then
there are natural bijections between:
(i) The elements of Tp X , i.e., k -linear maps mp {m2p Ñ k
(ii) Morphisms of schemes over k ,
Spec krϵs{ϵ2 ÝÝÑ X
with image p.

Proof We may assume that X is affine, say X “ Spec A, where A is a k -algebra. Let
m “ mp be the maximal ideal, and let ρ : A Ñ k “ A{m be the quotient map corresponding
to p. As p is a k -point, the composition of the structure map k Ñ A and ρ : A Ñ k is the
identity map.
First of all, any map of k -algebras A Ñ krϵs{ϵ2 factors uniquely via A{m2 Ñ krϵs{ϵ2 .
Hence there is a natural bijection
HomAlg{k pA, krϵs{ϵ2 q “ HomAlg{k pA{m2 , krϵs{ϵ2 q
Next, consider the exact sequence
0 ÝÝÑ m{m2 ÝÝÑ A{m2 ÝÝÑ A{m ÝÝÑ 0

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This sequence splits, as the structure map k Ñ A{m2 provides a section, so we may write
A{m2 “ k ‘ m{m2 . Given a map of k -algebras h : A{m2 Ñ krϵs{ϵ2 , the restriction of h
defines a k -linear map m{m2 Ñ kϵ » k . This defines a map of k -vector spaces
HomAlg{k pA{m2 , krϵs{ϵ2 q ÝÝÑ Homk pm{m2 , kq
The inverse is constructed as follows: given α : m{m2 Ñ k , define h by hpa ` tq “ a ` αptq
where a P k and t P m{m2 . One checks that this is a map of k -algebras.
There is also a description of Tp X as the vector space of derivations of OX,p into k , i.e.,
k -linear maps D : OX,p Ñ k satisfying the Leibniz rule (see Exericise 20.11.2).

11.2 Nonsingular schemes

We say that a Noetherian scheme X nonsingular or regular, at a point p if OX,p is a regular
local ring. Here we recall that a Noetherian local ring A with maximal ideal m is defined to
be regular if the maximal ideal can be generated by n elements, where n “ dim A is the
Krull dimension of A.
For instance, if p P Ank is the origin, corresponding to the maximal ideal m “ px1 , . . . , xn q,
then the local ring is isomorphic to
OAnk ,p “ krx1 , . . . , xn spx1 ,...,xn q
which is regular, as it has dimension n and m is generated by n elements. This is the intuition
of regularity: the n generators for m define n ‘coordinate functions’ on X near the point.
It is can certainly happen that mp requires more generators than the dimension of OX,p .
For instance, if X “ Spec krx, ys{px2 ´ y 3 q, and p is the origin, then OX,p has dimension
1, but mp “ px, yq cannot be generated by a single element. If this happens, we say that p is
a singular point. The set of singular points of X is denoted by singpXq. The scheme X is
said to be nonsingular if every point is nonsingular.
As it turns out, for a Noetherian local ring A, the condition of regularity can be formulated
in terms of the quotient m{m2 , which is naturally a vector space over the residue field A{m.
As such, it is finite-dimensional, because m is finitely generated. If x1 , . . . , xr are elements
from A that generate m{m2 as a A{m-vector space, then Nakayama’s lemma implies that the
x1 , . . . , xr generate m. Furthermore, by Krull’s Principal ideal theorem, we have
dim A “ ht m ď r “ dimA{m m{m2 ă 8 (11.10)
In other words, A is regular if and only if equality holds in the inequality (11.10).
On the geometric side, this means that a scheme X is nonsingular at a point p if and only
if the cotangent space mp {m2p has dimension equal to the Krull dimension of OX,p .
Here are three important facts about regular local rings.

Theorem 11.5. Let A be a regular local ring. Then

(i) For each prime ideal p Ă A, the ring Ap is also a regular local ring.
(ii) A is an integral domain.
(iii) A is a UFD.

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Proof For (i), see [Matsumura, Chapter 7, Theorem 45]. For (ii), see (Atiyah, 2018, p.
123).
The first item (i) is a rather non-trivial result; it is not even obvious that a localization of a
polynomial ring is regular.
The second item tells us that in a nonsingular scheme, any point is contained in a unique
irreducible component, and this component is an integral scheme. Hence the scheme is a
disjoint union of integral schemes. In particular, any connected nonsingular scheme is integral.
This observation is sometimes useful when proving that a ring is an integral domain (see
Exercise 11.6.17).
The third item (iii) is known as the Auslander–Buchsbaum theorem. This will be used
several times in this book, in particular when we discuss divisors in Chapter 17.
Another consequence of (i) is the following:

Corollary 11.6. A Noetherian scheme is nonsingular if and only if it is nonsingular at

every closed point.

Example 11.7. The schemes AnZ and PnZ are nonsingular. This is because the local rings are
localizations of the polynomial ring Zrt1 , . . . , tn s, and these are regular. More generally, AnA
and PnA is nonsingular if Spec A is. △
Example 11.8. The C-scheme X “ SpecpCrxs{px ` x2 qq is nonsingular, because it is
isomorphic to the two-point scheme Spec C \ Spec C.
On the other hand, the scheme Spec Crxs{px2 q is singular, because Crxs{px2 q is a local
ring of dimension 0, and the maximal ideal m “ pxq is nonzero. Alternatively, Crxs{px2 q is
clearly not an integral domain. △
Example 11.9. More generally, any regular local ring of dimension 0 must be a field (because
it is both Artinian and an integral domain.) Hence, X “ Spec A is a nonsingular scheme of
dimension 0, then A is a product of fields. △
Example 11.10 (Complete intersections). If A is a regular local ring of dimension n, and
f P m is a a nonzero element, then the quotient A{pf q is regular if and only if f R m2 .
Indeed, A is an integral domain, so A{pf q has dimension n ´ 1. If n “ m{pf q denotes the
maximal in A{pf q, then
n m
2
» 2
n m ` pf q
which has dimension n ´ 1 if and only if f is non-zero in m{m2 , i.e., f R m2 .
More generally, an inductive argument shows that if f1 , . . . , fr P m are elements of A
such that the classes f1 , . . . , fr are linearly independent in m{m2 , then A{pf1 , . . . , fr q is a
regular local ring. △
Example 11.11. Consider the ring A “ Zrxs{px2 ` 4q and the maximal ideal m “ p2, xq.
The scheme X “ Spec A is singular at the point x corresponding to m. Indeed, here the
cotangent space is given by
m{m2 “ p2, xq{p4, 2x, x2 q “ p2, xq{p4, 2x, ´4q

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which is a F2 -vector space of dimension 2 (the elements are: 0, 2, x, x ` 2). △

11.3 The Jacobian criterion and smoothness

Using the description of the tangent space in terms of the Jacobian matrix (11.8), we get the
following very useful criterion to check if a scheme is singular at a point.

Proposition 11.12 (Jacobian criterion). Let k be a field and let X “

Spec krx1 , . . . , xn s{pf1 , . . . , fr q. Let p P X be a k -point and let Jppq “ J bA κppq,
where J is the Jacobian matrix (11.8). Then
rank Jppq ď n ´ dim X. (11.11)
and X is nonsingular at p if and only if equality holds.

In light of Corollary 11.6, this means that if X is a scheme of finite type over an alge-
braically closed field k , then we can determine whether X is singular by computing the
Jacobian matrices at k -points in local affine charts. Here are a few examples of this.

Example 11.13. Let X “ Spec Crx, ys{py 2 ´ x3 q. The Jacobian matrix is given by

J “ p´3x2 2yq
which has rank 1 at every C-point except the origin. At the origin, the local ring is given by
` ˘
OX,p “ Crx, ys{py 2 ´ x3 q px,yq

and the cotangent space mp {m2p » Cx̄ ‘ Cȳ has dimension 2. △

Example 11.14. Consider the C-scheme X “ SpecpCrx, y, zs{pxy ´ z 2 qq. The Jacobian
is given by the matrix
` ˘
J “ y x ´2z
which has rank 1 at every point, outside the origin p “ p0, 0, 0q, so the tangent space Tp X
has dimension 2 at these points. At p, the cotangent space has dimension 3: it is given by

mp {m2p “ px, y, zq{px2 , xy, y 2 , xz, z 2 q » k x ‘ k y ‘ k z.

△
Example 11.15. Let X Ă A3k be the curve defined by the ideal I “ pxz ` y 2 , x2 ` y 2 ` 2zq.
The Jacobian is given by
ˆ ˙
y x 2z
J“
2x 2y 2
The singular locus singpXq is defined by the ideal generated by the 2 ˆ 2-minors of J and
I . If char k ‰ 2, we find that there is a unique singular point the origin p “ p0, 0, 0q. If
2 2 2
char k “ 2, then X is ?actually non-reduced, because x ` y ` 2z “ px ` yq , but x ` y R I .
In fact, in this case, I “ px ` z, y ` zq. △

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Unfortunately, the Jacobian criterion is not sufficient to check nonsingularity in general.

We will see simple counterexamples shortly. In general, if X is scheme of finite type over
a field k and p P X is a point, then the property that the Jacobian matrix at p has maximal
rank at p is called ‘smoothness’. Smoothness and nonsingularity are equivalent notions over
perfect fields, but not in general.

Definition 11.16 (Smoothness over a field). Let X be a scheme of finite type over a
field k . We say that a point p P X is smooth at p if there is an affine neighbourhood
U “ Spec A of p, where A “ krx1 , . . . , xn s{pf1 , . . . , fr q and the Jacobian J of the
f1 , . . . , fr satisfies
rank Jppq “ n ´ dimp X.
where dimp X is the dimension of X at p.

One key difference between ‘smoothness’ and ‘nonsingularity’ is that smoothness should
really be considered as a property of the morphism X Ñ Spec k , whereas nonsingularity
is an absolute property of X (this will be explained in Exercise 20.11.14). Smoothness is
furthermore a property which is stable under base change, wheras nonsingularity is not. The
next result tells us that we can relate the two notions by going to the algebraic closure of the
base field.

Proposition 11.17. Let X be a scheme of finite type over a field k and let p P X be a
closed point. Then the following are equivalent:
(i) X is smooth at p.
(ii) For each closed point p̄ P Xk̄ which maps to p, Xk̄ is smooth at p̄.
(iii) For each closed point p̄ P Xk̄ which maps to p, Xk̄ is nonsingular at p̄.

Proof Over an algebraically closed field, the conditions of nonsingularity and smoothness
are equivalent, by the Jacobian criterion, hence (ii) and (iii) are equivalent.
To prove (i)ô(ii), note that the statement is local around p, so we may assume that
X “ Spec A is affine, where A “ krx1 , . . . , xn s{pf1 , . . . , fr q.
The base change morphism Xk̄ Ñ X is surjective (by the Lying Over Theorem). Let
p̄ P Xk̄ be a point which maps to p. Then as k̄ is algebraically closed, Xk̄ is nonsingular at p̄
if and only if Xk̄ is smooth at p̄, and this happens if and only if rank Jpp̄q “ n ´ dim X .
But as rank Jppq “ rank Jpp̄q, this happens if and only if X is smooth at p.

Proposition 11.18. Let X be a scheme of finite type over a field k and let p P X be a
closed point. If X is smooth at p, then X is nonsingular at p. The converse holds if κppq
is separable over k .

Proof The statement is local on X , so we may assume that X “ Spec A where A “

krx1 , . . . , xn s{pf1 , . . . , fr q. We may also assume that X is irreducible, so dimp X “
dim X .
The statement is clear if κppq “ k and p is a k -point, by the Jacobian Criterion. For
a general closed point p, we reduce to this case as follows. Consider the base change

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Xκppq “ Spec B , where

B “ A bk κppq “ κppqrx1 , . . . , xn s{pf1 , . . . , fr q
As k Ñ κppq is a finite field extension, hence integral, also A Ñ B is integral, and so
Xκppq Ñ X is surjective by the Lying Over Theorem. Let p̄ P Xκppq be a κppq-point which
maps to p.
The Jacobian matrices Jppq and Jpp̄q of f1 , . . . , fr at the points p and p̄ have the same
rank (we may compute the rank by going to k ). Therefore, Xκppq is smooth at p̄ if and only
if X is smooth at p. Since p̄ is a κppq-point, we see that Xκppq is nonsingular at p̄ by the
Jacobian Criterion. This means that the classes of f1 , . . . , fr are linearly independent in
mp̄ {m2p̄ . Considering the map mp {m2p Ñ mp̄ {m2p̄ , we see that f1 , . . . , fr must be linearly
independent in mp {m2p as well, and hence OX,p is a regular local ring by Example 11.10.
For the converse, we will consider the base change Xk̄ to the algebraic closure k̄ . Suppose
that k Ă κppq is separable of degree d. By the Primitive Element Theorem, we have
κppq “ kpξq “ krts{pP ptqq for some polynomial P with P 1 pξq ‰ 0 in κppq. As P is
separable, we have, as rings
κppq bk k̄ “ k̄ ˆ ¨ ¨ ¨ ˆ k̄.
This implies that the scheme-theoretic fiber of Xκppq Ñ X over p consists of d reduced
points.
Let n “ m bk k̄ Ă Abk k̄ . Then n is the ideal describing p1 , . . . , pd as a closed subscheme
of Xk̄ . We have
n mp mp
2
“ 21 ‘ ¨ ¨ ¨ ‘ 2d
n mp1 mpd
Note that n{n2 “ m{m2 bk k̄ has dimension d ¨ dimk pm{m2 q as a k̄ -vector space. Therefore,
we have
d
ÿ mp
d ¨ dimk pm{m2 q “ dimk̄ p 2i q (11.12)
i“1
mpi

Recall that dimk̄ pmpi {m2pi q ě dim X for every i. If this is an equality for every i, then
(11.12) implies that p is nonsingular. Conversely, some inequatlity is strict, then the sum on
the right-hand side is ą d dim X and hence dimk pm{m2 q ą dim X . In other words, p is
nonsingular if and only if pi is nonsingular for every i. Therefore, assumping p is nonsingular,
the pi are nonsingular, hence smooth, and hence p is also smooth by Proposition 11.17.

Here is a concrete example where the converse fails:

Example 11.19 (A nonsingular, but non-smooth curve). Let k “ Fp ptq where p ą 2 is a
prime number and let X be the plane curve defined by the polynomial f “ y 2 ` xp ` t.
Note that X is integral. In fact, f is irreducible even in k̄rx, ys, so also Xk̄ is integral.
The Jacobian of f is given by
` ˘ ` ˘
J “ pxp´1 2y “ 0 2y .
The closed subset of X defined by the vanishing of the Jacobian matrix is given by tP u,

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where P P X is the point corresponding to the maximal ideal m “ pxp ` t, yq (this is a

prime ideal, because xp ` t is irreducible over k ). This means that X is nonsingular at every
point in X ´ tP u. However, X is not smooth at P , as the Jacobian is identically 0 there.
To show that X is still nonsingular at P , note that f P krx, ys is not contained in
m “ px2p ` 2t ` t2 , ypxp ` tq, y 2 q, and so OX,P is regular by Example 11.10. Hence X
2

is a nonsingular scheme.
The point of this example is that ‘smoothness’ behaves well under base change, whereas
nonsingularity does not. If we go to the algebraic closure k̄ , then t has a p-th root s, and then
Spec k̄rx, ys{py 2 ` xp ` sp q is no longer nonsingular (e.g., at the point px, yq “ p´s, 0q).
△

Example 11.20. Let k be a field of characteristic 2 containing an element α P k which

is not a square. Let P P A2k be the closed point of A2k corresponding to the maximal ideal
m “ pv, u2 ` αq in kru, vs.
The plane affine curve X Ă A2k defined by f “ v 2 ´ upu2 ` αq has a Jacobian given
by J “ pu2 ` α, 0q. This has rank 1 except at the point P corresponding to the maximal
ideal m “ pu2 ` α, vq. Therefore, X is smooth at all points of X ´ tP u, but not at P . X is
however nonsingular at P , as f does not belong to m2 “ pv 2 , vpu2 ` αq, u4 ` α2 q.
If K “ kpβq is the extension of k obtained by adjoining a square root β of α, then
the curve XK “ X ˆk K acquires a singular point. In terms of β , f takes the form
f “ v 2 ´ upu2 ` β 2 q “ v 2 ´ upu ` βq2 , and XK has a singular point at pu, vq “ p´β, 0q.
△

Proposition 11.21. Let X be a scheme of finite type over field k . Then the smooth locus
X sm “ t p P X | f is smooth at p u is an open subset of X .

Proof Let p P X be a point and choose an open set U “ Spec A as in the definition of a
smooth point. If the Jacobian matrix J of the defining polynomials has maximal rank n ´ r
at x, then some pn ´ rq ˆ pn ´ rq-minor of J is non-zero, and it will continue to be non-zero
in some open neighborhood W of p. Therefore W Ă X sm and hence X sm is open.

Corollary 11.22. For a reduced scheme of finite type over a perfect field k , the smooth
locus X sm is a dense open subset of X .

Proof We may assume that X is irreducible, hence integral. By the previous proposition, it
suffices to show that X sm is nonempty, i.e., X contains a single smooth point. To see this,
it is simplest to use the fact that X is birational to a hypersurface Y Ă Ank (Proposition
10.28). Write Y “ V pf q, where f is an irreducible polynomial in krx1 , . . . , xn s. For Y ,
the statement of the proposition is clear, because if the Jacobian matrix is zero everwhere
Bf
on Y , then Bx i
P IpY q “ pf q for each i. But by degree reasons this means that the partial
Bf Bf
derivatives Bx1 , . . . , Bx n
must all be the zero polynomial. This can only happen if k has
characteristic p and f is a polynomial in xp1 , . . . , xpn . Over an algebraically closed field, this
even implies that f “ g p is the p-th power of a polynomial g . However, this is not possible,
as we assumed f to be irreducible.
Finally, let U Ă X and V Ă Y be open sets related by an isomorphism ι : U Ñ V . By

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what we just showed, the smooth locus Y sm is dense in Y , and hence intersects V . As ι is an
isomorphism, it induces an isomorphism on tangent spaces, and hence U contains smooth
points as well.

Corollary 11.23. Let X be a scheme of finite type over a perfect field k . Then X is
nonsingular if and only if it is smooth over k .

Proof X is nonsingular if and only if it is nonsingular at each closed point X , which

happens if and only if it is smooth at every closed point. On the other hand, the smooth locus
of X is open, so if it contains all closed points of X , it must be equal to X , and hence X is
smooth.
Example 11.24. For varieties over non-perfect fields, it can happen that the smooth locus
is empty. For instance, if k “ Fp ptq for an odd prime p and X is the affine scheme defined
by the polynomial f “ xp ´ ty p in A2k , then the Jacobian matrix is the zero matrix, so
X sm “ H. This is ‘explained’ by the fact that the base change Xk̄ is non-reduced, hence
singular at every point: if s is a p-th root of t, then f “ px ´ syqp . △

11.4 Normal schemes

Recall that an integral domain A is said to be normal if it is integrally closed in its fraction
field K “ KpAq. This means that any element w P K which satisfies a monic equation
wn ` an´1 wn´1 ` ¨ ¨ ¨ ` a0 “ 0 with coefficients in A, is already contained in A.
Here are a few examples of normal rings:
Example 11.25. Any unique factorization domain A is normal. This includes rings such as
Z or krx1 , . . . , xn s where k is a field.
To prove this, suppose uv P K is integral over A. Since A is a UFD, we may assume
without loss of generality that u and v are coprime (i.e., they share no common prime factors).
By assumption, uv satisfies a monic polynomial equation:
´ u ¯n ´ u ¯n´1
` an´1 ` ¨ ¨ ¨ ` a0 “ 0,
v v
where ai P A. Multiplying by v n gives:
un ` an´1 un´1 v ` ¨ ¨ ¨ ` a0 v n “ 0.
This implies that v divides un , which by our assumption means that v must be a unit in A.
Therefore, uv P A, and hence A is normal. △
Example 11.26. Any localization S ´1 A of a normal integral domain A is normal. In fact,
there is a partial converse: an integral domain A is normal if and only if Ap is normal for all
prime ideals p, if and only if Am is normal for all maximal ideals m. △
Example 11.27. If A is a normal, then so is Arxs. △
? ?
Example 11.28. The ring Zr 5s is not ?
normal. The fraction field equals K “ Qp 5q, and
this contains the golden ratio ϕ “ 1`2 5 . Note that ϕ satisfies the monic equation
ϕ2 ´ ϕ ´ 1 “ 0,

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Motivated by the many desirable properties of normal rings, we make the following
definition for schemes.

Definition 11.29. We say that a scheme X is normal if for each point x P X , the local
ring OX,x is an integrally closed integral domain.

The primary example of a normal scheme is X “ Spec A, where A is a normal integral

domain. Note however, that in the definition of a normal scheme we do not make the
assumption that X is integral. For instance, X “ SpecpC ˆ Cq is a normal scheme. However,
if X is normal at a point x P X , the local ring OX,x is an integral domain, which implies
that there is a unique irreducible component of X containing x. This component, with its
scheme structure as an open subscheme, is integral. Hence any normal scheme is a disjoint
union of integral normal schemes. In any case, a normal scheme is always reduced.

Example 11.30. AnZ and PnZ are normal schemes, because the local rings are isomorphic to
localizations of the polynomial ring Zrx1 , . . . , xn s. △

Lemma 11.31. Let X be an integral normal scheme. Then OX pU q is normal for every
open subscheme U Ă X .

Proof Let A “ OX pU q. Suppose first that U is affine. Then all of the localizations Ap are
normal, by assumption, and so A is normal as well by Example 11.26.
If U Ă X is a general open subset, take elements u, v P A and suppose that u{v P K
satisfies a monic relation

pu{vqn ` an´1 pu{vqn´1 ` ¨ ¨ ¨ ` a0 “ 0 (11.13)

where ai P A. If V Ă U is an affine open, then restricting (11.13) to V shows that

pu|V q{pv|V q satisfies a monic equation. As OX pV q is normal, we have u|V {v|V P OX pV q,
i.e., there is a regular element sV P OX pV q so that v|V sV “ u|V . If W Ă V is another
affine, then we similarly have an element sW so that v|W sW “ u|W , and clearly sV |W “ sW
as OX pW q is an integral domain. Therefore the sections sV glue to a section s P OX pU q,
which has the property that as “ b, i.e., a{b “ s P OX pU q.

Although it is not obvious from the definition, the notion of normality is related to
nonsingularity. This is because of the algebraic fact that a regular local ring is normal. In fact,
by the Auslander–Buchsbaum theorem local regular rings are unique factorization domains,
and hence they are normal by Example 11.25.

Proposition 11.32. Any nonsingular scheme is normal.

While normal schemes are more general than regular schemes, they still have several nice
properties. For instance, if X is normal and of finite type over a field, then:
(i) The singular locus of X has codimension at least 2 in X (Theorem ??).
(ii) Any finite birational morphism Y Ñ X is an isomorphism (Proposition 11.54).

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(iii) Any rational function defined outside a closed set of codimension at least 2,
extends to a regular function on all of X (Theorem 11.42).

Normalization
A useful fact is that any integral scheme has a normalization. More precisely, if X is an inte-
gral scheme, then its normalization is a normal scheme X together with a dominant morphism
π : X Ñ X , which is universal among dominant morphisms from normal schemes.

Theorem 11.33 (Normalization). For an integral scheme X , there is a normal scheme

X , and a morphism π : X Ñ X satisfying the following universal property: For any
dominant morphism h : Y Ñ X from a normal scheme Z , there is a unique morphism
h : Z Ñ X such that h “ π ˝ h.
X
h π

h
Z X

Proof If X “ Spec A is affine, define X “ Spec A where A Ą A is the integral closure

of A in K “ KpXq, and π : Spec A Ñ Spec A is the morphism induced by the inclusion.
Note that A is an integral domain, and that Spec A is normal.
Next we verify the universal property. Let h : Z Ñ X be a dominant morphism from
an integral normal scheme Z . This means that the map h7 : A Ñ OZ pZq is injective (by
Example 7.17). As the ring OZ pZq is normal, the ring map A Ñ OZ pZq factors uniquely as
A Ñ A Ñ OZ pZq. This means that h factors via X , as we want.
The proof for a general scheme involves gluing the morphisms Ui Ñ Ui for an affine
cover tUi u of X . See Example B.17 for details.

Passing from X to its normalization X is a quite common operation in algebraic geometry.

This is because X it has better properties than X , and we can hope to study X by performing
computations on X .

Proposition 11.34. For a Noetherian integral scheme X , the normalization X has the
following properties:
(i) π : X Ñ X is surjective.
(ii) X and X have the same dimension.
(iii) There is a dense open subset U Ă X so that π restricted to π ´1 pU q is an
isomorphism.
(iv) If X is of finite type over a field or over Z, then π : X Ñ X is a finite
morphism.

Proof All of these properties are ‘local on X ’, so by the gluing construction used in the
construction of X , we reduce to X “ Spec A and X “ Spec A and π is induced by the
inclusion A Ă A. The first two statements (i) and (ii) follow from the Going-Up theorem.

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The statement (iii) holds because by construction, X and X have the same fraction field K ,
and π maps the generic point η “ Spec K of X maps to the generic point of X . Finally,
the statement (iv) follows from Theorem A.20, which tell us that with our assumptions, A is
finite as an A-module.

For a general scheme X , the normalization morphism π : X Ñ X may fail to be a finite.

In fact, there are examples of Noetherian integral domians A whose integral closure is not
finite over A.

Example 11.35 (The Cuspidal cubic). Consider the cuspidal cubic curve X “ Spec A
where A “ krx, ys{py 2 ´ x3 q. In Example ??, we showed that there is an isomorphism
of k -algebras A Ñ krt2 , t3 s defined the assignment x ÞÑ t2 and y ÞÑ t3 . It is clear that
krt2 , t3 s is an integral domain with fraction field K “ kptq. On the other hand, this ring is
visibly not normal, because t R krt2 , t3 s, but yet it satisfies the monic equation T 2 ´ t2 “ 0.
As krts is integrally closed in K (being a UFD), any element in K which is integral over A,
can be written as a polynomial in t. Therefore the integral closure of A is given by A “ krts.
The corresponding normalization morphism Spec A Ñ Spec A is exactly the morphism
π : A1k Ñ Spec A from Example ??. △

Example 11.36 (The Nodal cubic). Let X “ Spec A, where A “ krx, ys{py 2 ´ x3 ´ x2 q,
where k now is a field of characteristic not equal to 2 (if the characteristic is 2, we are back
in previous cuspidal case). This is the nodal cubic curve in A2k . Here it is less obvious what
the normalization should be, but it helps to think about it geometrically.
If we think of the corresponding affine algebraic set t px, yq | y 2 “ x3 ` x2 u Ă A2 pkq,
we see that the origin p0, 0q is a special point: any line l defined by an equation y “ tx,
intersects the curve in p0, 0q P X and one more point (namely x “ t2 ´ 1 and y “ tpt2 ´ 1q).
The polynomial map t ÞÑ pt2 ´ 1, t3 ´ tq gives a parameterization of the curve, which is
one-to-one when t ‰ 0.
This geometric observation can be turned into algebra as follows. Introduce the parameter
t “ y{x in the function field K of X . The equation y 2 “ x3 ` x2 then reduces to t2 “ 1 ` x,
after dividing by x2 . Moreover, the element t is integral, because it satisfies the monic equation
T 2 ´ x ´ 1 “ 0 (which has coefficients in A). As x “ t2 ´ 1 and y “ x ¨ y{x “ t3 ´ t,
we see that
A “ krt2 ´ 1, t3 ´ ts Ă krts Ă K “ kptq.
Moreover, since krts is integrally closed in K , we see that A “ krts. The normalization map
π : Spec A Ñ Spec A is an isomorphism outside the origin p0, 0q P X . Geometrically, the
map π identifies two points pt ` 1q and pt ´ 1q in A1k to the origin in X .
△
? ?
Example 11.37. Let A “ Zr ´7s. Then A is not normal, because the element α “ 21 ` 2´7
?
belongs to the fraction field Qp ´7q, and satisfies the integral equation x2 ´ 2x ` 8 “ 0,
yet α R A. In fact, we have A “ Zr2αs and the normalization is given by A “ Zrαs. This is
because Zrαs Ă A, and Zrαs is integrally closed by Exercise 11.6.8.
Note that there are two ring maps Zrαs Ñ Z{2; one which sends α to 0 and one that sends

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α to 1. Their kernels are the two prime ideals p1 “ p2, αq and p2 “ p2, α ´ 1q. Note that
p1 X Zr2αs “ p2, 2αq “ p2, 2α ´ 2q “ p2 X Zr2αs
?
In other words, p1 and p2 both map to the same prime ideal, q “ p2, 2αq “ p2, 1 ` ´7q
via the normalization map
π : Spec Zrαs ÝÝÑ Spec Zr2αs.
This example shares many similarities with the nodal cubic example. △
?
Example 11.38. Let A “ Z?r 8s. As in the previous example, we compute that the normal-
ization is given by A “ Zr 2s (which is normal by Exercise 11.6.8). Like in the cuspidal
cubic example, the normalization map
? ?
π : Spec Zr 2s ÝÝÑ Spec Zr 8s
is bijective on points, but it is not an isomorphism. △
Example 11.39 (The quadratic cone). Consider the affine scheme X “ Spec A where
A “ Crx, y, zs{pxy ´ z 2 q. Note that A is not a UFD as xy “ z 2 and one easily checks that
x, y and z all are irreducible elements, so we cannot immediately conclude that A is normal.
However, there is an isomorphism of rings
ϕ : A Ñ Cru2 , uv, v 2 s,
and the latter algebra is normal in its fraction field K “ Cpu2 , uv, v 2 q. Indeed, suppose
that T “ p{q P Cpu2 , uv, v 2 q satisfies a monic equation with coefficients in Cru2 , uv, v 2 s,
then in particular T P Cru, vs is a polynomial in u and v (as Cru, vs is integrally closed).
Therefore, q divides p and so T P Cru2 , uv, v 2 s. (For another proof, see Exercise 11.6.9.)

Q
A2

△
Example 11.40 (Toric varieties). Let C Ă Rn be a cone generated by finitely many lattice
points in Zn , and let X “ Spec krCs be the associated toric variety. Then X is normal.
To prove this, we begin by writing C “ H1 X ¨ ¨ ¨ X Hs where the Hi are half-spaces of
the form
Hi “ t v P Rn | ρi ¨ v ě 0 u,
where ρ P Zn is a primitive lattice vector, that is, ρi is not a positive integer multiple of
another vector in Zn .
Then, inside KpXq “ kpx1 , . . . , xn q, we have
s
č
krCs “ krHi s
i“1

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The intersection of normal subrings is again normal, so it suffices to prove that each k -
algebra krHi s is normal. To see this, since ρ is primitive, it can be extended to a basis of
Zn . After a change of cooordinates, we may therefore assume that ρ “ e1 . Then krHs “
krx1 , x˘1 ˘1
2 , . . . , xn s, which is clearly normal, being the localization of a polynomial ring.
This implies that all projective toric varieties, as defined in Section 6.1 are normal. △

11.5 Properties of normal schemes

The following is another nice property of normal schemes. It essentially says that rational
functions can be extended over codimension 2 subsets, that is, any rational function which is
regular over an open subset U Ă X can be extended uniquely to a regular function on all of
X , provided that X ´ U has has no codimension 1 components. This is sometimes referred
to as the Algebraic Hartogs’s Theorem, even though it merely is an algebraic version of a
much deeper result from complex function theory due to Friedrich Hartogs.
We will need the following algebraic fact, which says that a normal ring A is equal to the
intersection (inside the fraction field) of all its localizations at height 1 prime ideals.

Proposition 11.41. Let A be a normal integral domain. Then

č
A“ Ap (11.14)
htp“1

This has the following geometric consequence, which holds for all Noetherian normal
schemes.

Theorem 11.42 (“Algebraic Hartogs’s theorem”). Let X be a Noetherian normal

scheme, and let U Ă X be an open subset with codimX pX ´ U q ě 2. Then the
restriction map
OX pXq ÝÝÑ OX pU q (11.15)
is an isomorphism.

Proof As X is a disjoint of integral normal schemes, we may without loss of generality

assume that X is integral, so that we may view the rings of regular functions OX pV q as
subrings of the function field KpXq and the restriction map (11.15) is simply the inclusion
of subrings OX pXq Ă OX pU q.
For f P OX pU q, we need to show that f is in fact regular on all of X . In light of
Proposition ??, this is equivalent to saying that f P OX,p for every p P X . As the latter
condition only involves the local rings, we reduce to proving the proposition for X affine,
say X “ Spec A, where A is a normal integral domain.
Now, as X ´ U is assumed to be of codimension at least 2, U contains all points p
corresponding to prime ideals p of height 1. This means that OX pU q Ă OX,p “ Ap for
every such p. Therefore, by (11.14), we conclude that
č
OX pU q Ă Ap “ A “ OX pXq.
htp“1

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In particular, if X is integral, and a rational function f P KpXq is regular outside a closed

subset Z of codimension at least 2, then it is regular everywhere.
Example 11.43. The assumption that the codimension is at least 2 can not be removed: for
the open set Dptq Ă A1k we have OA1k pDptqq “ krt, t´1 s, whereas OA1k “ krts.
By the way, the above proposition gives another way to see the why OU pU q “ kru, vs for
the non-affine open set U “ A2k ´ V pu, vq in A2k from Chapter 5. △
Example 11.44. Consider the subring of krx, ys given by
A “ krx2 , xy, y 2 , x3 , x2 y, xy 2 , y 3 s
We can see directly that A is not normal, as x the monic equation T 2 ´ x2 “ 0 but x R A.
Similarly, y satisfies T 3 ´ y 3 “ 0. This implies that the extension A Ă krx, ys is an integral
extension.
Consider X “ Spec A and let m “ px, yq X A “ px2 , xy, . . . , y 3 q. We claim that the
morphism π : A2k Ñ X induced by A Ă krx, ys is an isomorphism outside tmu. To check
this, we can work over the distinguished open sets Dpx2 q and Dpy 2 q. Localizing at x2 , we
have Ax2 “ krx, ysx2 “ krx, ysx , so π is an isomorphism over Dpx2 q. Similarly, π is an
isomorphism over Dpy 2 q.
If p Ă A is a prime ideal of height 1, we must have p Ă X ´ tmu. This means that π
gives a bijection between the points of codimension 1 in X and the points of codimension 1
in A2k . Therefore,
č č č
Ap “ OX,p “ OA2 ,p “ krx, ys Ľ A.
htp“1 codim tpu“1 codim tpu“1

In this example, the singular locus of X is given by tmu, which is of codimension 2. △

Corollary 11.45. Let X and Y be affine schemes of finite type over a field k , and let
ϕ : X ´ Z Ñ Y be a morphism, where Z has codimension at least 2. Then if X is
normal, then ϕ extends to a morphism ϕ : X Ñ Y .

Proof Let Y Ă Ank be an embedding of Y into affine space. The composition X ´ Z Ñ

Y Ñ Ank is defined by n regular functions defined on X ´ Z . By the proposition above,
these extend to regular functions defined on all of X , and the extended map X Ñ Ank factors
via Y .
Example 11.46. The morphism π : A2k ´t0u Ñ P1k does not extend to a morphism π : A2 Ñ
P1k (it does not even extend as a continuous map). △
We saw that a Noetherian ring of Krull dimension 0 is normal if and only if it is a product
of fields. The next result gives a characterization of 1-dimensional normal rings. In short, it
says that a ring of dimension 1 is normal if and only if its localizations are discrete valuation
rings. Discrete valuation rings, in turn, have excellent properties: the structure of the ideals is
particularly simple, as the only non-zero ideals are powers of the maximal ideal. This fact
will be important later on when we study divisors on curves.

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Proposition 11.47. Let A be a Noetherian local domain of dimension 1 with maximal

ideal m. The following statements are equivalent:
(i) A is a DVR.
(ii) The maximal ideal m is principal.
(iii) Every non-zero ideal is a power of m.
(iv) dimA{m m{m2 “ 1.

Proof See Proposition A.48.

Corollary 11.48. Let X be a Noetherian integral scheme of dimension 1. Then X is

normal if and only if it is nonsingular.

Proposition 11.49. Let X be an integral scheme of finite type over an algebraically

closed field k . If X is normal, then the singular locus singpXq has codimension at least
2.

Proof It suffices to prove this in the case X “ Spec A is affine, where say A “ krx1 , . . . , xn s{a
for some a “ pf1 , . . . , fr q. By definition, the singular locus singpXq consists of all the
points x P X for which OX,x is not a regular local ring. Suppose that Z is a component of
singpXq of codimension 1 and let ζ be its generic point. Then as X is normal, the local ring
OX,ζ is a normal integral domain, of dimension 1, hence regular, by the above proposition.
Therefore, ζ R singpXq. However, with our assumptions, the singular locus singpXq is
a closed subset in X , being defined by the ideal b generated by a and the r ˆ r-minors
of the Jacobian matrix of the fi . Therefore singpXq must contain all the generic points of
its components, including ζ . This is a contradiction, and so singpXq cannot contain any
codimension 1 components.
There is a converse to this result, known as Serre’s Criterion. It gives a more geometric
characterisation of the property of ‘normality’ (which is fundamentally an algebraic notion).

Theorem 11.50 (Serre’s Criterion). Let X be a Noetherian integral scheme. Then X is

normal if and only
(i) X is nonsingular at all codimension 1 points, that is, OX,ζ is a regular local
ring for every point ζ of codimension 1.
(ii) Whenever U Ă X is an open set whose complement has codimension at
least 2, the restriction map (11.15) is an isomorphism.

Example 11.51 (Curves). A curve X is normal if and only it is nonsingular. △

Example 11.52 (Hypersurfaces). Let X be a nonsingular affine variety and let Y Ă X
be a hypersurface defined by f P OX pXq. Then the condition (ii) in Theorem 11.50 is
automatically satisfied (this is a non-trivial fact; see Eisenbud (2013)). Hence Y is normal if
and only if singpXq has codimension at least 2. △
Example 11.53. Let X be the scheme obtained from A2k by identifying two points p, q P A2k
(see Example B.23 on page 509). Then X is an integral scheme of dimension 2, and the

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singular locus consists of a single point p. However, consider now the complement U “ X´p,
which consists of two disjoint copies of A2k ´ p. The regular function f P OX pU q which
takes the value 0 on one component and 1 on the other clearly does not extend to all of X . △

The second important property of normal schemes concerns birational morphisms. Recall
that a morphism f : X Ñ Y is birational if it induces an isomorphism between open sets
U Ă X and V Ă Y . We have already seen several birational morphisms which are not
isomorphisms. For instance, when f is the blow-up of A2k at a point, there is a whole P1k
which collapses to a point. Even when f is finite, i.e., when it does not collapse a positive-
dimensional subscheme to a point, f can still fail to be an isomorphism.

f : Spec krxs ÝÝÑ Spec krx, ys{py 2 ´ x3 q

described in Example ?? is both birational and bijective on points, but not an isomorphism
in any neighborhood containing the origin. However, our next result shows that this type of
phenomenon does not occur if the target is a normal scheme. This previous is not normal, and
the failure of being an isomorphism is entirely concentrated at the singular point at the origin.

Proposition 11.54. Let X and Y be integral schemes, and let f : X Ñ Y be a finite,

birational morphism. If Y is normal, then f is an isomorphism.

Proof Since the property of being an isomorphism is local on the target, and finite mor-
phisms are affine, we may reduce to the case where both X and Y are affine, say X “ Spec B
and Y “ Spec A, and f is induced by an injective ring map ϕ : A Ñ B , where B is finite
as an A-module. Now, as f is birational, ϕ induces an isomorphism of the function fields
ϕK : KpAq Ñ KpBq. Since A is integrally closed, the finite extension A Ñ B must be an
isomorphism. Therefore, f is an isomorphism as well.

If X is a curve, being normal is the same thing as being nonsingular, so the normalization
π : X Ñ X produces a nonsingular curve which is birational to X .
In light of the relationship between the properties of normality and nonsingularity, one
can ask whether this holds in general i.e., whether the exists a ‘resolution of singularities’
π : Y Ñ X where Y is nonsingular, and π is birational. This is true if the ground field has
characteristic zero, by the following famous theorem of Hironaka:

Theorem 11.55 (Hironaka). Let k be a field of characteristic zero, and let X be a reduced
scheme of finite type over k . Then there exists a projective morphism π : Y Ñ X , where
Y is nonsingular and an open dense set U Ă X such that the restruction π ´1 pU q Ñ U
is an isomorphism.

Hironaka’s result is in fact more precise, saying that one can construct the morphism π
through a series of blow-ups along nonsingular closed subschemes, which are systematically
chosen to eliminate singularities. While the proof is highly technical, this process ultimately
yields a nonsingular scheme. The analogous statement in positive characteristic is unsolved,
and remains one of the most important open problems in algebraic geometry.

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11.6 Exercises
Exercise 11.6.1 (Tangent space to a projective variety). Continuing the previous exercise, let
X Ă Pnk be a closed subscheme given by homogeneous polynomials F1 , . . . , Fr . Show that
for a k -point p “ pa0 : ¨ ¨ ¨ : an q P X , the tangent space of X at p is given by
Tp X “ pKer Jppqq {kpa0 , . . . , an q
where J is the Jacobian matrix of the Fi . H INT: Consider the affine cone CpXq Ă An`1
and the map CpXq ´ 0 Ñ X .
Exercise 11.6.2. Compute the singular points of the scheme Spec Crx, ys{pxyq.
Exercise 11.6.3. Compute the singular points for the Whitney umbrella A “ krx, y, zs{py 2 `
x3 ´ x2 z 2 q.
Exercise 11.6.4. Let k be a field of characteristic 2 or 3 and let X Ă A2k be a curve defined
by the equation y 2 “ x3 ` ax ` b, where a, b P k . Show that X is nonsingular if and only
if 4a3 ` 27b2 ‰ 0. What happens if k has characteristic 2 or 3?
Exercise 11.6.5. Let k be a field and let C be the image of the morphism
f : A1k ÝÝÑ A3k ; t ÞÑ pt3 ´ 1, t4 ´ t, t5 ´ t2 q.
Explicitly, C is defined by the ideal I “ py 2 ´ xz, x2 y ` xy ´ z 2 , x3 ` x2 ´ yzq.
a) Show that the singular locus of C consists of the origin p “ p0, 0, 0q and show
that Tp X is 3-dimensional.
b) Show that the curve C does not lie on any nonsingular surface in A3k .
Exercise 11.6.6. Let A be an integral domain with fraction field K . Let x P K be an element.
Show that the following are equivalent:
a) x is integral over A
b) Arxs is a finite A-module
c) There exists a subalgebra A1 Ă A such that x P A1 and R is a finite A1 -module.
Exercise 11.6.7. Let K{Q be a finite field extension of Q and let α P K . Show that α is
integral over Z if and only if its minimal polynomial f pxq P Qrxs has integer coefficients.
?
Exercise 11.6.8. Let d be a square-free integer and let K “ Qp dq. Let OK denote the ring
of integers in K , that is, the integral closure of Z in K . ?
a) Show that the minimal polynomial of an element w “ a`b d, where a, b P Q
and b ‰ 0 is given by
W 2 ´ 2aW ` pa2 ´ b2 dq “ 0
b) Show that if w is integral over Z, then 2a, a2 ´ b2 d P Z. H INT: Use Exercise
11.6.7 ? ?
c) Show that OK “ Zr 1`2 d s if d ” 1 mod 4 and OK “ Zr ds otherwise.
Exercise 11.6.9. Prove directly that A “ Crx, y, zs{pz 2 ´ xyq is normal as follows. Let
B “ Crx, ys, so that A “ Brzs{pz 2 ´ xyq.
a) Show that A is a finite B -module of rank 2, with basis 1, z .

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b) Show that KpBq “ Cpx, yq and the field extension KpBq Ă KpBq has
degree 2.
c) Show that w “ u ` vz P A satisfies the monic polynomial
T 2 ´ 2uT ` pu2 ´ xyv 2 q “ 0.
d) Show that if w is integral over B , then u P Crx, ys; xyv 2 P Crx, ys; and hence
v P Crx, ys. Conclude that w P A.
Exercise 11.6.10. Let X be a Noetherian integral scheme. Show that X is normal if and
only if OX,x is normal for every closed point x P X . H INT: If y P X is any point, there is
a closed point in tyu.
Exercise 11.6.11. Show that the projective scheme X “ V` px21 ´ 15x0 x2 q Ă P2Z is normal.
Exercise 11.6.12 (The cone over a rational quartic curve). Consider X “ Spec A, where A
is the C-algebra
A “ Cru4 , u3 v, uv 3 , v 4 s » Crt0 , t1 , t3 , t4 s{pt0 t4 ´ t1 t3 , t31 ´ t20 t3 , t33 ´ t1 t34 q.
a) Show that X is a variety of dimension 2.
b) Show that X is nonsingular outside the origin p “ V pt0 , t1 , t3 , t4 q.
c) Show that
t21 t2
“ u2 v 2 “ 3
t0 t4
defines a regular function on X ´ p, but it does not extend to all of X . Conclude
that X satisfies (i) but not (ii) of Serre’s criterion.
d) Show that the ideal pt0 q is not principal in A. H INT: A primary decomposition
of pt0 q is given by
pt0 q “ pt0 , t21 q X pt0 , t4 q
Exercise 11.6.13. Show that the normalization of the scheme X “ Spec Zr6is is given by
Spec Zris.
Exercise 11.6.14. Let A be a local principal ideal domain which is not a field. Show that A
is a discrete valuation ring.
?
Exercise 11.6.15. Describe the normalization of the scheme Spec Zr ´3s.
Exercise 11.6.16 (Tangent space to projective space). Let k be a field and consider the
quotient map π : An`1k ´ 0 Ñ Pnk . Let P “ pa0 , . . . , an q P Ank ´ 0 and p “ pa0 : ¨ ¨ ¨ :
an q P Pk be two k -points. Show that π induces a map TP Ank Ñ Tp Pnk , and show that the
n

kernel can be naturally identified with k ¨ pa0 , . . . , an q Ă k n`1 . Conclude that the tangent
space of Pnk at p is given by k n`1 {pk ¨ pa0 , . . . , an qq.
Exercise 11.6.17. Let X Ă P3k be the twisted cubic curve, defined by the ideal I “
py 2 ´ xy, xw ´ yz, z 2 ´ ywq in krx, y, z, ws.
a) Show directly that X is nonsingular.
b) Show that X is connected. H INT: Look in the affine charts.
c) Deduce that I is a prime ideal.

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Exercise 11.6.18 (Nonsingular Toric Varieties).

a) Let C Ă Rn be a cone generated by finitely many vectors v1 , v2 , . . . , vn P Zn .
Assume that the vi form a basis for the lattice Zn . Show that the associated
toric variety X “ Spec krCs is nonsingular.
b) Let P Ă Rn be a lattice polytope. Suppose that for each vertex v P P , the cone
Cv at v can be generated by n vectors that form a basis for Zn . Show that the
associated toric variety X is nonsingular.
c) * Prove that the converses of parts (a) and (b) hold: if X is nonsingular, then, up
to a change of coordinates, C is generated by the standard basis vectors of Zn ,
and for each vertex v P P , the cone Cv can be generated by n basis vectors for
Zn . H INT: See (?, Theorem 1.3.12).
Exercise 11.6.19 (Involutions). Let B be an integral domain and let σ : B Ñ B be a ring
map satisfying σ 2 “ idB . The map σ extends to an involution of the fraction field K of B by
the assignment σ px{yq “ σpxq{σpyq. Define
A “ B σ “ tx P B | σpxq “ xu
be the ring of invariants and let L “ KpAq be its field of fractions.
a) Show that L “ K σ “ tz P K | σpzq “ zu.
b) Show that B is integral over A and if B is normal, A will be normal. H INT:
Any x P B will satisfy a quadratic equation with coefficients in A.
c) Consider B “ krx, ys and the involution σpx, yq “ p´x, ´yq. Show that
B σ “ krx2 , xy, y 2 s. Deduce that the ring A “ kru, v, ws{pv 2 ´ uwq is
normal.

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The functor of points

What is the best way to specify a scheme? By definition, a scheme X can be described by a
collection of rings tAi uiPI , and for each pair i, j P I , a ring isomorphism ϕij : pAi qgij »
pAji qgij between the localizations. This is certainly an efficient way of presenting a scheme,
but not the most conceptual. Often, schemes are better understood in terms of the problems
they are designed solve. The fiber product of two schemes is a good example of this; the
explicit construction of XˆS Y is mainly used to show that the product exists, most arguments
use only the formal properties of the product. Likewise, Projective space Pn is a scheme
whose k -points parameterize 1-dimensional subspaces of k n`1 . This geometric description is
much more intuitive and useful than the description in terms of gluing together n ` 1 affine
spaces. However, a priori, there could be other schemes with the same property, so we have
to be more precise about the ‘universal property’ that characterizes Pn .
The precise way of specifying such universal properties is through the functor of points.
This is an important concept in algebraic geometry, which provides a different way of thinking
about schemes. Instead of focusing on the internal structure of a scheme, the functor of points
focuses how a scheme interacts with other schemes.
Specifically, the functor of points of a scheme X assigns to each scheme T the set of
morphisms T Ñ X . We have already seen that it is natural to study maps from a fixed
scheme into X : we think of the points of X as morphisms Spec k Ñ X and we think of
tangent vectors as morphisms Spec krϵs{pϵ2 q Ñ X . By looking at all possible morphisms
T Ñ X at once, we get a functor from schemes to sets which contains all the information
about X ; in particular X can be recovered by its functor.
The functor of points perspective is particularly useful for proving formal properties of
schemes, such as Proposition 8.9.

12.1 The functor of points

Recall from Section 4.9, that for a scheme X and a ring R, the set of R-valued points XpRq
is the set of all scheme maps Spec R Ñ X . There is a generalization of this, where we
consider the set of all morphisms T Ñ X from a fixed scheme T into X . Formally, we
define the functor of points associated with a scheme X to be the functor hX : Schop Ñ Sets
defined by

hX pT q “ HomSch pT, Xq.

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This functor sends a morphism f : S Ñ T to the map of sets

hX pf q : hX pT q ÝÝÑ hX pSq
g ÞÝÑ g ˝ f.
If f : X Ñ Y is a morphism of schemes, there is for each T an induced map hf pT q : hX pT q Ñ
hY pT q defined by sending g : T Ñ X to f ˝ g .
A natural question is whether the scheme X is determined by the functor hX . The answer
is ‘yes’, and this is essentially the content of Yoneda’s lemma.
Let us formulate Yoneda’s Lemma in its natural generality. If C is a category, and X is an
object in C, we get a contravariant functor from C to Sets by
HomC p´, Xq
A natural transformation η between two such functors is a collection of morphisms
ηT : HomC pT, Xq ÝÝÑ HomC pT, Y q, (12.1)
one for each object T in C such that whenever h : S Ñ T is a morphism in C there is a
commutative diagram (of sets)
ηT
HomC pT, Xq HomC pT, Y q

ηS
HomC pS, Xq HomC pS, Y q
and where the vertical arrows are given by σ ÞÑ h ˝ σ .

Lemma 12.1 (Yoneda’s Lemma). Let C be a category and let X and Y be objects of C.
Given a natural transformation
η : HomC p´, Xq ÝÝÑ HomC p´, Y q, (12.2)
there is a unique morphism f : X Ñ Y that induces η , that is, for every object T in C,
the map of sets
ηT : HomC pT, Xq ÝÝÑ HomC pT, Y q, (12.3)
is given by ηT pσq “ f ˝ σ .
Furthermore, η is an isomorphism if and only if f is an isomorphism.

Proof Applying η to the object X , we have a map of sets

ηX : HomC pX, Xq ÝÝÑ HomC pX, Y q.
If there is a morphism f : X Ñ Y that induces η , then we must have
ηX pidX q “ f ˝ idX “ f.
Therefore f is determined by η , so it is unique if it exists.
For the existence, we define f “ ηX pidX q and will check that f induces η . This means
that for any object T , the map of sets
ηT : HomC pT, Xq ÝÝÑ HomC pT, Y q

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is equal to the map of sets that sends σ : T Ñ X to f ˝ σ .

Since η is a natural transformation, we have for any σ : T Ñ X , a commutative diagram
ηX
HomC pX, Xq HomC pX, Y q

ηT
HomC pT, Xq HomC pT, Y q

Going through the diagram clockwise, we see that idX gets sent to σ ˝ f , while going
counterclockwise, idX gets sent to ηT pσq. Hence
ηT pσq “ σ ˝ f.
and so η is induced by f .
In particular, we have the following consequences:

Corollary 12.2. For two schemes X and Y , we have:

(i) hX and hY are isomorphic (as functors from Schop to Sets), if and only if
X »Y:
(ii) If a functor F is isomorphic to hX for some scheme X , then X is determined
up to isomorphism.

Replacing the scheme X with its associated functor of points hX , may at this point seem
like just yet another jump in abstraction, but the nice thing is that you can work with functors
whose values are sets. For instance, by the Yoneda lemma, we see that giving a morphism
f : X Ñ Y of schemes, is the same thing as for each scheme Y giving a map of sets
f pT q : XpT q Ñ Y pT q which is functorial in T (i.e. a natural transformation). In fact, using
that schemes are locally affine, and that morphisms of schemes glue together, it is even
sufficient to test this condition on affine schemes T “ Spec B .
Another important consequence of this is that instead of specifying a scheme explictly, say
by giving a projective embedding and a homogeneous ideal, we can simply specify a functor
equivalent to hX , and this will precisely pin down what scheme we are talking about. Many
schemes are in the first place defined as solutions to universal problems (e.g. fiber products),
so computations involving the functors is both more natural and simpler than say, explicit
equations.
We say that a functor F : Schop Ñ Sets is representable if there exists a scheme X so
that F » hX .
Example 12.3 (The functor of points of A1 ). The functor
F pT q “ OT pT q
is representable by the affine line A1Z “ Spec Zrts. This follows by Theorem ?? on page ??,
which says that to give a morphism T Ñ A1Z is the same thing as specifying an element of
OT pT q.
More generally, An represents the functor F pT q “ ΓpT, OT qn . This is just a fancy way
of saying that a morphism X Ñ An is the same thing as an n-tuple of regular functions. △

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Example 12.4 (The functor of points of Spec A). More generally still, let A be a ring and
consider the functor F : Schop Ñ Sets given by
F pT q “ HomRings pA, ΓpT, OT qq.
Then Theorem 4.17 implies that F is represented by Spec A. △
Example 12.5 (Group schemes). A group scheme is a scheme G so that hG : Schop Ñ Sets
takes values in the category of groups (viewed as a subcategory of Sets). By Yoneda’s
Lemma, being a group scheme implies that there are scheme morphisms m : G ˆ G Ñ G
and i : G Ñ G satisfying the usual group axioms (associativity, identity, and inverses). △
1
Example 12.6. The scheme GLn “ Spec Zrx11 , . . . , xnn , detpx ij q
s is a group scheme. It
represents the functor
GLn pT q “ t M P Matnˆn pOT pT qq | DN P Matnˆn pT q such that M N “ N M “ id u.
If A and B are n-matrices with entries in OT pT q, then pA, Bq ÞÑ AB defines a natural
transformation GLn pT q ˆ GLn pT q Ñ GLn pT q, and so we get a morphism of schemes
m : GLn ˆ GLn ÝÝÑ GLn .
which on R-points corresponds to multiplying the corresponding matrices. Likewise, the
assignment A ÞÑ A´1 defines a natural transformation, and hence a morphism of schemes
i : GLn ÝÝÑ GLn
It would be possible to write down formulas for the ring homomorphisms that define m
and i, but even checking associativity by composing the relevant ring maps would be quite
cumbersome. However, as we know that matrix multiplication is associative, Yoneda’s Lemma
ensures that the corresponding scheme morphisms satisfy these properties. △
Example 12.7. For each n P N, the group of n-th roots of 1 is defined by
µn “ Spec Zrt, t´1 s{ptn ´ 1q.
It represents the functor F pT q “ t a P OT pT q | tn “ 1 u. △
Example 12.8 (Group actions). If G is a group scheme, and X is a scheme, it is clear what it
should mean to have an action of G on X ; for each T , the group GpT q should act on XpT q
in a natural way. This translates into existence of a scheme morphism σ : G ˆ X Ñ X
satisfying the usual axioms for a group action. For instance, σpghqpxq “ gphpxqq for every
x P XpT q and g, h P GpT q. △
Example 12.9. As a concrete example, we can consider the group Gm acting on the scheme
U “ An`1 ´ V px0 , . . . , xn q. Then the scheme morphism π : U Ñ Pn is Gm is Gm -
invariant. In fact, Pn is universal among such Gm -invariant morphisms, and so it can really
be considered as a quotient of U by Gm in the category of schemes
Pn “ U {Gm
△
Exercise 12.1.1. Let X be a scheme and let hX be its functor of points.

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a) Show that hX is determined by its restriction to the subcategory of affine

schemes AffSch.
b) Deduce that X is completely determined by the functor R ÞÑ XpRq from
Rings to Sets.
Exercise 12.1.2. a) Let n ą 0 be an integer. Find a scheme which represents the
functor
F pXq “ ts P OX pXq|sn “ 0u.
b) Show that the functor defined by
F pXq “ ts P OX pXq | s is nilpotentu
is not representable.
Exercise 12.1.3 (Surjectivity of maps of schemes). A morphism of schemes f : X Ñ Y is
said to be surjective if it is surjective on the level of topological spaces.
a) Show that f is surjective if and only if the following condition holds: for every
field K and each K -valued point y P Y pKq there exist a field extension
L of K and x P XpLq such that f pLqpxq P X is the image of y under
Y pKq Ñ Y pLq.
b) Show that surjective morphisms are stable under composition and base change.
c) Suppose f : X Ñ Y is of finite type. Show that f is surjective if and only if the
induced map Xpkq Ñ Y pkq is surjective for every algebraically closed field k .
Exercise 12.1.4. The aim of this exercise is to investigate the functor of points of projective
space Pn . We will associate to a scheme T , its set of data pL, s0 , . . . , sn q where L is
an invertible sheaf L, with an pn ` 1q-tuple of sections s0 , . . . , sn that locally generate
L everywhere. We declare pL, s0 , . . . , sn q „ pM, t0 , . . . , tn q if there is an isomorphism
f : L Ñ M so that f ˚ psi q “ λ ¨ ti for some λ P OTˆ pT q.
a) Show that „ is an equivalence relation.
b) Consider the assignment
␣ ˇ (
F pT q “ pL, s0 , . . . , sn qˇs0 , . . . , sn P ΓpT, Lq generate L everywhere { „
Show that F is a functor.
c) Show that there is a natural transformation
ΦpXq : HompT, Pn q Ñ F pT q
sending a morphism f : T Ñ Pn to the equivalence class of the data
pL, s0 , . . . , sn q “ pf ˚ Op1q, f ˚ x0 , . . . , f ˚ xn q
d) Construct an inverse to Φ and deduce that F is represented by Pn .
e) Show that elements of F pSpec kq are in correspondence with pn ` 1q-tuples
pa0 , . . . , an q P k n`1 , so that not all ai are zero. Thus we recover the usual
description of the k -points of projective space as ‘1-dimensional subspaces of
k n`1 ’.
f) Show that the previous exercise also holds for a local ring.

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g) Show that for a ring R, the set F pSpec Rq is in bijection with the set of rank
1 summands of Rn`1 , i.e., modules of rank 1 such that M ‘ E » Rn`1 for
some module E . This is the right generalization of a ”line in k n ” for general
rings.

Exercise 12.1.5. This is a continuation of Exercise 12.1.4. We will consider the product
X “ Pm ˆ Pn and give a new interpretation of the Segre embedding X Ñ Ppm`1qpn`1q´1
in terms of the functor of points.
a) Let T be a scheme and let pL, s0 , . . . , sm q and pM, t0 , . . . , tn q be elements
of hPm pT q and hPn pT q respectively. Show that the pm ` 1qpn ` 1q tensor
products uij “ pr1˚ si b pr2˚ tj generate pr1˚ L b pr2˚ M on T ˆ T .
b) Show that
ppr1˚ L b pr2˚ M, u00 , . . . , umn q (12.4)

defines an element of hPpm`1qpn`1q´1 pT q, and that this defines a functor from

Schop Ñ Sets.
c) Deduce that there is a morphism ϕ : X Ñ Ppm`1qpn`1q´1 .
d) Show that ϕ is an embedding. H INT: Show that the morphism ϕ has the
property that ϕ´1 pD` puij qq “ D` px0 q ˆ D` py0 q, and show that ϕ restricts
to an embedding on distinguished subsets.

Exercise 12.1.6. This is a continuation of Exercise 12.1.4. We will consider the projective
space Pn and give a new interpretation of the Veronese embedding X Ñ PN in terms of the
functor of points.
a) Let T be a scheme and let `pL, s˘0 , . . . , sn q be an element of hPm pT q. Show that
for each d ě 1, the N “ n`d d
monomials

sbe
0
0
b sbe
1
1
b ¨ ¨ ¨ b sbe
n
n
(12.5)

for e0 ` ¨ ¨ ¨ ` en “ d, generate Lbd .

b) Show that Lb together with the N sections in (12.5) defines an element in
hPN ´1 pT q and that this defines a functor from Schop Ñ Sets.
c) Deduce that there is a morphism ϕ : X Ñ PN ´1 .
d) Show that ϕ is an embedding. H INT: Consider distinguished open sets.

Exercise 12.1.7. Show that the functor of points of the diagonal morphism ∆X{S : X Ñ
X ˆS X is given by
∆X{S pT q “ t pu, vq P XpT q ˆ XpT q | f puq “ f pvq u
where f : X Ñ S is the structure morphism.

12.2 The fiber product in terms of the functor of points

There is a nice way to explain the universal property of fiber products of two S -schemes X
and Y in terms of the functors of points hX , hY and hS . For a scheme T , it translates into the
following: the set HomSch pT, X ˆS Y q is the fiber product of the two sets HomSch pT, Xq

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and HomSch pT, Y q over HomSch pT, Sq. In other words, there is a natural bijection of sets (!)
hXˆS Y pT q ÝÝÑ hX pT q ˆhS pT q hY pT q. (12.6)
By uniqueness, these bijections are functorial in T , and we conclude that the functor of points
of the fiber product X ˆS Y is isomorphic to the fiber product functor hX ˆhS hY , which
assigns the set hX pT q ˆhS pT q hY pT q to a scheme T . Thus the fiber product of schemes is
not so mysterious after all – it is essentially forced upon us by the universal property of fiber
products of sets.
Once we know the functor of points of X ˆS Y , Yoneda’s Lemma implies that many
computations involving fiber products reduce to ones involving sets only. To illustrate this,
we give a proof of Proposition 8.9
Proof of Proposition 8.9 By Yoneda’s Lemma, it suffices to verify the corresponding state-
ments for sets, and this is elementary: note that the assignments pb, aq ÞÑ b; pb, cq ÞÑ pc, bq;
and ppb, cq, dq ÞÑ pb, pc, dqq give natural bijections of sets
B Â A » B pb, aq ÞÑ b
B Â C » C Â B pb, cq ÞÑ pc, bq
pB Â Cq ˆC D » B Â pC ˆC Dq ppb, cq, dq ÞÑ pb, pc, dqq.
These translate into natural isomorphisms of functors
hXˆS S » hX
hXˆS Y » hXˆS Y
hpXˆS Y qˆS Z » hXˆS pY ˆS Zq
and by Yoneda’s lemma, we have the isomorphisms between the corresponding fiber products
as well.
n`m
Example 12.10. To show AnZ ˆZ Am Z » AZ , we can show that the two functors of points
are isomorphic. Note for a scheme T , there is a bijection of sets
pAnZ ˆZ Am
Z qpT q » A
n`m
pT q
defined by
pa1 , . . . , an q ˆ pb1 , . . . , bm q ÞÑ pa1 , . . . , an , b1 , . . . , bm q.
for a1 , . . . , an , b1 , . . . , bn P OT pT q. This assignment is natural in T , so we get an isomor-
phism between the functors of points and hence an isomorphism of schemes. △

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Quasi-coherent sheaves

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13.1 Kernels and images

For a map of sheaves ϕ : F Ñ G , we define its kernel as follows:

Definition 13.1. The kernel Ker ϕ of ϕ is the subsheaf of F defined by

pKer ϕqpU q “ Ker ϕU
for each open U Ă X . In other words, pKer ϕqpU q consists of the sections in FpU q that
are mapped to zero by ϕU : FpU q Ñ GpU q.

The restriction maps of F induce restriction maps of Ker ϕ. Indeed, if s P FpU q maps to
zero by ϕU , then for any V Ă U ,

ϕV ps|V q “ ϕU psq|V “ 0

and hence s|V P pKer ϕqpV q.

The Locality axiom for Ker ϕ follows automatically from the Locality axiom for F . For
the Gluing axiom, let tUi uiPI be an open cover of an open set U and let si P pKer ϕqpUi q
be sections that agree on the overlaps. As F is a sheaf, we may glue together the si ’s to a
section s P FpU q. Now we have ϕpsq|Ui “ ϕps|Ui q “ ϕpsi q “ 0 for all i. By the Locality
axiom for G , it follows that ϕpsq “ 0, and hence s P pKer ϕqpU q.

Lemma 13.2. For each point x P X , one has pKer ϕqx “ Ker ϕx .

Proof To see the inclusion pKer ϕqx Ă Ker ϕx , let sx P pKer ϕqx . By definition of the
stalk, we may find an open set V and s P Ker ϕV so that sx is the germ of s at x. As
ϕV psq “ 0, we also have ϕx psx q “ pϕV psqqx “ 0 in Gx , and hence sx P Ker ϕx .
Conversely, let sx P Ker ϕx and let s P FpV q be a section representing sx . As ϕx psx q “

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0, we have pϕV psqqx “ 0. This means that there exists some open set W Ă V so that
ϕU psq|W “ 0, and hence s|W P pKer ϕqpW q. But this means that sx P pKer ϕqx .

Images
Defining the image of a map ϕ : F Ñ G between sheaves is more subtle than defining the
kernel. One can define the image presheaf by setting
" ˇ *
Im pϕU q “ ϕU psq P GpU q ˇ s P FpU q . (13.1)
ˇ

The restriction maps G induce restriction maps of Im ϕ, as ϕ is compatible with restrictions,

but it will not be a sheaf in general. The issue arises from the Gluing axiom: given an open
cover Ui of U and sections of the form ti “ ϕUi psi q, we can glue the ti to a section t P GpU q.
However, there is no guarantee that t is of the form ϕpsq for some s P F pU q. For that, the
sections si would need to agree on the overlaps Ui X Uj , and there is no reason for why they
should do so. Example 13.4 below illustrates this.
To define the image sheaf, we need to add in all sections that can be obtained by gluing
together local sections of the form ϕUi psi q as above. In other words, we include the sections
of GpU q which are ‘locally images of ϕ’. For a later applications, we allow F to be simply a
presheaf.

Definition 13.3. For a map of presheaves ϕ : F Ñ G , where G is a sheaf, we define the

image sheaf Im ϕ by
" *
ˇ there is a cover Ui of U and sections
ˇ
pIm ϕqpU q “ t P GpU q ˇ .
si P FpUi q such that t|Ui “ ϕUi psi q

The restriction maps of G induce restriction maps of Im ϕ. The Locality axiom holds for
free because G is a sheaf. As for the Gluing axiom, suppose we are given an open cover
tUi uiPI of an open set U and sections ti P pIm ϕqpUi q that agree on the overlaps. Since G is
a sheaf, the ti ’s glue together to a section t P GpU q, and t is by construction locally an image
because each ti is. The sheaf Im ϕ is therefore a subsheaf of G : it is the smallest subsheaf of
G containing the images of ϕ.
Unlike the situation for kernels, pIm ϕqpU q is not always equal to Im ϕU . We always have
the inclusion Im ϕU Ă pIm ϕqpU q, as any section of the form ϕpsq clearly lies in Im ϕ, but
there may also be additional elements of pIm ϕqpU q which are only locally images of ϕ.
Here is a concrete example where this happens:

Example 13.4. Let Z be the closed subscheme given by the ‘x-axis’ in A2k , that is, Z “
Spec krxs inside A2k “ Spec krx, ys. Let ι : Z Ñ A2k denote the inclusion, and consider the
associated map of sheaves
ι7 : OA2k ÝÝÑ ι˚ OZ .

We will show that the naive image presheaf G defined by GpW q “ Im pι7 pW qq is not a
sheaf. To see this, let U “ Dpxq “ Spec krx, ysx and V “ Dpyq “ Spec krx, ysy . Then

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U X Z “ Z ´ V pxq and V X Z “ H. Over these open sets, the map ι7 is given by

ι7U : OA2 pU q “ krx, ysx ÝÝÑ OZ pι´1 U q “ krxsx

7
ιV : OA2 pV q “ krx, ysx ÝÝÑ OZ pι´1 V q “ 0
7
ιU XV : OA2 pU X V q “ krx, ysxy ÝÝÑ OZ pι´1 U X V q “ 0.
ι7U YV : OA2 pU Y V q “ krx, ys ÝÝÑ OZ pι´1 pU Y V qq “ krxsx .

Here we have used Example 4.26 for OA2 pU Y V q “ krx, ys.

Now note that the elements x´1 P GpU q and 0 P GpV q both restrict to 0 in GpU XV q “ 0.
However, they do not glue together to a section of G over U Y V , because there is no element
of krx, ys that maps to x´1 in krxsx . In fact, the image sheaf is equal to all of ι˚ OZ (see
also Example 13.11). △

Example 13.5. There is one situation when the formula pIm ϕqpU q “ Im ϕU in fact holds
for every U , namely when each ϕU : FpU q Ñ GpU q is injective.
To see this, let t P pIm ϕqpU q be a section and let si P FpUi q be elements so that
t|Ui “ ϕpsi q for every i. Then si |Ui XUj and sj |Ui XUj map to the same element in GpUi XUj q
for every i and j , so by injectivity, this forces si |Ui XUj “ sj |Ui XUj . Therefore, the si glue to
an element s P FpU q. We must have ϕpsq “ t, because ϕpsq|Ui “ ϕps|Ui q “ ϕpsi q “ t|Ui
for all i. Hence t “ ϕpsq and so t P pIm ϕqpU q. △

In general, the image sheaf behaves well when it comes to stalks:

Lemma 13.6. For each x P X , we have pIm ϕqx “ Im ϕx .

Proof Let tx P Im ϕx and pick an sx P Fx with ϕx psx q “ tx . Choose sections s P

FpV q and t P GpV q representing sx and tx respectively, for some neighbourhood V of
x. As ϕx psx q “ tx , we have ϕV psq “ t after possibly shrinking V . This implies that
t P pIm ϕqpV q, and hence tx P pIm ϕqx .
Conversely, let tx P pIm ϕqx . By definition, there exists an open set U and a section
t P pIm ϕqpU q so that tx is the germ of t at x. As t P pIm ϕqpU q, there exists an open cover
tUi uiPI of U and sections si P FpUi q such that t|Ui “ ϕUi psi q for each i. Choosing an
index i such that x P Ui , we have

tx “ pt|Ui qx “ pϕUi psi qqx “ ϕx ppsi qx q

This means that tx P ϕx .

13.2 Injective and surjective maps of sheaves

We say that a map of sheaves ϕ : F Ñ G is injective if Ker ϕ “ 0, and that ϕ is surjective if
Im ϕ “ G .
The next two lemmas say that being injective or surjective are ‘local in nature’; they can
be checked on stalks.

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Lemma 13.7. For a map of sheaves ϕ : F Ñ G , the following are equivalent:

(i) ϕ is injective.
(ii) ϕU : FpU q Ñ GpU q is injective for all U Ă X .
(iii) ϕx : Fx Ñ Gx is injective for all x P X .

This is a consequence of the formula pKer ϕqpU q “ Ker ϕU , and the fact that a sheaf is
zero if and only if its stalks are all zero.
When it comes to surjectivity, the formula pIm ϕqpU q “ Im ϕU does not hold in general,
and in fact the item (ii) above does not hold with ‘injective’ replaced by ‘surjective’. Indeed,
surjectivity of ϕ means that for every open set U Ă X , and t P GpU q, there exists an open
cover tUi u of U and elements si P FpU q so that ϕpsi q “ t|Ui . The best we can say is:

Lemma 13.8. For a map of sheaves ϕ : F Ñ G , the following are equivalent:

(i) ϕ is surjective.
(ii) ϕx : Fx Ñ Gx is surjective for all x P X .

This follows from the following small lemma, which will be useful later:

Lemma 13.9. Two subsheaves H, G of a sheaf F are equal if and only if Hx “ Gx (as
subgroups of Fx ) for all x P X .

Proof The ‘only if’-direction is trivial. For the ‘if’ direction, let U Ă X be an open set and
let s P GpU q be a section. By assumption, for each x P X , sx lies in Gx “ Hx . This means
that for each x P U , there exists a neighbourhood Ux of x and a section tx P HpUx q such
that tx represents the germ sx . As the tUx uxPU form a cover of U , and the sections tx and ty
agree over Ux X Uy (they are both germs of s), they glue together to a section t P HpU q. By
construction s and t have the same germs at every point x P U , so by the Locality axiom,
s “ t. This shows that GpU q Ă HpU q for every open set U , and so G is a subsheaf of H.
By symmetry, we also have H Ă G and hence H “ G .
Example 13.10. If X “ Spec A is an affine scheme, and ϕ : M ĂÑN r is a map of sheaves
of ‘tilde-type’, then the following are equivalent:
(i) ϕ : M
ĂÑN r is injective (resp. surjective)
(ii) ϕp : Mp Ñ Np is injective (resp. surjective) for every p P Spec A.
(iii) ϕX : M Ñ N is injective (resp. surjective).
This follows because the stalk maps of ϕ are exactly the localized maps in item (ii). △
Example 13.11. The map ι7 : OA2k Ñ ι˚ OZ of Example 13.4 is not surjective when evaluated
over U “ A2k . The map ι7 is however surjective as a map of sheaves. To see this, note that
A2k is covered by the two opens U “ Dpxq and U 1 “ Dpx ´ 1q. We already showed that ι7U
is surjective, as this is given by the quotient map krx, ysx Ñ krxsx . By Example 13.10, ι7p is
surjective for all p P U . A similar argument applies to U 1 . Therefore ι7 is surjective on every
stalk, and so it is surjective. △
For a map ϕ : F Ñ G to be an isomorphism, i.e., it admits an inverse map ψ : G Ñ F , we
have the following clean statement.

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Proposition 13.12. Let ϕ : F Ñ G be a map of sheaves. Then the following four

conditions are equivalent.
(i) The map ϕ is an isomorphism.
(ii) For every x P X , the map on stalks ϕx : Fx Ñ Gx is an isomorphism.
(iii) One has Ker ϕ “ 0 and Im ϕ “ G .
(iv) For all open subsets U Ă X the map on sections ϕU : FpU q Ñ GpU q is an
isomorphism.

Proof (i) ñ (ii). This implication is clear.

(ii) ñ (iii). Ker ϕ “ 0 follows by Lemma 13.2 and Im ϕ “ G follows by Lemma 13.6
and Lemma 13.9.
(iii) ñ (iv). As Ker ϕ “ 0, ϕ is injective. In that case, taking images commutes with
taking sections (Example 13.5), and so we have Im ϕU “ pIm ϕqpU q. But by assumption,
Im ϕ “ G , so ϕU is also surjective.
(iv) ñ (iii). If ϕU is an isomorphism for every U , the inverse maps ψU “ ϕ´1
U define an
inverse morphism ψ : G Ñ F .

13.3 Exact sequences

A sequence of maps of sheaves
ϕ ψ
F G H (13.2)

is said to be exact if Im ϕ “ Ker ψ as subsheaves of G .

A short exact sequence is an exact sequence of the form
ϕ ψ
0 F G H 0 (13.3)

where we have exactness at all stages. This is just a convenient way of simultaneously saying
that ϕ is injective, that ψ is surjective and that Im ϕ “ Ker ψ .
Exactness for a sequence of sheaves is a purely local condition; the sequence (13.2) is
exact if and only if for each x P X the sequence induced on stalks
ϕx ψx
Fx Gx Hx (13.4)

is exact. This follows from Lemma 13.9 applied to Ker ϕ and Im ϕ.

The following proposition will be very important:

Proposition 13.13 (Taking sections is left exact). Given a short exact sequence as in
(13.3), then for each open subset U Ă X , the sequence
ϕU ψU
0 FpU q GpU q HpU q
is exact.

Proof As ϕ is injective, we have that ϕU is injective, and also that pIm ϕqpU q “ Im ϕU by

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Example 13.5. By definition, we have Ker ϕU “ pKer ϕqpU q. Combining these, we obtain
Im ϕU “ pIm ϕqpU q “ pKer ψqpU q “ Ker ψU ,
and hence the above sequence is exact.
One way of phrasing Proposition 13.13 is to say that taking sections over an open set U
is a left exact functor. This functor, however, is not right exact in general. This failure of
exactness is a fundamental problem in algebraic geometry. We will explore this in greater
detail in Chapter 18 where we discuss cohomology.
Example 13.14. Consider the two points p “ p0 : 1q and q “ p1 : 0q in P1k and let
ι : Z Ñ P1k be the closed embedding given by their union. Let I be the kernel of the map
ι7 : OP1k Ñ ι˚ OZ . The sheaf I fits into the following sequence
ι7
0 I OP1k ι˚ OZ 0. (13.5)

We claim that this sequence is exact, i.e., that ι7 is surjective. For this, it suffices to check that
the map is surjective locally. If U0 “ P1k ´ p » Spec krss, then pι˚ OZ qpU q “ krss{s and
the map ι7 pU q is given by the quotient map krss Ñ krss{s, which is surjective. A similar
argument shows that ι7 is surjective over U1 “ P1k ´ q . Hence the sequence (13.5) is exact.
Now, consider the global sections of this sequence. We have ΓpP1 , OP1k q “ k by Proposi-
tion ?? and ι˚ OZ pP1k q “ OZ pZq “ k ‘ k and the sequence becomes
0 ÝÝÑ ΓpP1k , Iq ÝÝÑ k ÝÝÑ k ‘ k.
Here the right-most map cannot be surjective, so the sequence is not exact. △

13.4 The sheaf associated to a presheaf

Essentially any construction for abelian groups, such as forming kernels, cokernels, tensor
products, direct sums etc. has an analogue for sheaves. For these constructions, one typically
starts by writing down a naive presheaf and then proceeds to show that it satisfies the two
sheaf axioms. This works well in some cases (e.g., for the kernel sheaf), but in general, it can
fail to be a sheaf (as for the image presheaf). To obtain an actual sheaf, we sometimes need
to replace the naive presheaf with a sheaf which in some sense best approximates it; as one
says, we sheafify it.
What prevents a presheaf F in being a sheaf is of course the failure of one of the sheaf
axioms:
(i) There are non-zero sections which are locally zero, i.e., become zero when
restricted to the open sets of a covering. These are sections s P FpU q so that
sx “ 0 for every x P U .
(ii) There are collections of local sections which are compatible over the overlaps,
but do not glue to a global section of F .
In light of this, it is clear what the sheafifcation should do: mod out by the sections in the
item (i) and add in all the missing sections in item (ii).
The sheafification should also satisfy some sort of uniqueness property in the form of a

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universal property. More precisely, to any presheaf F , we will construct a sheaf F ` and a
map of presheaves
κF : F ÝÝÑ F `
which is universal among maps from F into a sheaf.
The main properties of F ` and κF are summarised in the following proposition.

Proposition 13.15. Given a presheaf of abelian groups F on X , there is a sheaf F ` and

a map of presheaves κF : F Ñ F ` satisfying the following properties:
(i) κF is functorial in F : for each map of presheaves ϕ : F Ñ G induces a map
of sheaves ϕ` : F ` Ñ G ` such that following diagram commutes:
ϕ
F G
κF κG (13.6)
ϕ`
F` G`
(ii) If F is a sheaf, then κF is an isomorphism.
(iii) κF and F ` satisfy the universal property that any map of presheaves F Ñ G
where G is a sheaf, factors through F ` in a unique way. In other words, if G
is a sheaf, there is a natural isomorphism

HomPAbpXq pF, Gq » HomAbpXq pF ` , Gq, (13.7)

where on the left-hand side G is considered as a presheaf. This property

characterizes F ` up to a unique isomorphism.
(iv) κF induces an isomorphism on stalks: Fx » Fx` for every x P X .

We will now explain how to construct F ` and κF from F . As in the construction of a

sheaf from a sheaf on a basis, the details of the explicit construction will not be terribly
important. All arguments involving F ` in this book rely mainly on the four properties in the
Proposition 13.15. This is a good illustration of the principle: “ask not what the thing is, but
what it does”.
The sectionsś of F ` can be thought of as sequences psx qxPU of germs of F , i.e., elements
of the product xPU Fx , but only those sequences that arise from local sections of F are
1
allowed.śMore precisely, for an open set U Ă X , and we say that a sequence of germs
psx q P xPU Fx is locally induced by sections of F if there is an open cover tUi u of U and
sections ti P FpUi q so that pti qx “ sx for x P Ui . We then define F ` by setting
ź
F ` pU q “ t psx qxPU | psx q is locally induced by sections of F u Ă Fx
xPU
ś ś
The projection maps xPU Fx Ñ xPV Fx give restriction maps F pU q Ñ F ` pV q for `

each U Ą V , making F ` into a presheaf.

Lemma 13.16. F ` is a sheaf.

1 The notation is not ideal: sx is a germ at x, but at the same time, x serves as an index.

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Proof Locality holds: if tUi u is an open cover of U , and t “ psx qxPU is a section of F ` pU q
such that t|Ui “ 0 for each i, then sx “ 0 for every x P Ui . Hence, if t|Ui “ 0 for all i, it
follows that t “ 0.
Gluing holds: Suppose we are given an open cover tUi u of U and sections ti “ psix qxPUi
of F ` over Ui matching on the intersections Ui X Uj . Saying that the sections agree over
the overlaps, means that for any x P Ui X Uj , the components six and sjx are equal in Fx .
Therefore, we get a well-defined section t P F ` pU q by using this common component as
the component of t at x. It is clear that t|Ui “ ti . Moreover, t is locally induced by sections
of F because the ti are.
There is a canonical map of presheaves
κF : F ÝÝÑ F `
that sends a section s P FpU q to the sequence of all its germs; that is, to the element psx qxPU .
The kernel of this map consists exactly of the sections with all germs equal to zero, that is,
the sections of F which are ‘locally zero’.
For a map of presheaves ϕ : F Ñ G , we may define a map of sheaves ϕ` : F ` Ñ G ` over
an open set U using the product of all the stalk maps Fx Ñ Gx with x P U . In other words,
ϕ`U sends psx qxPU to pϕx psx qqxPU . With this definition, the diagram (13.6) is commutative.
It is not hard to check that id` ` ` `
F “ idF ` and that pψ ˝ ϕq “ ψ ˝ ϕ for two composable
`
morphisms between presheaves on X , so that F ÞÑ F is a functor from the category of
presheaves on X to the category of sheaves on X .
Proof of Proposition 13.15 The statement (i) was proved in the paragraph above.
As for (ii), let us assume that F is a sheaf. Then the Locality axiom implies that κF is
injective. For surjectivity, let t P F ` pU q be an element and let si P FpUi q be a collection of
sections that locally induces t. Then since si and sj have the same germs for every point in
Ui X Uj , the restrictions si |Ui XUj and sj |Ui XUj are equal. Hence the si glue to an element
s P FpU q, (because F is a sheaf). By construction, s has the same germs as the si over Ui ,
so we have κF psq “ t.
For (iii), if G is a sheaf, then the map (13.7) sends ϕ : F Ñ G to the map κ´1 `
G ˝ ϕ , which
` `
is the composition F Ñ G Ñ G . This defines a bijection, as the inverse sends a map
ψ : F ` Ñ G to the map of presheaves F Ñ G given by the composition ψ ˝ κF .
Finally, let us prove claim (iv), starting with injectivity. Let tx P Fx , and represent it as
the germ of a section t P FpV q defined in some neighbourhood of x. Then over V , the map
κF sends t to the collection of germs pty qyPV . If t maps to zero in F ` pV q, then in particular,
tx “ 0.
For surjectivity, take an element in pF ` qx and represent it by a section t P F ` pV q over
some neighbourhood V of x. Taking a smaller V if necessary, we may assume that t is
induced from a section of F , that is, there is a section s P FpV q such that for each y P V ,
the y -th component of t is equal to the germ sy . But then over V , κF sends s to t, and so the
germ sx P Fx maps to tx P pF ` qx .

Example 13.17. A presheaf F which is contained in a sheaf G is particularly easy to sheafify.

The sheafification F ` equals the image sheaf of the inclusion map F Ñ G . In other words,

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the sections in F ` pU q are the sections in GpU q that locally lie in F ; that is, sections s so
that s|Ui P FpUi q for some open cover tUi u of U . △
Example 13.18. Let X “ A1C and let p, q be two distinct points. Consider the presheaf F
defined by
FpU q “ t f P OX pU q | f ppq “ f pqq u.
Then F is not a sheaf. The reason is that F|U “ OU for any open set U which contains p but
not q (as the condition f ppq “ f pqq becomes redundant there). The same holds for open sets
containing q but not p. Therefore, if we take any f P OX pXq such that f ppq ‰ f pqq, we can
restrict it to the open sets U “ A1C ´ p and V “ A1C ´ q , which form an open cover of X .
The restrictions f |U and f |V give two sections of FpU q “ OX pU q and FpV q “ OX pV q,
which are equal to f |U XV , over U X V , but the element they glue to, namely f , is not an
element of FpXq.
Note that F is a subpresheaf of OX . The above argument shows that every section of OX
locally lies in F , so that when we sheafify, we get F ` “ OX . △

13.5 Cokernels and quotients

The main reason to introduce sheafification is to be able to define cokernels and quotient
sheaves. For a map of sheaves ϕ : F Ñ G , we define the cokernel Coker ϕ to be the sheaf
associated to the presheaf
pCoker1 ϕqpU q “ GpU q{Im ϕpU q.
For a subsheaf G Ă F of a sheaf G , the quotient sheaf F{G is the sheaf associated to the
presheaf
1
pF{Gq pU q “ FpU q{GpU q.
In other words, F{G is the cokernel of the inclusion map G Ñ F .
Note that over an open set U , the cokernel presheaf is simply given by Coker ϕU . Com-
posing ϕ with the canonical map Coker1 ϕ Ñ Coker ϕ we obtain a map G Ñ Coker ϕ. It
sits in the sequence
ϕ
F G Coker ϕ 0. (13.8)

Example 13.19. In the sequence (13.5) the subsheaf I Ă OP1k identifies with the sections
of OP1l vanishing along the subscheme Z . By the uniqueness of the cokernel, we get an
isomorphism of sheaves O{I » ι˚ OZ . Even in this example it is necessary to sheafify,
as the ‘naive’ quotient sheaf on global sections satisfies OP1 pP1 q{IpP1 q “ k , whereas
pι˚ OZ qpP1 q “ OZ pZq “ k ‘ k . △

13.6 The inverse image sheaf

For a morphism f : X Ñ Y and a sheaf G on Y , we can define the inverse image sheaf
f ´1 G , which is a sheaf on X . This sheaf has a universal property which is closely related
to the pushforward functor f˚ : for any sheaf F on X , the sheaf morphisms f ´1 G Ñ F are

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in one-to-one correspondence with sheaf morphisms G Ñ f˚ F . The inverse image sheaf

is trickier to define than the pushforward sheaf, but it has better properties. For instance,
applying f ´1 preserves both stalks and exact sequences.
To define the sheaf f ´1 G over an open set U Ă X , it may be tempting to consider
something like Gpf pU qq. However, this does not work in general, because the sheaf G only
knows about the open sets in Y , and typically subsets of the form f pU q will not be open in
Y . The solution is to adapt the definition of the stalk, and look at “germ-like” equivalence
classes of sections GpV q as V runs over the collection of open sets containing f pU q. More
precisely, we define the inverse image presheaf fp´1 G by setting, for an open set U Ă X ,
ž
pfp´1 GqpU q “ GpV q{ „ . (13.9)
V Ąf pU q

The elements in the disjoint union are indexed as pairs ps, V q, where V Ă Y is an open set
in Y containing f pU q and s P GpV q. Two such sections ps, V q and ps1 , V 1 q are defined to
be equivalent if they agree over some smaller open set W with f pU q Ă W Ă V X V 1 .
The restriction maps are defined as follows: if U 1 Ă U , then any open set V contain-
ing f pU q also contains f pU 1 q. This implies that a pair ps, V q representing an element in
pfp´1 GqpU q naturally determines an element in pfp´1 GqpU 1 q.
A different way of expressing (13.9) is as a direct limit
fp´1 pGqpU q “ lim
ÝÑ GpV q, (13.10)
V Ąf pU q

where V runs over open sets in Y containing f pU q. The restriction maps then arise directly
from the universal property of the direct limit, because if U 1 Ă U , the set of opens containing
f pU q is contained in the set of open sets containing f pU 1 q.
The definition of fp´1 G has many similarities with that of the stalk of a sheaf. If V Ă Y
is a subset which contains f pU q and s P GpV q, then the equivalence class of s defines an
element in fp´1 pGqpU q. Conversely, any section of fp´1 G over U arises in this way from a
section of GpV q over some V containing f pU q.
Example 13.20. Let ι : Y Ñ X be the inclusion of a closed subscheme in a scheme X .
Then ι´1 ´1
p OX is a presheaf of rings on Y . A section of ιp OX over V Ă Y , can be thought
of as a regular function on V obtained by restriction from some open neighbourhood in X
containing V . △
Unfortunately, the presheaf (13.9) is not a sheaf in general, so we take the associated sheaf:

Definition 13.21. Let f : X Ñ Y be a continuous map and G a presheaf on Y . The

inverse image f ´1 G is the sheaf associated to the presheaf (13.9).

Unraveling the definition, the sections of f ´1 G can be explicitly described as ś

follows. For
an open set U Ă X , the group pf ´1 GqpU q is the set of sequences psx qxPU P xPU Gf pxq
that satisfy the following property: for every x P U , there is an open set V containing f pxq,
a section t P GpV q, and an open set U 1 Ă U so that f pW q Ă V and sx “ tf pxq for every
x P U 1.
Given a morphism of sheaves on Y , ϕ : G Ñ H, we obtain a collection of maps

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ϕV : GpV q Ñ HpV q for open sets V containing f pU q. These maps are compatible with
restrictions, so they pass to the direct limit and induce a map of presheaves fp´1 G Ñ fp´1 H
and consequently to a map of sheaves f ´1 G Ñ f ´1 H . Therefore the inverse image defines
a functor f ´1 : AbpY q Ñ AbpXq.
While the pushforward f˚ F is easier to define and the sections are more intuitively
understood, the inverse image f ´1 G has better formal properties. For instance, the stalks are
easy to compute:

Proposition 13.22. The stalk of f ´1 G at a point x P X is isomorphic to Gf pxq .

Proof As sheafification preserves stalks, it suffices to verify this on the level of presheaves:
pfp´1 Gqx “ lim ´1
ÝÑ fp GpU q “ lim ÝÑ GpV q “ lim
ÝÑ lim ÝÑ GpV q “ Gf pxq .
U Qx U Qx V Ąf pU q V Qf pxq

Note that there is no such description of the stalk of a pushforward. Also, unlike the
pushfoward, the inverse image is an exact functor:

Corollary 13.23. If 0 Ñ G 1 Ñ G Ñ G 2 Ñ 0 is an exact sequence on Y , then

0 ÝÝÑ f ´1 G 1 ÝÝÑ f ´1 G ÝÝÑ f ´1 G 2 ÝÝÑ 0
is an exact sequence of sheaves on X .

Proof A sequence of sheaves is exact if and only if it is exact on stalks, so the lemma
follows by Proposition 13.22.
Example 13.24 (Restriction to an open set). Let ι : U Ñ X be the inclusion of an open set
in X . Then for a sheaf F on X , we have
ι´1 F “ F|U .
Indeed, if V Ă U is an open set, then V is also open in X , and the direct limit limW ĄιpV q FpW q
ÝÑ
simply evaluates to FpV q. Note that the inverse inverse presheaf ι´1p F is a sheaf in this case.
△
Example 13.25 (General restrictions). If Z Ă X is an arbitrary subset, then the naive
restriction of sections does not directly give a sheaf on Z , because an open subset V Ă Z
will typically not be open in X . The sections of the inverse image sheaf are instead determined
by sections of FpU q as U runs over the open sets in X containing Z .
In particular, if Z “ txu, we recognize the definition of the stalk, and ι´1 F “ Fx , as a
constant sheaf on txu. △
Example 13.26. The presheaf defined by (13.9) is not in general a sheaf. For instance, if
f : X Ñ Y is the constant map with image y P Y , then for each open set U Ă X , the direct
limit in (13.9) will be the stalk Gy . Therefore, (13.9) defines the constant presheaf with value
Gy , and as we observed in Example 3.9 on page 52, this is not always a sheaf.
For an explicit example, we can let X “ Spec k \ Spec k , Y “ Spec k and f : X Ñ
Spec k . Then f ´1 OY “ OX , which is the constant sheaf on k on X , whereas fp´1 OY is the
constant presheaf on k .

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More generally, if f : X Ñ Y is any morphism, and A is a constant presheaf with value

A on Y , then fp´1 A is the constant presheaf on X and f ´1 A is a constant sheaf on X (both
with value A.). △
Example 13.27. As fp´1 OY is a presheaf of rings, f ´1 OY is a sheaf of rings on X . To see
this, we can either apply Exercise 13.7.11, or use the explicit description of the sections of
f ´1 OY in terms of germs. △
Here is the aforementioned universal property of the inverse image sheaf.

Theorem 13.28. Let f : X Ñ Y be a map of schemes and let G be a sheaf on Y . Then

there exist functorial bijections
HomAbpXq pf ´1 G, Fq ÝÝÑ HomAbpY q pG, f˚ Fq (13.11)
for each sheaf F on X .

Functoriality here means that commutative diagrams on the left induce and are induced by
diagrams on the right:

f ´1 G F G f˚ F
ÐÑ

f ´1 G 1 F1 G1 f˚ F 1

Proof As F is a sheaf, the universal property of sheafification tells us that there is a one-to-
one correspondence between sheaf maps f ´1 G Ñ F and maps of presheaves fp´1 G Ñ F .
Therefore, it suffices to establish a bijection between:
(i) The set of sheaf maps ϕ : G Ñ f˚ F
(ii) The set of presheaf maps ψ : fp´1 G Ñ F .
We will show that both sets are in bijection with
(iii) The set of maps
ΛU,V : GpV q ÝÝÑ FpU q
so that for each inclusion U Ą U 1 in X and V Ą V 1 in Y with V Ą f pU q and
V 1 Ą f pU 1 q, then the following diagram commutes
ΛU,V
GpV q FpU q
(13.12)
ΛU 1 ,V 1
GpV 1 q FpU 1 q.

(ii) ô (iii): Let U Ă X be an open set. As pfp´1 GqpU q is defined as a direct limit,
a map ψU : fp´1 GpU q Ñ FpU q is specified by a collection of maps ΛU,V : GpV q ÝÝÑ
FpU q so that whenever V Ą V 1 Ą f pU q, we have ΛU 1 ,V “ ΛU,V ˝ ρV V 1 . That ψ is
compatible with the restriction maps translates is exactly the condition that (13.12) commutes
for all U, U 1 , V, V 1 . Conversely, any such collection ΛU,V determines a map of presheaves
ψ : fp´1 G Ñ F by letting ψU be the map induced by ΛU,V in the direct limit.

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(i) ô (iii): Given a collection ΛU,V , we obtain a sheaf map ϕ : G Ñ f˚ G by defining

ϕV “ Λf ´1 V,V . The diagrams (13.12) ensure that ϕ is a map of sheaves.
Conversely, we need to show that every map of sheaves ϕ : G Ñ f˚ G arises from a unique
collection ΛU,V in this way. Given ϕ, we define ΛU,V to be the composition
ϕ
GpV q ÝÝVÑ Fpf ´1 V q ÝÑ FpU q.
This assignment makes the diagrams (13.12) commute. Moreover, for an open set of the form
U “ f ´1 V , we have ΛU,V “ ϕV , so we see that the collection ΛU,V induces ϕ.
The collection ΛU,V is in fact uniquely determined by the maps of the form Λf ´1 V,V
for V Ă Y . Indeed, the diagram (13.12) implies that if U Ă f ´1 V , then ΛU,V factors as
Λf ´1 V,V : GpV q Ñ Fpf ´1 V q followed by the restriction Fpf ´1 V q Ñ FpU q. In particular,
there is one and only one ΛU,V that induces ϕ.
Despite of the name, the functors f˚ and f ´1 are not inverses: it is not true that f˚ f ´1 G »
G or f ´1 f˚ F » F in general. However, there is a relationship between these functors.
Applying the bijection (13.11) to the identity maps f ´1 G Ñ f ´1 G and f˚ F Ñ f˚ F , we
obtain canonical sheaf maps
η : G ÝÝÑ f˚ f ´1 G and ε : f ´1 f˚ F ÝÝÑ F.
These can be understood a bit more explicitly as follows. For an open set in X of the form
f ´1 V , the direct limit (13.10) is canonically isomorphic to GpV q. Therefore, a section GpV q
defines a section of fp´1 G over f ´1 pV q. Then η is induced by the composition
GpV q ÝÝÑ f˚ fp´1 GpV q ÝÝÑ f˚ f ´1 GpV q.
In particular, there is a natural map
ΓpY, Gq ÝÝÑ ΓpX, f ´1 Gq. (13.13)
Likewise, we define the map ε as follows. For an open set U Ă X , taking the direct limit of
the restriction maps Fpf ´1 V q Ñ FpU q over open sets V with V Ą f pU q, we get a map of
presheaves fp´1 f˚ F Ñ F . Then sheafifying, we get ε.
Example 13.29. Let f : A2k “ Spec krx, ys Ñ Spec k be the structure morphism. Then
f ´1 OSpec k is the constant sheaf on k on A2k . The morphism OSpec k Ñ f˚ f ´1 OA2k is the
identity map. On the other hand, f˚ OA2k is the constant sheaf with value krx, ys on Spec k and
f ´1 f˚ OA2k is also the constant sheaf on A2k with value krx, ys. The map f ´1 f˚ OA2k Ñ OA2
is given by the inclusion krx, ys Ñ OA2k pV q for every V Ă A2k . △
Example 13.30. In contrast, for the structure morphism f : P1k Ñ Spec k and F “
OP1k p´1q, we have f˚ F “ ΓpP1k , Fq “ 0, and f ´1 f˚ F “ 0. Hence f˚ f ´1 F ‰ F .
△
Example 13.31. Let p P A2k be a a point in A2k “ Spec krx, ys and let ι : tpu Ñ A2k
be the inclusion. Then ι´1 OA2k is equal to the stalk OX,p “ krx, ysmp , seen as a constant
sheaf on tpu. For an open set V Ă A2k , the morphism OA2k Ñ ι˚ ι´1 OA2k is the germ map
OA2 pV q Ñ krx, ysmp if p P V , and the zero map otherwise.
The sheaf ι˚ OSpec k is the skyscraper sheaf at p with value k on open sets containing p
and zero elsewhere. The map ι´1 ι˚ OSpec k Ñ OSpec k is the identity map. △

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The sheaf f ´1 G is uniquely determined by the property (13.11). This follows from the
following version of Yoneda’s Lemma:

Lemma 13.32 (Yoneda’s Lemma). Let C be a category and let X and Y be objects of
C. Given a natural transformation
η : HomC pY, ´q ÝÝÑ HomC pX, ´q, (13.14)
there is a unique morphism f : X Ñ Y that induces η , that is, for every object W in C,
the map of sets
ηW : HomC pY, W q ÝÝÑ HomC pX, W q, (13.15)
is given by ηW pσq “ σ ˝ f .

Proof The proof is very similar to Lemma 12.1.

13.7 Exercises
Exercise 13.7.1. Let F be a presheaf on a scheme X and let F ` be the sheaf associated to
F . Show that there is a natural map FpXq Ñ F ` pXq. Find examples where this fails to be
injective or surjective.
Exercise 13.7.2. Let T “ tx, y, z, vu be a topological space with open subsets txu, tyu,
O “ tx, y, zu, P “ tx, y, vu, and O X P “ tx, yu.
Define a presheaf G on T as follows:
‚ Gptxuq “ Gptyuq “ Z{2,
‚ GpOq “ GpP q “ GpO X P q “ GpT q “ Z,
with the natural quotient maps as the restrictions.
a) If G ` denotes the sheafification, show that G ` pT q “ tpm, nq P Z ‘ Z | m ” n
mod 2u.
b) Show that the element p0, 2q P G ` pT q cannot come from a global section of G
over T . Conclude that G is not a sheaf.
Exercise 13.7.3. Define a sheaf on a topological space X by FpXq “ Z and FpU q “ 0 for
all other open sets. Show that F is a presheaf and describe the associated sheaf F ` .
Exercise 13.7.4. Suppose ϕ : F Ñ G is a map of sheaves with the property that there is a
covering tUi u of X so that each ϕUi is injective. Must ϕ be injective? If so, prove it, if not,
give a counterexample.
Exercise 13.7.5. Let ϕ : F Ñ G be a map of sheaves on X and let B be a basis for the
topology on X . Show that ϕ is an isomorphism if and only if ϕV is an isomorphism for every
V P B.
Exercise 13.7.6. Show that the sheaf associated to the ‘constant presheaf’ FpU q “ A of
Example 3.9 is the sheaf AX described in (??).
Exercise 13.7.7. Prove that the sheafification is unique up to a unique isomorphism.

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Exercise 13.7.8. Let X be a topological space and let j : U Ñ X be an open set. For a
sheaf G on U , we define a sheaf j! G on X , the extension by zero by the sheaf associated to
the presheaf
#
GpV q V Ă U
jp! pV q “
0 otherwise

a) Show that the stalk pj! Gqx is equal to Gx if x P U , and 0 otherwise.

b) Show that j! is a left adjoint to j ! .
c) Assume U “ X ´ Z , where i : Z Ñ X is a closed subset. If F is a sheaf on
X , show that there is an exact sequence
0 Ñ j! pF|U q Ñ F Ñ i˚ pF|Z q Ñ 0. (13.16)
d) Compute the exact sequence (13.16) for X “ Ank , Z “ p0, . . . , 0q and F “
OAnk .
Exercise 13.7.9. Let f : X Ñ Y and g : Y Ñ Z be two morphisms.
a) Show that pg ˝ f q˚ “ g˚ ˝ f˚
b) Show that pg ˝ f q´1 “ f ´1 ˝ g ´1 . H INT: There are sheafifications involved.
Define a map from one side to the other and show that it induces an isomorphism
on stalks.
Exercise 13.7.10. Let X be a scheme and let F be a presheaf on X . Show that sheafification
commutes with restriction, i.e., F ` |U “ pF|U q` . H INT: Use the universal property and
check the stalks.
Exercise 13.7.11. Let F be a presheaf of rings on X . Show that the associated sheaf F ` is a
sheaf of rings.
Exercise 13.7.12. Let X “ tx, y, zu be a topological space with three elements and open
sets H, X, V “ txu, U1 “ tx, yu, U2 “ tx, zu. Let F be the constant presheaf on X .
a) Show that F is a sheaf.
b) Let G be the presheaf defined by GpHq “ 0, GpXq “ Z ‘ Z, and GpV q, GpU1 q,
GpU2 q all equal to Z, and let the restriction maps be ρU1 ,V “ ρU2 ,V “ idZ and
let GpXq Ñ GpVi q be the i-th projection, for i “ 1, 2. Show that G is a sheaf.
c) Consider the map ϕ : F Ñ G defined by ϕX “ Z Ñ Z ˆ Z, x ÞÑ px, xq, and
all the other maps equal to the identity map. Show that ϕ is a map of sheaves.
d) Compute the values of the image presheaf and conclude that it is not a sheaf.
Exercise 13.7.13. Consider the adjunction between the inverse image functor f ´1 and the
direct image functor f˚ . Recall the maps η : G Ñ f˚ f ´1 G and ε : f ´1 f˚ F Ñ F .
a) Given a morphism ϕ : f ´1 G Ñ F , show that the corresponding morphism on
the right-hand side of (13.11) is obtained by applying η : G Ñ f˚ f ´1 G and
then composing with the morphism f˚ ϕ : f˚ f ´1 G Ñ f˚ F .
b) Given a morphism ψ : G Ñ f˚ F , show that the corresponding morphism on
the left-hand side is obtained by first applying f ´1 ψ : f ´1 G Ñ f ´1 f˚ F and
then composing with the map ε : f ´1 f˚ F Ñ F .

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Exercise 13.7.14. Show that the map ΓpY, Gq ÝÝÑ ΓpX, f ´1 Gq of (13.13) needs not be
injective or surjective in general, even in the basic case in Example 13.24.

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Quasi-coherent sheaves

14.1 Sheaves of modules

A module over a ring is an additive abelian group equipped with a multiplicative action of
the ring. Loosely speaking, we can multiply elements of the module by elements from the
ring. In a similar way, an OX -module is a sheaf F whose sections over open sets U can be
multiplied by sections of OX pU q.
More formally, we define an OX -module as a sheaf F equipped with multiplication maps
FpU q ˆ OX pU q Ñ FpU q, one for each open subset U of X , making the group of sections
FpU q into an OX pU q-module in a manner which is compatible with restriction maps. In
other words, for every pair of open subsets V Ă U , the diagram below is required to commute

FpU q ˆ OX pU q FpU q
(14.1)

FpV q ˆ OX pV q FpV q.

Here vertical arrows represent restrictions maps and horizontal ones are multiplication maps.
A map of OX -modules or an OX -linear map is a map of sheaves α : F Ñ G between
two OX -modules F and G such that for each open U the map αU : FpU q Ñ GpU q is
OX pU q-linear. The OX -modules on a scheme X form a category, which we denote by
ModOX .
We write HomOX pF, Gq for the subgroup of HomAbpXq pF, Gq consisting of sheaf maps
F Ñ G which are OX -linear.
Example 14.1 (Pushforwards). For a morphism f : X Ñ Y and an OX -module F , the
pushforward f˚ F is naturally an OY -module via the natural map f 7 : OY Ñ f˚ OX . That
is, for a section s P f˚ FpV q and a P OY pV q, we define a ¨ s P f˚ FpV q to be section
f 7 paq ¨ s P Fpf ´1 V q. △
Example 14.2 (Ideal sheaves). Ideal sheaves are important examples of OX -modules. A
sheaf I is an ideal sheaf if IpU q Ă OX pU q is an ideal for each open set U Ă X . For an
ideal sheaf I , the quotient sheaf OX {I associated to an ideal sheaf I is an OX -module.
The primary example is the following. Let ι : Y Ñ X be a closed embedding, then the
kernel I of the map ι7 : OX Ñ ι˚ OY is an ideal sheaf of OX , and there is an exact sequence
0 ÝÝÑ I ÝÝÑ OX ÝÝÑ ι˚ OY ÝÝÑ 0
We also see that ι˚ OY » OX {I as OX -modules.

265

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See Example 13.14 for a concrete example of an ideal sheaf. △

Example 14.3. If F is a sheaf obtained by gluing together sheaves Fi defined on a cover
U “ tUi uiPI , and each Fi is an OUi -module, then F is an OX -module. Indeed, for V Ď X
open, s P FpV q, and a P OX pV q, the product a ¨ s is defined by gluing the local products
a|Ui XV ¨ s|Ui XV in Fi pUi X V q. △
Example 14.4. Write P1k for the projective line over a field k , and consider the sheaves
OP1k pnq from Section 6.3. That is, OP1k pnq is the sheaf obtained by gluing OU0 to OU1
using the isomorphism OU1 |U0 XU1 Ñ OU0 |U0 XU1 on U0 X U1 “ Spec kru, u´1 s given
by multiplication by un . Then OP1k pnq is an OP1k -module by Example 14.3. The map
ϕ : OP1k p´1q Ñ OP1 is a map of OP1k -modules, and the image of ϕ is an ideal sheaf of
OP1 . △
Example 14.5 (Modules on spectra of DVR’s). Modules on the prime spectrum of a discrete
valuation ring R are particularly easy to describe. Recall that the scheme X “ Spec R has
only two non-empty open sets: the whole space X itself and the tηu consisting of the generic
point. The singleton tηu is the underlying set of the open subscheme Spec K , where K
denotes the fraction field of R.
We claim that giving an OX -module is equivalent to giving an R-module M , a K -vector
space N and an R-module homomorphism ρ : M Ñ N .
Indeed, given an OX -module F , we get the R-modules M “ FpXq and N “ Fptηuq,
and the latter is a vector space over K “ OX ptηuq. The homomorphism ρ is just the
restriction map FpXq Ñ Fptηuq. Conversely, given the data M , N and a map ρ : FpXq Ñ
Fptηuq, we can define a presheaf F by setting FpXq “ M and Fptηuq “ N and use
ρ as the restriction map. If we also set FpHq “ 0, we have a presheaf F which satisfies
the two sheaf axioms. Furthermore, since M and N are modules over OX pXq “ R and
OX ptηuq “ K respectively, this makes F into an OX -module.
Note that the restriction map can be any R-module homomorphism M Ñ N . In particular,
it can be the zero homomorphism, and in that case M and N can be completely arbitrary
modules. △

14.2 The tilde of a module

The most important examples of OX -modules are the sheaves of the form M Ă, which we
introduced in Section 4.2. Let us briefly recall the construction. If A is a ring, and M is an
A-module, the sheaf MĂ on X “ Spec A is the sheaf extending the following B -sheaf

M
ĂpDpf qq “ Mf .

The restriction maps are the canonical localization maps, which are described as follows:
when Dpgq Ă Dpf q, we may write g r “ af for some a P A and some r P N, and the
localization map Mf Ñ Mg sends mf ´n to an mg ´nr .
Ă is an OX -module. Over a distinguished open set U “ Dpf q,
It is almost immediate that M
the group M pDpf qq “ Mf is a module over Af , and if U Ă X is any open subset, we may
Ă
cover it by distinguished open sets Dpf q and define an OX pU q-module structure on M
ĂpU q

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by means of the exact sequence in claim 4.2 of Proposition ??. In the same way, one verifies
that the restriction maps are OX -module homomorphisms.
Sending an A-module M to the OX -module M Ă defines the tilde functor, from ModA to
ModX . This functor has very good properties, as we are going to see. We start by explaining
the universal property of MĂ among OX -modules.

Proposition 14.6. Let X “ Spec A be an affine scheme. For an A-module M and an

OX -module F , there is a natural isomorphism

Ă, Fq
HomOX pM » HomA pM, FpXqq

Ă Ñ F to ϕX : M Ñ FpXq. It is functorial in both M and F .

that sends ϕ : M

Proof Let f P A, and consider the commutative diagram

ϕX
M FpXq

ϕDpf q
Mf FpDpf qq

where the vertical maps are restriction maps. This gives the following relation:
ϕDpf q pm{1q “ ϕX pmq|Dpf q .
Note that FpDpf qq is an Af -module, because F is an OX -module. Therefore, in the local-
izations at f , we have the following relation
ϕDpf q pmf ń q “ ϕX pmq|Dpf q ¨ f ń , (14.2)
where mf ń P Mf . This means that the maps ϕDpf q are completely determined by ϕX :
M Ñ FpXq. By Proposition 3.14, the map of sheaves ϕ is completely determined once it is
specified over the Dpf q’s. Hence ϕ is determined by ϕX , and the map in the proposition is
injective.
For the surjectivity, suppose we are given a map of A-modules α : M Ñ FpXq. As usual,
to define a map MĂ Ñ F it suffices to tell what it does to sections over the distinguished open
sets Dpf q. Inspired by (14.2), we define αDpf q by
αDpf q pmf ń q “ αpmq|Dpf q ¨ f ń .
(Note that αpmq is a section of FpDpgqq, so the multiplication makes sense because F is an
OX -module). Hence αDpf q is simply the composition of the two maps of Af -modules
αf
Mf FpXqf FpDpf qq,

where the right-hand map is induced from the restriction map FpXq Ñ FpDpf qq by
localization (note that FpDpf qq is an Af -module). This is compatible with the restriction
Ă Ñ F . Taking f “ 1, we see that we
maps, so we get a well-defined map of sheaves ϕ : M
recover α from ϕ on global sections.

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The statement about the functoriality also follows from formula (14.2). The details are left
to the reader.

If we apply Proposition 14.6 to M “ FpXq and consider the preimage of the identity
map FpXq Ñ FpXq, we obtain the following corollary:

Corollary 14.7. For each OX -module F on an affine scheme X , there is a unique

OX -module homomorphism
βF : FpXq
Č ÝÝÑ F (14.3)
that induces the identity on the spaces of global sections. The map βF is functorial in F .

In concrete terms, the map βF is defined over a distinguished open subset Dpf q as
Č over Dpf q is an element of the form s{f n where
follows. A section of the sheaf FpXq
s P FpXq. Regarding f ´n as a section of OX pDpf qq “ Af , we may send s{f n to the
product s|Dpf q ¨ f ´n , which, because F is an OX -module, defines a section of FpDpf qq.

Proposition 14.8. Let A be a ring and let X “ Spec A. Then:

(i) The functor M ÞÑ MĂ is exact.
(ii) If M and N are A-modules, the map α ÞÑ α r gives an isomorphism
HomA pM, N q » HomOX pM
Ă, N
r q,

with inverse ϕ ÞÑ ϕX .

Proof Let
0 M1 M M2 0. (14.4)

be an exact sequence of A-modules. This gives the sequence OX -modules

0 M
Ă1 M
Ă M
Ą2 0. (14.5)

To check that (14.5) is exact, it suffices to check that it is exact on stalks for every point
x P X . But if x P X corresponds to the prime ideal p Ă A, the stalks of (14.5) is simply the
localization of (14.4) at p (which is exact, because localization is an exact functor).
The last statement follows from Proposition 14.6 with F “ N r and the fact that by
αqX “ α.
definition pr

Item ?? above says that the tilde functor is fully faithful. Hence it establishes an equivalence
between the category ModA of A-modules and a subcategory of ModX . This subcategory is
usually a strict subcategory; most OX -modules are not of tilde-type.

14.3 Quasi-coherent sheaves

The following is the most important definition in this chapter.

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Definition 14.9. Let X be a scheme and F be an OX -module. We say that F is quasi-

coherent if for every affine U “ Spec A Ă X , the restriction F|U is of tilde type, that
is, there exists an A-module M such that F|U » MĂ as OU -modules.

We let QCohX denote the category of quasi-coherent sheaves on X , that is, the subcategory
of ModX where the objecs are quasi-coherent sheaves and the morphisms are morphisms of
OX -modules.
The next theorem is an important result in the theory of quasi-coherent sheaves. It says that
to verify that a sheaf F is quasi-coherent, it suffices to check that F is locally of tilde-type for
the opens in a single open affine cover; this is a much easier condition to check in practice.

Theorem 14.10. Let X be a scheme and F an OX -module. Suppose that there exists an
open affine covering tUi uiPI of X , where Ui “ Spec Ai , and Ai -modules Mi such that
F|Ui » M Ăi as OUi -modules for each i. Then F is quasi-coherent.

Proof Let P be the following property of an open affine U in X : the canonical map

βF |U : FpU
Čq ÝÝÑ F|U (14.6)

from Corollary 14.7 is an isomorphism. We will show that P is a distinguished property (as
defined in Section 9.1). This, together with Proposition 9.1, will imply the theorem.
To verify (D1), assume (14.6) is an isomorphism. We need to show that βF |Dpgq : FpDpgqq
Č Ñ
F|Dpgq is also an isomorphism. It suffices to check this over any distinguished Dpf gq con-
tained in Dpgq. This reduces to showing that

FpDpgqqf g ÝÝÑ FpDpf gqq (14.7)

is an isomorphism.
Consider the restriction maps FpU q Ñ FpDpgqq Ñ FpDpf gqq. As f g acts invertibly
on FpDpf gqq, they induce

FpU qf g ÝÝÑ FpDpgqqf g ÝÝÑ FpDpf gqq.

The composition is exactly the map (14.6) evaluated over the open set Dpf gq, and hence
it is an isomorphism by assumption. The first map is also an isomorphism, because it is a
localization of the map (14.6) evaluated over the open set Dpgq. Therefore the second map,
which is exactly the map (14.7), is also an isomorphism.
The condition (D2) requires more work. Write U “ Spec A for a ring A. We want to
show that the canonical map
β : FpU
Čq ÝÝÑ F (14.8)

is an isomorphism. It suffices to check this over every distinguished open set Dpf q Ă U .
This reduces to showing that the map

FpU qf ÝÝÑ FpDpf qq (14.9)

which sends s{f n to s|Dpf q ¨ f ´n is an isomorphism.

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We assume that U is covered by two distinguished opens Dpg1 q and Dpg2 q both having
property P . For Dpg1 q, this means that the canonical map

βF |Dpg1 q : FpDpg
Č1 qq ÝÝÑ F|Dpg q
1
(14.10)

is an isomorphism. In particular, over the open set Dpf g1 q Ă Dpg1 q, this means that the
map
FpDpg1 qqf ÝÝÑ FpDpf g1 qq (14.11)

which sends s{f n with s P FpDpg1 qq to s|Dpf g1 q ¨ f ´n , is an isomorphism. Similarly for

Dpg2 q.
Consider the sheaf exact sequence for the cover consisting of Dpg1 q and Dpg2 q:

0 FpU q FpDpg1 qq ‘ FpDpg2 qq FpDpg1 g2 qq. (14.12)

There is a similar sequence for the cover of Dpf q consisting of Dpf g1 q and Dpf g2 q. These
fit into the following diagram:

0 FpU qf FpDpg1 qqf ‘ FpDpg2 qqf FpDpg1 g2 qqf

0 FpDpf qq FpDpf g1 qq ‘ FpDpf g2 qq FpDpf g1 g2 qq

The top row is the localization of (14.12) with respect to f , so both rows are exact. The three
vertical maps are direct sums of the appropriate β -maps. The two vertical maps to the right
are isomorphisms; the middle one by assumption and the rightmost one by the property (D1)
which we just showed. By the 5-lemma, the left-most vertical map, which equals (14.9) is
also an isomorphism.

For an affine scheme X “ Spec A, the assignment M ÞÑ M Ă defines a functor p´q

Ą
from A-modules to the category of quasi-coherent sheaves QCohX . By definition, every
Ă, and we can recover the module M by M “ ΓpX, Fq.
quasi-coherent sheaf is of the form M
In fact, this induces an equivalence of categories:

Theorem 14.11. Let X “ Spec A be an affine scheme. Then the tilde functor
Ą : ModA ÝÝÑ QCohX
p´q
is an equivalence of categories. The global sections functor Γ : QCohX Ñ ModA that
sends F to FpXq defines a quasi-inverse.

Note that we always have an equality of A-modules ΓpX, M Ăq “ M . On the other

hand, the functorial map βF : FpXq Ñ F from Corollary 14.7 on page 268 is merely an
Č
isomorphism, hence the term ’quasi-inverse’ in the theorem.
This result has several imporant consequences for quasi-coherent sheaves on affine schemes.
The first is that the global section functor preserves exact sequences.

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Corollary 14.12. Let X “ Spec A be an affine scheme. If

0 F1 F F2 0
is an exact sequence of quasi-coherent sheaves, then the sequence on global sections

0 F 1 pXq FpXq F 2 pXq 0 (14.13)

is also exact. In other words, the global section functor is exact.

Proof Since the global section functor is left exact, we need only show that (14.13) is right
exact, i.e., that the cokernel C “ CokerpFpXq Ñ F 2 pXqq is zero. In any case, there is an
exact sequence
FpXq ÝÝÑ F 2 pXq ÝÝÑ C ÝÝÑ 0
Applying the tilde functor, which is exact (Proposition 14.8), we get an exact sequence
F ÝÝÑ F 2 ÝÝÑ C
r ÝÝÑ 0

By assumption, the map F Ñ F 2 is surjective, so C

r “ 0 and hence C “ ΓpX, Cq
r “ 0 as
well.
Example 14.13 (Quasi-coherent sheaves on P1 ). Consider the projective line P1k over k , with
the usual covering by U0 “ Spec krus and U1 “ Spec kru´1 s, glued together along their
common open set Spec kru, u´1 s.
The sheaves OP1k pnq, as defined in Chapter 5, are all quasi-coherent. This follows because
OP1k pnq|Ui » OUi for each i “ 1, 2 and OUi is of course quasi-coherent.
More generally, we can classify all quasi-coherent sheaves on P1k as follows. A quasi-
coherent sheaf on P1k is given by a triple pM0 , M1 , τ q, where M0 is a module over OX pU0 q “
krus, where M1 is a module over OX pU1 q “ kru´1 s and where
τ : M1 bkru´1 s kru, u´1 s ÝÝÑ M0 bkrus kru, u´1 s.
is an isomorphism of modules over kru, u´1 s.
In terms of this description, the sheaves OP1k pnq are given by the triples M0 “ krus,
M1 “ kru´1 s and the map τ : kru, u´1 s Ñ kru, u´1 s is multiplication by un .
This description of the sheaf is not unique: given any isomorphisms ψ0 : M0 Ñ M0 and
ψ0 : M1 Ñ M1 , we can define τ 1 “ pψ1 b idq ˝ τ ˝ pψ0 b idq which defines an isomorphic
sheaf.
△

Quasi-coherent sheaves and localization

There is a big difference between quasi-coherent sheaves and OX -modules in general. One
way to explain the term ‘coherence’ is that if U is an affine subset, then the groups of sections
of F over smaller affines Dpgq Ă U are completely determined its sections over U itself,
through the localization map
FpU qg ÝÝÑ FpDpgqq. (14.14)

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given by s{g n ÞÑ s|V ¨ g ´n . The fact that (14.14) is an isomorphism is very special, that
sections of F over the small open sets Dpgq are determined from more globally defined
sections in FpU q. In contrast, Exercise (14.12.13) shows that the sections of a general
OX -module can vary rather wildly.
In fact, an OX -module F is quasi-coherent if and only if the localization maps (14.14)
are isomorphisms for every U and Dpgq. If F is quasi-coherent, then F|U » M Ă for some
A-module M and (14.14) follows by the description of the sections of MĂ over distinguished
open sets. Conversely, if (14.14) holds then the natural map βF |U : FpU q Ñ F|U is an
Č
isomorphism, and so F is quasi-coherent.
Example 14.14 (Quasi-coherent sheaves on spectra of DVR’s). Let us continue Example
14.5 of the spectrum X of a discrete valuation ring A. An OX -module F given by the data
M, N, ρ is F quasi-coherent if and only if ρ b idK : M bA K Ñ N is an isomorphism (of
K -vector spaces).
If F is quasi-coherent, then every point has a neighbourhood on which F is the tilde
of some module. The only neighbourhood of the unique closed point is X itself, and so
F “ M Ă. Therefore, N “ FpU q “ Mp0q “ M bA K and ρ b id is an isomorphism.
Conversely, if ρ b idK : M bA K Ñ N is an isomorphism, then F is given by FpXq “ M
and Fptηuq “ M bA K , and so F » M Ă, and it is quasi-coherent. △

14.4 Kernels, Images and Cokernels

Most of the constructions for modules over a ring have analogues for OX -modules and
quasi-coherent sheaves. For instance, given a map of OX -modules ϕ : F Ñ G , the kernel,
image and cokernel of α, have natural OX -modules structures. Here it is clear that the image
and cokernel presheaves have OX -module structures, and then Exercise 14.12.4 shows that
also the associated sheaves are OX -modules.
If ϕ : F Ñ G is a map between quasi-coherent sheaves, then Ker ϕ, Im ϕ and Coker ϕ
are also quasi-coherent. To see this, we may restrict to an open affine U “ Spec A. On U ,
the restriction of ϕ is of the form α
r for some map of A-modules α : M Ñ N . Since p´q Ą is
an equivalence of categories, we have:

Ker ϕ “ Ker
Č α, Im ϕ “ Im
Ć α, Coker ϕ “ Coker
Čα.

In particular, the global sections over an affine open set U are computed as follows:
(i) ΓpU, Ker ϕq “ Ker αU ,
(ii) ΓpU, Im ϕq “ Im αU ,
(iii) ΓpU, Coker ϕq “ Coker αU .
The same applies to quotients: if G Ă F is a subsheaf of an OX -module F , then the quotient
sheaf F{G is naturally an OX -module. If F and G are both quasi-coherent, then so is F{Q
and for every affine open subset U Ă X , we have
ΓpU, F{Gq “ FpU q{GpU q.
It is important to note that this formula does not hold for arbitrary open subsets (see Example
13.14).

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Proposition 14.15 (The 2-out-of-3 property). Let X be a scheme and consider a short
exact sequence of OX -modules

0 F G H 0 (14.15)
If two of the sheaves F, G, H are quasi-coherent, then the third is quasi-coherent as well.

Proof If F and G are quasi-coherent, then so is H, being the cokernel of F Ñ G . Likewise,

if G and H are quasi-coherent, then so is F , being a kernel.
To prove that G is quasi-coherent if F and H are is trickier. We can deduce this using
a future result in Chapter 18: that in Corollary 14.12 it is actually sufficient that only the
leftmost sheaf F 1 is quasi-coherent in order for the sequence (14.13) to be right exact. Given
this fact, we can conclude that the sequence (14.15) is exact when evaluated over an affine
open U “ Spec A. Applying tilde, this means that the upper horizontal sequence in the
diagram below is exact:

0 FpU
Čq GpU
Ćq HpU
Čq 0

0 F|U G|U H|U 0

The three vertical maps are the natural β -maps from Corollary 14.7 on page 268. Since F
and H both are quasi-coherent, the two outer vertical maps are isomorphisms. The Snake
Lemma then implies that the middle vertical map is an isomorphism as well, and hence G is
quasi-coherent.

14.5
Direct sums and products
Àn
For a finite collection of sheaves F1 , . . . , Fn , their direct sum i“1 Fi is defined by:

n
à n
à
ΓpU, Fi q “ Fi pU q.
i“1 i“1

This is a sheaf withÀ restriction maps defined componentwise. Locality holds because if
n
s “ ps1 , . . . , sn q P i“1 Fi pU q restricts to 0 on a covering, then all s1 “ ¨ ¨ ¨ “ sn “ 0
by Locality for the Fi ’s. Likewise, given local sections matching on the overlaps, one can
glue componentwise.
It is also possible to define direct sums of an arbitrary collection of sheaves tFi uiPI , see
Exercise 14.12.10.
Àn
If the Fi are OX -modules, the direct sum i“1 Fi is also an OX -module in a natural way,
with multiplication being defined componentwise. Moreover, if the Fi are quasi-coherent,
the direct sum is also quasi-coherent:

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Proposition
À14.16. If F1 , . . . , Fn is a collection of quasi-coherent sheaves, then the
n
direct sum i“1 Fi is quasi-coherent. If X “ Spec A, then for all A-modules Mi , we
have
àn
Č àn
Mi » M
Ăi . (14.16)
i“1 i“1

Proof We may reduce to the affine case and write X “ Spec A. It suffices to prove
the formula (14.16); quasi-coherence follows from this. As direct sums commute with
localization, we have for every f P A, a natural isomorphism of Af -modules
n
à n
à
p Mi qf ÝÝÑ pMi qf (14.17)
i“1 i“1

These maps are compatible with the restriction maps, so there is an induced map of OX -
Àn ÀČ n
modules ϕ : Č i“1 Mi Ñ i“1 Mi . The map ϕ is an isomorphism, as it induces an isomor-
phism on stalks.
ś
For a collection of sheaves tFi uiPI , we define the direct product iPI Fi by:
˜ ¸
ź ź
Γ U, Fi “ Fi pU q. (14.18)
iPI iPI

ś is a sheaf with componentwise restriction maps. Moreover, if the Fi are OX -modules,

This
iPI Fi is naturally an OX -module. śn
If F1 , . . . , Fn are quasi-coherent sheaves,
Ànthen the product i“1 Fi is also quasi-coherent.
In fact, it is isomorphic to the direct sum i“1 Fi Moreover, if X “ Spec A, then
˜ ¸Ă
źn źn
Mi “ MĂi
i“1 i“1

For an infinite collection of sheaves, however, it can happen that the direct product is not
quasi-coherent. The reason is that direct products do not commute with localization in general
(see Exercise 14.12.10).

14.6 Tensor products

For two OX -modules F and G , we define the tensor product F bOX G to be the sheaf
associated to the presheaf
T pU q “ FpU q bOX pU q GpU q (14.19)
We will sometimes write simply F b G for this tensor product, provided that the context is
clear.
In this definition, it is necessary to sheafify to obtain a sheaf (see Example 14.18 below).
Luckily, the next proposition allows us to understand the sections of tensor products of
quasi-coherent sheaves, at least over affine open subsets.

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Proposition 14.17. If F and G are quasi-coherent sheaves, then so is F bOX G . Moreover,

for any open affine subset U there is a canonical isomorphism
pF bOX GqpU q “ FpU q bOX pU q GpU q (14.20)
If X “ Spec A, we have for two A-modules M and N :
MČ
bA N » M
Ă bOX N
r. (14.21)

Proof As in Proposition 14.16, we reduce to the affine case assume X “ Spec A and to
proving (14.21). The main point is that tensor products commute with localization. More
precisely, for each f P A, there is a natural isomorphism of Af -modules

Mf bAf Nf ÝÝÑ ΓpU, MČ

bA N q “ pM bA N qf (14.22)

given by the assignment m{f a bn{f b ÞÑ pmbnq{f a`b . These isomorphisms are compatible
with the restriction maps, so there is an induced map of OX -modules ϕ : MČ bA N Ñ
M bA N . As the maps (14.22) are isomorphisms, ϕ induces an isomorphism on stalks, and
Č
hence it is an isomorphsm.

Over non-affine subsets, the formula (14.20) does not neccesarily hold, as seen in the next
two examples.

Example 14.18. Let F “ OP1k p1q and G “ OP1k p´1q. Then the global sections of the tensor
product presheaf (14.19) is given by

FpP1k q bk GpP1k q “ k 2 bk 0 “ 0.

However, there is an isomorphism of sheaves

F b G “ OP1k p1q bOP1 OP11 p´1q » OP1k ,

which has global sections equal to k . This means that the presheaf (14.19) is not a sheaf. △

Example 14.19. Let A “ krx, y, zs, and X “ A3k “ Spec A. Consider the A-modules

M “ krx, y, zs{pxq and N “ krx, y, zs{pyq.

Then M bA N “ krx, y, zs{px, yq » krzs.

Let F “ M Ă and G “ N r . If we let U “ A3 ´ tp0, 0, 0qu, then arguing as in Example
k
4.26, we find that FpU q “ M and GpU q “ N , so

FpU q bOX pU q GpU q » krzs.

On the other hand,

Ą » krz, z ´1 s.
pF bOX pU q GqpU q “ ΓpU, krzsq

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14.7 Finiteness conditions for OX -modules

Just as with modules over a ring, we can impose finite finiteness conditions on OX -modules
on a scheme X .

Definition 14.20. A quasi-coherent sheaf F is finitely generated, or of finite type, if for

each open affine U Ă X , the group FpU q is a finitely generated OX pU q-module.

As in the case of quasi-coherence, the conditions hold for any open affine cover provided
that it holds for one:

Proposition 14.21. Let X be a scheme and let F be a quasi-coherent sheaf on X . If

there is an affine cover tUi u with Ui “ Spec Ai such that for each i, F|Ui » M
Ăi for
some finitely generated Ai -module, then F is of finite type.

Proof The main idea is to consider the property P for affine subsets U Ă X : FpU q is a
finitely generated OX pU q-module. By Exercise 14.12.49, P is a distinguished property, so
we conclude using Proposition 9.1.
Example 14.22. If X “ Spec A, then an OX -module F is of finite type if and only if
F »MĂ for some finitely generated A-module. △
Example 14.23. Let X “ Spec Z. If F is an OX -module of finite type, then F “ M Ă for
some finitely generated Z-module M , and by the structure theorem for finitely generated
abelian groups, we may write M “ Zr ‘ T , where T is a finite direct product of groups of
the form Z{nZ. Hence we may write
r
F “ OX ‘T (14.23)
where T is a sheaf having stalks Tp “ 0 for all but finitely many p and Tp0q “ 0. (T is a
torsion sheaf, see Exercise 14.12.40.) △
Example 14.24. The argument of the previous example in fact applies over any PID A: for
X “ Spec A, any OX -module of finite type sheaf must have the form M Ă for M “ Ar ‘ T
where T is a finitely generated torsion module. In particular, any finite type sheaf on the
affine line A1k “ Spec krxs decomposes as
r
F “ OX ‘T (14.24)
where T is a torsion sheaf. △
Example 14.25. Let X “ Spec Crxs, Y “ Spec C, and let f : X Ñ Y be the structure
morphism A1C Ñ Spec C. Then f˚ OX is not of finite type, as it equals the tilde of Crxs
which is not finitely generated as a C-module.
In fact, it follows by definition that for an affine morphism f : X Ñ Y , f˚ OX is of finite
type if and only if f is of finite type. △

Definition 14.26 (Coherent sheaves). Let X be a Noetherian scheme. We say that a

quasi-coherent sheaf is coherent if it is of finite type.

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Remark 14.27. There is a more general definition of ‘coherence’ for OX -modules on arbitrary
schemes, but it is more complicated and less commonly encountered. Specifically, an OX -
module F is defined to be coherent if it is of finite type, and for every open U Ă X and
every morphism of OU -modules ϕ : OUn Ñ F|U , the kernel of ϕ is of finite type. The main
reason for introducing this stronger notion of coherence, is that the category of coherent
sheaves forms an abelian category even in the non-Noetherian setting. The downside is that
the coherence condition is very difficult to check in general, and in fact, for some schemes,
even affine ones, the structure sheaf OX itself is not coherent.
In this book, we will be interested in coherent modules exclusively in the Noetherian
setting, and here the notion is equivalent to being of finite type.

14.8 The Hom-sheaf

Recall that we defined the Hom-sheaf of two sheaves F and G as the sheaf
HompF, GqpU q “ HomAbpU q pF|U , G|U q.
For two OX -modules F and G , we define the Hom-sheaf Hom OX pF, Gq, by
Hom OX pF, GqpU q “ HomOU pF|U , G|U q. (14.25)
where the right-hand side means all the OU -linear maps.
The sheaf Hom OX pF, Gq is an OX -module is a natural way, but it is not always quasi-
coherent, even if both F and G are. This is because of the fact that Hom does not commute
with localization in general. However, if X is Noetherian and F is of finite type, then it is
quasi-coherent:

Proposition 14.28. Let F and G be quasi-coherent sheaves on a Noetherian scheme X ,

and assume that F is coherent.
(i) Hom OX pF, Gq is quasi-coherent.
(ii) Hom OX pF, Gq is coherent if F and G are.
(iii) For every open affine U “ Spec A,
Hom OX pF, Gq|U “ HomOX pU q pFpU q, GpU qq r .
(iv) The stalk at a point x P X is given by
Hom OX pF, Gqx “ HomOX,x pFx , Gx q. (14.26)

Proof The key point is that if A is Noetherian, and M is finitely generated, then HompM, ´q
commutes with localization (see Proposition A.8). More precisely, for every f P A, there is a
canonical isomorphism
HomA pM, N qf “ HomAf pMf , Nf q (14.27)
for each A-module N . These isomorphisms are compatible with restriction maps, so we get
a map of sheaves HomČ A pM, N q Ñ HomOX pM , N q. By (14.27), the induced map is an
Ă r
isomorphism on stalks, and so it is an isomorphism of sheaves, giving us (iii). The remaining
statements follow from this formula.

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Example 14.29. Let X “ Spec Z and consider the Z-modules M “ Q and N “ Z. Then
the global sections of Hom OX pM
Ă, N
r q is given by:

Hom OX pM
Ă, N
r qpXq “ HomZ pQ, Zq “ 0.

On the other hand, the stalk at the generic point η is equal to

Hom OX pM r qη “ HomS ´1 Z pS ´1 Q, S ´1 Zq “ HomQ pQ, Qq “ Q.

Ă, N

Here S “ Z ´ 0. From this it follows that Hom OX pM r q is not a quasi-coherent sheaf.△

Ă, N

14.9 Pushforwards
For a morphism of schemes f : X Ñ Y , and an OX -module F , the pushforward sheaf f˚ F
defined by f˚ FpU q “ Fpf ´1 U q, is an OY -module (see Example 14.1). In this section, we
will explore when the pushforward f˚ F of a quasi-coherent sheaf is again quasi-coherent.
The simplest situation is the following. Consider a morphism f : Spec B Ñ Spec A
induced by a ring map ϕ : A Ñ B . Via ϕ, any B -module M can also be considered as
a module over A. We write MA for M as an A-module. This works well with respect to
localization: if g P A, then there is an isomorphism of Ag -modules Mϕpgq “ pMA qg (where
Mϕpgq is considered as an Ag -module).

Proposition 14.30.
f˚ M
Ă“M
Ą A. (14.28)

Proof Recall Proposition 2.29 which says that f ´1 Dpgq “ Dpϕpgqq. This means that we
have equalities

pf˚ M Ăpf ´1 Dpgqq “ M

ĂqpDpgqq “ M ĂpDpϕpgqqq “ Mϕpgq “ pMA qg ,

where the last equality holds by the above paragraph. These equalities are compatible with the
restriction maps of the sheaves involved, and so by Exercise ?? on page ??, we are done.
Example 14.31. If X “ Spec B , Y “ Spec A, and f : X Ñ Y is a morphism induced by
ϕ : A Ñ B , then canonical map f 7 : OY Ñ f˚ OX is the map A
rÑBĂA , where we consider
7
B as an A-module. Over the open set U “ Dpgq Ă Y , fU is given by the localized map
Ag Ñ Bϕpgq . △
Recall that a morphism f : X Ñ Y is affine if f ´1 pU q is affine whenever U Ă X is an
affine open subset. As quasi-coherence can be checked locally, Proposition 14.30 implies that
f˚ F is quasi-coherent on Y for any affine morphism.

Corollary 14.32. Let f : X Ñ Y be an affine morphism and let F be a quasi-coherent

sheaf on X . Then f˚ F is quasi-coherent sheaf on Y .

While pushwards are in general not quasi-coherent for arbitrary morphisms of schemes,
they are for a large class of morphisms. For the most general statement, see Exercise 14.12.48.

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Theorem 14.33 (Quasi-coherence of pushforwards). Let f : X Ñ Y be a morphism

of schemes and assume that X is Noetherian. If F is a quasi-coherent sheaf on X , then
the pushforward f˚ F is quasi-coherent on Y .

Proof To show that f˚ F is quasi-coherent, it suffices to show that f˚ F|U is quasi-coherent

for any affine open subset U Ă Y . Therefore, we may assume that Y “ Spec A is affine.
Let M “ ΓpY, f˚ Fq “ FpXq. Note that f˚ F is an OY -module, so we may view M as
an A-module. We claim that the natural map
Ă ÝÝÑ f˚ F
M (14.29)
which sends s{g n with s P FpXq to s|f ´1 Dpgq ¨ g ´n , is an isomorphism. It suffices to check
this on distinguished open subsets Dpgq Ă Y for g P A.
As X is Noetherian, we can cover X by finitely many affine open subsets Ui “ Spec Bi .
Further, we can cover the intersections Ui X Uj by finitely many affines Uijk “ Spec Bijk .
With these definitions, a section of pf˚ FqpY q “ FpXq is given by a collection of
elements si P FpUi q that agree over the various intersections Uijk . More formally,
˜ ¸
à à
M “ pf˚ FqpY q “ Ker FpUi q Ñ FpUijk q
i i,j,k

Localization commutes with direct sums, so we get, using the quasi-coherence of F ,

˜ ¸
à à
Mg “ Ker FpUi qg Ñ FpUijk qg
i i,j,k
˜ ¸
à à
» Ker FpSpecpBi qg q Ñ FpSpecpBijk qg q (14.30)
i i,j,k

» Fpf ´1 Dpgqq “ pf˚ FqpDpgqq. (14.31)

To see that the third isomorphism holds, consider the covering tpSpec Bi qg u of f ´1 Dpgq.
These isomorphisms are compatible with the restriction maps, so we get (14.29).
Example 14.34. Consider the projective line P1k and the ‘n-th power map’ map f : P1k Ñ P1k ,
given by px0 : x1 q ÞÑ pxn0 : xn1 q. We claim that
f˚ OP1k “ OP1k ‘ OP1 p´1qn´1 .
Over the open set U0 “ Spec krus, where u “ x1 {x0 , the morphism f is induced by the
ring map krvs Ñ krus which sends v ÞÑ un . Restricted to V0 “ Spec krvs, the pushforward
f˚ OP1 is the tilde of krus as a krvs-module, or in other words, the tilde of
krvsrus{pun ´ vq “ krvs ‘ krvsu ‘ ¨ ¨ ¨ ‘ krvsun´1 . (14.32)
´1
Likewise, in the other affine chart, the morphism is induced by the ring map krv s Ñ
kru´1 s sending v ´1 ÞÑ uń , and when restricted to V1 “ Spec krv ´1 s, f˚ OP1 |V1 is the
tilde of
krv ´1 sru´1 s{puń ´ v ´1 q “ krv ´1 s ‘ krv ´1 su´1 ‘ ¨ ¨ ¨ ‘ krv ´1 suń`1 . (14.33)

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and on V0 X V1 “ Spec krv ˘1 s, it is the tilde of

krv ˘1 sru´1 s{pun ´ vq “ krv ˘1 s ‘ krv ˘1 su´1 ‘ ¨ ¨ ¨ ‘ krv ˘1 suń`1 .
In this ring, we have uiń “ v ´1 ui for i “ 1, . . . , n ´ 1 and we can use this to pass between
the two bases t1, u, . . . , un´1 u and t1, u´1 , . . . , uń`1 u in (14.32) and (14.33). From this,
we find that f˚ OP1 is obtained by gluing together OVn0 and OVn1 using the isomorphism
M
OVn01 ÝÑ OVn01 defined by the matrix
¨ ˛
1 0 0 ¨¨¨ 0
˚ .. ‹
˚0 0
˚ 0 . v ´1 ‹
‹
M “ ˚ ... .. . . ..
˚ ‹
˚ . . . 0 ‹ ‹
˚
. . .. .. ‹
˝0 0 . . . ‚
0 v ´1 0 ¨¨¨ 0
This means that f˚ OP1 “ OP1 ‘ OP1 p´1qn´1 .
△
The following example shows that even f˚ OX can fail to be quasi-coherent if X is not
Noetherian:
š 1
Example 14.35. Let X “ iPN Ak be the disjoint union of countably infinitely many
copies of A1k “ Spec krxs and let f : X Ñ A1k be the morphism that is the identity on each
component. Then f˚ OX is not quasi-coherent. Indeed, the global sections of f˚ OX satisfy
ź
ΓpA1k , f˚ OX q “ ΓpX, OX q “ krxs. (14.34)
iPN

On the other hand, over the open set Dpxq Ă A1k , we have
ź
ΓpDpxq, f˚ OX q “ Γpf ´1 Dpxq, OX q “ krx, x´1 s. (14.35)
iPN

If f˚ OX wereśquasi-coherent, then the group (14.35) would be would be isomorphic to the

localization p iPN krxsqx . ś However, this is not the case, as the elements here are of the form
n
p{x
ś where p “ pp q
i iPN P iPN krxs is a sequence of polynomials, whereas the elements of
´1
iPN krx, x s are sequences of the form ppi x´ni qiPN where we allow arbitrarily negative
powers of x. Therefore f˚ OX is not quasi-coherent. △

14.10 Pullbacks
Let f : X Ñ Y be a morphism of schemes. Recall that we defined the pushforward functor
which produces an OY -module f˚ F from an OX -module F . There is an opposite operation,
called the pullback, which produces an OX -module f ˚ G on X from an OY -module G on Y .
The sheaf f ˚ G satisfies a universal property similar to that of f ´1 G . Namely, maps of OX -
modules f ˚ G Ñ F are in one-to-one correspondence with maps of OY -modules G Ñ f˚ F .
The precise statement is the following theorem:

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Theorem 14.36. Let f : X Ñ Y be a map of schemes and let G be an OY -module.

Then there exists an OX -module f ˚ G on X along with functorial bijections
HomOX pf ˚ G, Fq ÝÝÑ HomOY pG, f˚ Fq (14.36)
for each OX -module F . The sheaf f ˚ G is unique up to isomorphism.

By Yoneda’s Lemma (Lemma 13.32), the pullback f ˚ G is determined up to isomorphism

by the universal property (14.36). We will therefore refer to a sheaf satisfying this condition
as the pullback of G by f .

Pullbacks for X and Y affine

We begin by studying the pullback in the most important special case, namely for quasi-
coherent sheaves via morphisms of affine schemes.
Suppose X “ Spec B , Y “ Spec A and f : X Ñ Y is induced by a ring map ϕ : A Ñ
B . Consider a quasi-coherent sheaf of the form G “ N
r on Y , where N is an A-module. As
B is an A-algebra, the tensor product N bA B is naturally a B -module, and we define the
pullback of G by the formula

f ˚N
r “ NČ
bA B. (14.37)

This defines a functor f ˚ : QCohY Ñ QCohX . Indeed, any a map of quasi-coherent sheaves
Nr Ñ N Ă1 is induced by a map of A-modules N Ñ N 1 . This in turn induces a map of
B -modules N bA B Ñ N 1 bA B and consequently a map of OX -modules f ˚ N r Ñ f ˚N Ă1 .
In this case, the identity (14.36) comes from the formal properties of Hom and the tensor
product. More precisely, we recall the following natural bijection, which holds for all A-
modules N and B -modules M :

HomB pN bA B, M q “ HomA pN, MA q (14.38)

This bijection sends a B -linear map ϕ on the left-hand side to the A-linear map N Ñ MA
given by n ÞÑ ϕpn b 1q. This map is functorial in M and N . (See Exercise 14.12.33.)
Using this, and Proposition 14.6, we have for any OX -module F ,

HomOX pf ˚ G, Fq “ HomOX pGpYČ q bA B, Fq

“ HomA pGpY q bA B, FpXqq
“ HomA pGpY q, FpXqA q
“ HomO pGpY
Ćq, f˚ Fq
Y

“ HomOY pG, f˚ Fq.

The fact that these isomorphisms are functorial in F and G follows from the functoriality of
the isomorphism in (14.38) (see Exercise ??) and the isomorphism βF in Proposition 14.6.

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General pullbacks
We will now define the pullback of an OY -module G for a general morphism of schemes
f : X Ñ Y . To do this, we use the construction of the image image sheaf in Section 13.6.
Recall that if G is a sheaf on Y , the inverse image sheaf f ´1 G is the sheaf associated to the
presheaf

fp´1 GpU q “ lim

ÝÑ GpV q.
V Ąf pU q

If G is an OY -module, each GpV q is an OY pV q-module for each V Ą f pU q. This means

that fp´1 GpU q inherits the structure of a module over the ring fp´1 OY pU q. Explicitly, over
an open set U Ă X , we can represent elements fp´1 OY pU q and fp´1 GpU q by a P OY pV q
and s P GpV q for some open set V Ă Y containing f pU q. Then the product a ¨ s P GpV q is
well-defined, and its class defines an element in fp´1 GpU q.
On the other hand, the ring OX pU q is a module over the ring fp´1 OY pU q via the natural
map fp´1 OY pU q Ñ OX pU q induced by sending pa, V q to f 7 paq|U . We then define the
presheaf fp´1 G by

fp˚ GpU q “ fp´1 GpU q bfp´1 OY pU q OX pU q

Note that fp˚ GpU q is an OX pU q-module. We define f ˚ G to be the associated sheaf. It is an

OX -module by Exercise 14.12.4.
This defines the sheaf f ˚ G , but the definition is not very practical for concrete computations.
Our next step will be to show that f ˚ G satisfies the universal property (14.12.31). This will in
turn imply that the sheaf f ˚ G restricts to the pullback sheaf defined earlier for quasi-coherent
sheaves; it is this characterization that is used in most computations.

Proof of Proposition 14.36 By the universal property of the inverse image presheaf (see
Theorem 13.28), we have a natural isomorphism

HomPAbpXq pfp´1 G, Fq » HomAbpY q pG, f˚ Fq.

Under this bijection, OY -linear maps G Ñ f˚ F on the right correspond to fp´1 OX -linear
maps fp´1 G Ñ F on the left. Furthermore, by the usual change-of-rings formula (14.38), we
have a natural bijection

Homfp´1 OY pfp´1 G, Fq “ HomOX pfp´1 G bfp´1 OX , Fq.

Finally, by sheafification, fp´1 OY -linear maps fp´1 G Ñ F are in one-to-one correspondence

with f ´1 OY -linear maps f ´1 G Ñ F .

Example 14.37 (Restrictions). Let i : U Ñ X be the inclusion of an open subset and let
F be an OX -module. Then the pullback is given by the restriction i˚ F “ F|U , which is
naturally an OU -module. △

Here are a few useful properties of the pullback:

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Proposition 14.38. Let f : X Ñ Y be a morphism of schemes. Then:

(i) f ˚ OY “ OX .
(ii) f ˚ is a right-exact functor.
(iii) If g : W Ñ X is a morphism, then
pf ˝ gq˚ “ g ˚ ˝ f ˚ .
(iv) For any two OX -modules G and H, we have
f ˚ pG ‘ Hq “ f ˚ G ‘ f ˚ H and f ˚ pG bOY Hq “ f ˚ G bOX f ˚ H.
(v) For x P X , we have
pf ˚ Gqx “ Gf pxq bOY,f pxq OX,x .

Proof (i): To show that f ˚ OY “ OX , it suffices to show that OX satisfies the required
universal property (14.36). This follows from the identities

HomOX pf ˚ OY , Fq “ HomOY pOY , f˚ Fq

“ f˚ FpY q “ FpXq
“ HomOX pOX , Fq

Also the item (iii) follows from the universal property. The sheaf f ˚ pg ˚ Gq is an OX -module,
and for any OX -module F , we have

HomOX pf ˚ pg ˚ Gq, Fq “ HomOY pg ˚ G, f˚ Fq

“ HomOZ pG, g˚ f˚ Fq
“ HomOZ pG, pg ˝ f q˚ Fq.

Here all equalities are canonical bijections. Therefore f ˚ pg ˚ Gq and pg ˝ f q˚ G are isomorphic.
Similar arugments show the identities in (iv).
(iv): the description of the stalks of f ˚ G follows because pf ´1 Gqx “ Gf pxq , and pf ´1 OY qx “
OX,x , so that
pf ˚ Gqx “ pGf pxq q bOX,x OX,x » Gf pxq .

(Here we have also used the fact that sheafification preserves stalks.)
(ii): If 0 Ñ G 1 Ñ G Ñ G 2 Ñ 0 is an exact sequence of OY -modules, then the induced
sequence obtained by applying f ˚ is obtained by applying fp´1 p´q (which is right-exact)
and then sheafifying (which is exact).

Proposition 14.39. Let f : X Ñ Y be a morphism of schemes, and let G be a quasi-

coherent sheaf on Y . Then f ˚ G is quasi-coherent on X and for each pair of affine open
subsets U Ă X , V Ă Y such that f pU q Ă V , we have

f ˚ G|U “ GpV q bO
Č
Y pV q
OX pU q (14.39)

Proof This follows formally from the universal property. Let i : U Ñ X , j : V Ñ Y

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denote the inclusions and write g “ f |U . Then we have bijections

HomOU pg ˚ j ˚ G, Fq “ HomOV pj ˚ G, g˚ Fq “ HomOY pG, j˚ g˚ Fq
“ HomOY pG, f˚ i˚ Fq “ HomOX pf ˚ G, i˚ Fq
“ HomOU pi˚ f ˚ G, Fq.
This shows that i˚ f ˚ G » g ˚ j ˚ G , which by Example 14.37 means that f ˚ G|U “ pf |U q˚ pG|V q.
But by the affine case, the pullback pf |U q˚ pG|V q is isomorphic to the sheaf to the right in
(14.39).

Pulling back sections

As for the inverse image sheaf, the universal property (14.36) shows that there are canonical
maps
η : G ÝÝÑ f˚ f ˚ G (14.40)
and
ε : f ˚ f˚ F ÝÝÑ F. (14.41)
Here η is map of OY -modules and ε is a map of OX -modules. These are obtained by applying
(14.36) to the two identity maps f ˚ G Ñ f ˚ G and f˚ F Ñ f˚ F .
Example 14.40. If f : Spec B Ñ Spec A is induced by A Ñ B , and F “ M Ă and G “ N
r,
we can understand the two adjunction maps (14.40) and (14.41) as follows. The map
r ÝÝÑ f˚ f ˚ N
η: N r

is the map of OY -modules induced by N Ñ pN bA BqA , sending n ÞÑ pn b 1q. Likewise,

ε : f˚ f ˚ M
Ă ÝÝÑ M
Ă

is induced by the map of A-modules MA bA B Ñ M sending m b b to bm. △

Example 14.41. Let X be a scheme over a field k and let f : X Ñ Spec k be the structure
morphism. Then for an OX -module F , we have f˚ F “ FpXq (the constant sheaf on
Spec k ), and f ´1 f˚ F “ FpXq is the constant sheaf on X . The canonical map f ˚ f˚ F Ñ F
is given by the ‘multiplication map’
FpXq bk OX pU q ÝÝÑ FpU q
induced by s b f ÞÑ f ¨ s. △
For a section s P GpV q, we define the pullback to be the section
f ˚ psq “ ηpsq P Γpf ´1 V, f ˚ Gq.
Example 14.42. In the special case when G “ OY , we have f ˚ OY “ OX and the pullback
of s P OY pV q is simply given by
f ˚ psq “ f 7 psq P OX pf ´1 V q.
△

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Example 14.43. When X “ Spec B , Y “ Spec A, and f is induced by a ring map

ϕ : A Ñ B , we can understand this pullback as follows. Given s P GpY q, the pullback f ˚ s
is given by the element s b 1 P pf ˚ GqpXq “ GpY q bA B . △
For a general morphism f : X Ñ Y and a section s P GpV q of a quasi-coherent sheaf G ,
we can compute f ˚ psq by covering V and f ´1 V by affine schemes, and then reducing to the
case of Example 14.43. The examples below illustrate this.
Example 14.44. Consider A1k and the squaring map f : A1k Ñ A1k induced by krys Ñ krxs
sending y to x2 . Then f ˚ OA1k “ OA1k , and the isomorphism is simply the tilde of the
isomorphism of krxs-modules
krys bkrys krxs “ krxs.
On global sections, the section y P OA1k pA1k q, pulls back to y b 1 P krys bkrys krxs, which
maps to x2 via this isomorphism. Hence we write f ˚ pyq “ x2 . △
Example 14.45. While the inverse image functor f ´1 is exact, the pullback f ˚ is only
right-exact. The reason is that tensoring by a module is only a right-exact operation in general.
Consider for instance the ideal sheaf sequence of the origin p P A1k “ Spec krts:

0 Ip OA1k ι˚ Op 0. (14.42)

where ι : Spec κppq Ñ A1k is the closed embedding. This sequence is obtained by applying
tilde to the sequence
t
0 krts krts krts{ptq 0.
Applying ι˚ to (14.42) corresponds to applying tilde to the following sequence, obtained by
tensoring by κppq “ krts{ptq:
tbid
0 krts bkrts κppq krts bkrts κppq krts{ptq bkrts κppq 0.

However the latter sequence is not exact because the map t b id equals the zero map. △
Example 14.46. Consider the closed subscheme X “ V px0 q Ă P2k and let ι : X Ñ P2k be
the inclusion. Then ι˚ OP2k p1q “ OP1k p1q. Hence
ΓpP2k , OP2k p1qq » kx0 ‘ kx1 ‘ kx2 and ΓpX, ι˚ OP2k p1qq » kx1 ‘ kx2
In particular, contrary to f˚ , the pullback functor does not always give an isomorphism on
global sections. △
Example 14.47. Consider the projective line P1k and the squaring map f : P1k Ñ P1k , which
restricts to the squaring map U0 Ñ U0 on each Ui » A1k . We claim that f ˚ OP1k p1q “
OP1k p2q.
Recall that OP1k p1q is obtained by gluing together OU0 and OU1 over U0 X U1 via the
isomorphism τ01 : OU1 |U0 XU1 Ñ OU0 |U0 XU1 given by multiplication by x. This means that
f ˚ OP1 p1q is obtained by gluing together OU0 and OU1 over U0 X U1 via the isomorphism
f ˚ pτ01 q : OU1 |U0 XU1 Ñ OU0 |U0 XU1 given by multiplication by f ˚ pxq “ x2 (see Example
14.44). Therefore f ˚ OP1k p1q “ OP1k p2q.

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We can also pull back sections of OP1 p1q. Let s P ΓpP1k , OP1 p1qq be the section given
locally by s0 “ ax ` b on U0 and s1 “ a ` bx´1 on U1 . Then f ˚ s is the section given by
f ˚ ps0 q “ ax2 ` b and f ˚ ps1 q “ a ` bx´2 on the respective open sets. △
Example 14.48. Consider the morphism
f : P1k ÝÝÑ P2k ; pu0 : u1 q ÞÑ pu20 : u0 u1 : u21 q.
Over the standard covering, f : P1k Ñ P2k is given by the two morphisms
f0 : U0 “ Spec krts ÝÝÑ V0 “ Spec krx, ys
given by t ÞÑ pt, t2 q and
f1 : U1 “ Spec krss ÝÝÑ V1 “ Spec kru, vs
given by s ÞÑ ps2 , sq.
Over the overlap U0 X U1 “ Spec krt, t´1 s, we have u “ xy ´1 , v “ y ´1 , so both
morphisms agree with the one induced by krx, y, x´1 y, xy ´1 s Ñ krt, t´1 s x ÞÑ t, y ÞÑ t2 .
Consider the ideal sheaf I of the closed subscheme given by the line V px0 q. Then
I » OP1k p´1q and
f ˚ I » f ˚ OP2k p´1q “ OP1k p´2q
The pullback f ˚ x0 “ u20 P OP2 p´2q defines the subscheme of P1k given by the ideal pu20 q;
this is the scheme-theoretic image of V pIq. △

14.11 Closed subschemes and closed embeddings

As a first application of the material in this chapter, we will give a more systematic treatment
of closed subschemes, as we discussed in Section 4.7.
The prototype example of a closed subscheme is the affine subscheme Spec A{I Ñ
Spec A defined by an ideal I Ă A. The general definition will involve ideal sheaves
rather than ideals. That is, closed subschemes will correspond to ideal sheaves I , so that
IpU q Ă OX pU q is an ideal for each U . In order to obtain a scheme, it is important that I is
quasi-coherent.
For a sheaf F on a space X we define the support of F , denoted by SupppFq, by
" ˇ *
SupppFq “ x P X ˇ Fx ‰ 0 .
ˇ

In a similar way, for a section s P FpU q we define the support of s P FpU q, denoted by
Supppsq, as the set of points x P U such that the germ sx P Fx of s is nonzero.
Note that if s P FpXq is a section and x is a point such that sx “ 0 in Fx , then there is
an open neighbourhood V Ă X containing x such that sy “ 0 for all y P V . It follows that
the support of s is a closed subset of X . In contrast, the support of a sheaf is in general not
closed (see Example 14.50 below).
Example 14.49. If X “ Spec A and M is some finitely generated A-module, then

SupppM
Ăq “ SupppM q “ t p | Mp ‰ 0 u “ V pAnnpM qq

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where AnnpM q is the annihilator ideal

AnnpM q “ t a P A | am “ 0 for all m P M u.
So for instance, if a Ă A is an ideal, then
SupppA{
Ąaq “ V paq

△
Example 14.50. Let S “ ta1 , a2 , a3 , . . . u Ă A1C “ Spec Crts be an infinite set of closed
points, and consider the OA1C -module M
Ă given by the Crts-module
à
M“ Crts{pt ´ ai q.
iPN

Then the stalks of M are C at each ai , and zero otherwise. Therefore, the support is equal to
S , which is not closed. △
Let X be a scheme and let ι : Y Ñ X be a closed embedding. Then ι is an affine
morphism, so the pushfoward ι˚ OY is a quasi-coherent sheaf on X (Corollary 14.32). If we
define I to be the kernel of ι7 , we have the ideal sheaf sequence
0 ÝÝÑ I ÝÝÑ OX ÝÝÑ ι˚ OY ÝÝÑ 0. (14.43)
Note that I is quasi-coherent being the kernel of a map of quasi-coherent sheaves (Proposition
14.15). It is even of finite type, as it is a quotient of OX . Therefore, every closed subscheme
determines a quasi-coherent ideal sheaf I .
Conversely, if I Ă OX is a quasi-coherent ideal sheaf, we can construct a closed sub-
scheme V pIq of X as follows. The underlying set of V pIq will be the support
Z “ SupppOX {Iq Ă X.
If U “ Spec A Ă X is any open affine, then I|U “ r a where a “ IpU q is an ideal in
A “ OX pU q is an ideal. Moreover, as the tilde functor is exact, we have

OX {I|U “ A{
Ąa (14.44)
Therefore,
Z X U “ SupppA{aq “ V paq Ă U.
This shows that Z is a closed subset of X . We equip Z with the subspace topology.
Next, we define the structure sheaf OZ . The OX -module OX {I is a sheaf on X , but can
in fact be considered as a sheaf on Z . Over each open set U , we may identify Z X U with
SpecpA{aq and then OX {I|U is identified with the structure sheaf on SpecpA{aq. We let
OZ be OX {I , but viewed as a sheaf on Z . Note that the stalks of OZ at a point x P Z , are
given by OX,x {Ix , which is a quotient of a local ring, hence a local ring. Hence pZ, OZ q
is a locally ringed space. It is by construction a scheme, as if U “ Spec A Ă X is affine,
then Z X U is isomorphic to the affine scheme SpecpA{aq. We shall denote this scheme by
V pIq.
It is not hard to check that the assignments ι ÞÑ I and I ÞÑ V pIq are inverses, so we get
the following important theorem:

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Theorem 14.51. For a scheme X , any closed subscheme is of the form V pIq “
pSupppOX {Iq, OX {Iq for some unique quasi-coherent ideal sheaf I .

The discussion above shows that for a general ideal sheaf I , the quotient OX {I is always
the structure sheaf of a locally ringed space. It is the quasi-coherence of I that guarantees
that this is locally affine, hence a scheme.
In the affine case, we get a new proof of Proposition 4.30:

Corollary 14.52 (Closed subschemes of affine schemes). The assignment I ÞÑ

SpecpA{Iq gives a one-to-one correspondence between the set of ideals of a ring A and
the set of closed subschemes of Spec A.

Example 14.53. If Y and Z are two closed subschemes defined by ideal sheaves I and J
respectively, then the scheme-theoretic intersection Y bX Z is defined by the ideal sheaf
I ` J Ă OX . △
The next result says that any closed subset Z Ă X can be equipped with a reduced scheme
structure.

Proposition 14.54. Each closed subset Z Ă X of a scheme is the support of a unique

reduced closed subscheme.

Proof Define a sheaf of ideals I Ă OX by the formula

IpU q “ t s P OX pU q | spxq “ 0 for all x P Z X U u,
where as usual spxq denotes the image of the germ sx in the residue field κpxq “ OX,x {mx .
It is straightforward to check that IpU q is an ideal, and that this assignment defines an ideal
sheaf. We claim that I is quasi-coherent.
If U “ Spec A is affine, then Z X U is a closed subset, and so it is of the form Ş V pIq
for a unique radical ideal I Ă A. The ideal I is equal to the intersection I “ IĂp p of
all the prime ideals containing I . This intersection is precisely the set of elements a P A
that vanish at all points in V pIq, and hence IpU q “ I . Being a radical ideal is a property
that localizes, so IpDpgqq “ I| r Dpgq for all distinguished open subsets Dpgq Ă U , and
consequently I|U “ Ir. This shows that I is quasi-coherent.
The corresponding subscheme Z has structure sheaf given by OX {I . Z is reduced, because
for each affine U Ă X , IpU q is radical, and so OX pU q{IpU q is a reduced ring.
The uniqueness statement is clear in the affine case, as the ideal defining Z is uniquely
determined by Z . In general, if I and I 1 are two quasi-coherent ideals as in the proposition,
their restrictions to each open affine subset must be equal. This implies that the inclusion
I Ă I ` I 1 is an equality (using Lemma 13.9), and hence I “ I 1 .

14.12 Exercises
Exercise 14.12.1. Let A “ C ˆ C ˆ C. Describe all OX -modules on X “ Spec A. Then
describe all quasi-coherent and coherent sheaves on X .

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Exercise 14.12.2. Let X “ A1C and let F be the constant sheaf on Z. Is F an OX -module?

Exercise 14.12.3. For each of the schemes below, describe the OX -modules on X .
a) X is the scheme obtained by gluing Spec Zp2q and Spec Zp3q along their com-
mon open subscheme Spec Q.
b) X is the scheme obtained by gluing two copies of Spec Zp2q along Spec Q.
c) Let X be the scheme obtained by gluing the schemes Xi “ Spec Zppi q together
along their common open subschemes Spec Q. Describe the OX -modules on
X.
Exercise 14.12.4. Suppose that F is a presheaf of OX -modules (i.e. a presheaf satisfying
the usual OX -module axioms). Show that the associated sheaf F ` is an OX -module in a
natural way. H INT: One can use the universal properties of sheafification, or the explicit
description of F ` .

Exercise 14.12.5. Let X be a scheme and let F be an OX -module on X . Show that

HomOX pOX , Fq “ FpXq.
Exercise 14.12.6. Show in detail that the kernel, cokernel and image of a map of OX -modules
indeed are OX -modules.

Exercise 14.12.7. Let ϕ : F Ñ G be a map of sheaves.

a) Show that Ker ϕ satisfies the following universal property: Any map of sheaves
ν : H Ñ F such that ν ˝ ϕ “ 0 factors via a unique map ν : H Ñ Ker ϕ.
b) Show that Im ϕ satisfies the following universal property: Given a map of
sheaves α : F Ñ H and β : H Ñ G such that β ˝ α “ ϕ, there is a unique
morphism t : H Ñ Im ϕ factoring β .
c) Show that Coker ϕ satisfies the following universal property: Given a map
ψ : G Ñ H with ψ ˝ ϕ “ 0, there is a unique map t : Coker ϕ Ñ H factoring
ψ.
d) Show that if ϕ is a map of OX -modules, the corresponding universal properties
in a),b) and c) hold in the category of OX -modules as well.
e) Show that a sequence of OX -modules is exact if and only it is exact as a
sequence of sheaves.
H INT: The arguments in a), b) and c) are rather different. For b), use the explicit description
of Im ϕ. For c), the universal property of sheafification may be helpful.

Exercise 14.12.8. Show that the sequence (13.8) is exact. H INT: Show that it is exact on
stalks.

Exercise 14.12.9
À (General direct sums). Let tFi uiPI be a collection of sheaves. We define
the direct sum iPI Fi as the sheaf associated to the presheaf
˜ ¸1
à à
Fi pU q “ Fi pU q. (14.45)
iPI iPI
À
a) Show that if the Fi are OX -modules, then iPI Fi is an OX -module.

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À property: for any family of maps ηi : Fi Ñ G

have the following universal
there is a unique map η : i Fi Ñ G such that ηi “ ηÀ ˝ ϵi .
c) Show that if the Fi are OX -modules, then the sheaves iPI Fi , and the maps
ηi and ϵi are maps of OX -modules.
À
d) Show that if X “ Spec A, then for all A-modules Mi , we have Č iPI Mi »
À Ă
iPI Mi .. À
e) Show that if the Fi are quasi-coherent, then iPI Fi is again quasi-coherent
and for each open affine U Ă X ,
˜ ¸
à à
Fi pU q “ Fi pU q.
iPI iPI

š8
f) Let X “ n“1 Spec C be the disjoint union of countably many copies of
Spec C. For each n P N, let ιn : pn “ Spec C Ñ X be the open embedding of
Fn “ ιn˚ C the skyscraper sheaf À
the n-th copy of Spec C and letÀ at pn . Consider
8 8
the direct sum presheaf S “ n“1 Fn . Show that SpXq ‰ n“1 Fn pXq.
Deduce that S is not a sheaf.
g) Let tFi u be a family of sheaves
À on X and
À U Ă X an open set. If U is
quasi-compact, show that p i Fi qpU q “ i Fi pU q.
h) Conclude that if X is Noetherian, then the presheaf defined in (14.45) is a sheaf.

Exercise 14.12.10 (Direct products). Let tFi uiPI be a collection of sheaves.

ś
ś direct product presheaf iPI Fi defined in the text is a sheaf.
a) Show that the
b) Show that iPI Fi is an OX -module if the Fi are OX -modules. ś
c) Show that the direct product has canonical projections πi : i Fi Ñ Fi having
the universal
ś property: for any collection of maps ϵi : F i Ñ G there is a map
η : Fi Ñ i Fi such that πi ˝ η “ ϵi .
d) Let X “ Spec A and let tMi uiPI be a collection of A-modules. Show that
ś ś Ă
i Mi satisfies the universal property of iPI Mi in the subcategory QCohX .
Č
ś Ă ś
e) Let X “ Spec Z and Mi “ Z for i P N. Show that Mi and Č Mi are
iPN iPN
not isomorphic. Deduce that a direct product of quasi-coherent sheaves need not
be quasi-coherent in general. H INT: Infinite direct products do not commute
with localization.

Exercise 14.12.11.
À Show that śthe direct sum sheaf can be defined as the image sheaf of
the natural map iPI Fi Ñ iPI Fi where the left-hand Àside is regarded as a presheaf.
H INT: Use Example 13.17 and the universal property of .

Exercise 14.12.12. Let X be a scheme. A directed system of sheaves is a family of sheaves

tFi uiPI indexed by a directed set I , together with morphisms of sheaves ϕij : Fi Ñ Fj
for each i ď j such that (i) ϕii is the identity map on Fi ; and (ii) For i ď j ď k , we have
ϕik “ ϕjk ˝ ϕij . We define the direct limit sheaf as the sheaf associated to the presheaf

ÝÑ Fi qpU q “ lim
plim ÝÑ Fi pU q. (14.46)

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a) Show that there are sheaf maps ϕi : Fi Ñ lim Fi satisfying the following
ÝÑ
compatability condition: For all i ď j , we have ϕj ˝ ϕij “ ϕi .
b) Show that lim Fi satisfies the following universal property: if there exists
ÝÑ
another sheaf G with morphisms ψi : Fi Ñ G satisfying the compatibility
condition, then there exists a unique morphism of sheaves u : F Ñ G such that
u ˝ ϕi “ ψi for all i P I .
c) Show that if the Fi are OX -modules, then lim Fi is naturally an OX -module.
ÝÑ
d) Show that if each Fi is quasi-coherent, show that lim Fi is quasi-coherent, and:
ÝÑ
(i) For each affine open set U Ă X , plim Fi qpU q “ lim Fi pU q.
ÝÑ ÝÑ
(ii) For each x P X , we have plim Fi qx “ limpFi qx .
ÝÑ ÝÑ
Exercise 14.12.13 (Godement sheaves). Given any collection of abelian groups tAx uxPX
indexed by the points x of X , we can define a sheaf A by
ź
ApU q “ Ax ,
xPU

and whose restriction maps to smaller open subsets are just the projections onto the corre-
sponding smaller products.
a) Show that A is a sheaf.
b) If we assume that each Ax be a module over the local ring OX,x , show that
sheaf A becomes an OX -module.
Exercise
ś 14.12.14. For a presheaf F consider the sheaf ΠpFq defined by ΠpFqpU q “
xPU Fx . This is sometimes called the ‘sheaf of discontinuous sections’.
Show that there is a canonical map of presheaves σ : F Ñ ΠpFq so that F ` “ Im σ .
Exercise 14.12.15. Let F and G be OX -modules. Show that the assignment U ÞÑ HomOX pFpU q, GpU qq
does not define a presheaf in general.
Exercise 14.12.16 (Stalks of the Hom-sheaf). Let F and G be two OX -modules.
a) For x P X , show that there is a natural map
Hom OX pF, Gqx ÝÝÑ HomOX,x pFx , Gx q (14.47)
b) Show that the map (14.47) may fail to be surjective. H INT: Let F be a
skyscraper sheaf and G a constant sheaf; then HompF, Gq is the zero-sheaf.
c) Show that the map (14.47) may fail to be injective. H INT: Let U “ X ´ x
and let F “ G “ i! Z, where i : U Ñ X is the inclusion; then Fx “ Gx “ 0.
Exercise 14.12.17. Let X be a scheme and let F be a sheaf of OX -modules. We define the
support as the set
SupppFq “ t x P X | Fx ‰ 0 u. (14.48)
Likewise, if s P FpU q, we define the support of s as the subset of points x P X such that
sx P Fx is non-zero.
a) Show that the support of a section s P FpU q is closed in U .
b) If s, t P FpU q, show that Supppstq Ă SupppsqXSuppptq and that Suppps`
tq Ă Supppsq Y Suppptq.

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c) Show that if ϕ : F Ñ G is a map of OX -modules, then Supppϕpsq Ă

Supppsq.
d) Find an example of an OX -module F so that SupppFq is not closed in X .
H INT: Consider infinite products.
e) Show that if F is a sheaf of OX -algbras, then SupppFq is closed. H INT: A
ring is the zero ring if and only if 1 “ 0.
Exercise 14.12.18 (Subsheaf with support). Let Z Ă X be a closed subset and F a sheaf
on X . For an open set U Ă X , define ΓZXU pU, Fq to be the subgroup of FpU q consisting
of sections s with Supppsq Ă Z X U . Show that this is a sheaf. This is usually denoted by
HZ0 pFq. Show that the functor F ÞÑ ΓZXU pU, Fq is left exact.
Exercise 14.12.19. Let X be the nodal cubic curve and consider the normalization morphism
f : A1k Ñ X of Example 11.36, which identifies the two points 0 and 1 in A1k .
a) Show that the presheaf fp´1 OX consists of the sections of OA1k which take the
same value on 0 and 1. Show that this is not a sheaf.
b) Compute f ´1 OX and f ˚ OX .
Exercise 14.12.20. Show that a sheaf F on a scheme X is quasi-coherent if and only if
there is an open cover tUi uiPI such that each of the restrictions F|Ui may be presented as the
cokernel of a map between free OX -modules; that is, they appear in exact sequences

OUJ i OUI i F|Ui 0,

À
where G I stands for the direct sum iPI G of copies of a sheaf G (and where I and J may
be infinite and dependent on i). Conclude that being quasi-coherent is a local property for an
OX -module.
Exercise 14.12.21. Let X “ Spec A be an affine scheme and let
¨ ¨ ¨ ÝÝÑ Fi´1 ÝÝÑ Fi ÝÝÑ Fi`1 ÝÝÑ ¨ ¨ ¨
be an exact sequence of quasi-coherent sheaves. Show that
¨ ¨ ¨ ÝÝÑ Fi´1 pXq ÝÝÑ Fi pXq ÝÝÑ Fi`1 pXq ÝÝÑ ¨ ¨ ¨
is also exact.
Exercise 14.12.22. Let X “ Spec A and consider an distinguished open subscheme Dpgq Ă
X . Let F be a quasi-coherent OX -module. Show that:
(i) If s P FpXq and s|Dpgq “ 0, then g n s “ 0 for some n P N.
(ii) If s is a section of MĂ over Dpgq, then for some n P N the section g n s extends
to X ; that is, there is a t P FpXq so that t|Dpgq “ g n s.
H INT: Use the formula FpDpgqq “ FpXqg .
Exercise 14.12.23. Show the following:
(i) The skyscraper sheaf of k on A1k “ Spec krts at the origin 0 is quasi-coherent.
(ii) The skyscraper sheaf of kptq on A1k “ Spec krts at the origin 0 is not quasi-
coherent. H INT: Consider sections over U “ Dptq.

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Exercise 14.12.24. Let X Ă A3k “ Spec krx, y, zs be the twisted cubic curve, defined the
ideal I “ py ´ x2 , z ´ x3 q. Consider the morphism π : X Ñ A1k “ Spec krzs induced by
the ring map krzs Ñ krx, y, zs{I .
(i) Show that π is a finite morphism.
(ii) Compute the sheaves π˚ OC , π ˚ OA1k and π ˚ J where J is the ideal sheaf of the
closed point 0 P A1k .
Exercise 14.12.25 (Flat morphisms). Let f : X Ñ Y be a morphism of schemes and let
x P X be a point. We say that f is flat if the map fx7 : OY,f pxq Ñ OX,x makes OX,x into a
flat OY,f pxq -module for every x P X .
(i) Show that open embeddings are flat. What about closed embeddings?
(ii) Show that a morphism of schemes Spec B Ñ Spec A is flat if and only if the
map of rings A Ñ B is flat. More generally, a quasi-coherent sheaf M Ă on
Spec B is flat over Spec A if and only if M is flat as an A-module.
(iii) Which of the morphisms in Exercise 2.7.28 are flat?
(iv) Prove that the blow-up morphism π : Bl0 A2 Ñ A2 is not flat.
Exercise 14.12.26 (Flat base change). Let A be a ring and let B be a flat A-algebra. Let X
be an A-scheme and let XB denote the base change to B . Show that for any quasi-coherent
sheaf F , there is a natural isomorphism
ΓpXB , FB q “ ΓpX, Fq bA B
˚
where FB “ q F and q : XB Ñ X is morphism obtained by base change. H INT: Tensor
the sheaf exact sequence.
Exercise 14.12.27 (Morphisms to a closed subscheme). Let Z be a closed subscheme of
X given by sheaf of ideals I . Suppose f : Y Ñ X is a morphism of schemes. Show that f
factors through a map g : Y Ñ Z if and only if
(i) f pY q Ă Z .
(ii) I Ă Kerpf 7 : OX Ñ f˚ pOY qq.
For a morphism of schemes f : Y Ñ X , we can define the scheme-theoretic image of
f as a subscheme Z Ă X satisfying the universal property that if f factors through a
subscheme Z 1 Ă Z , then Z Ă Z 1 . To define Z it is tempting to use the ideal sheaf
I “ KerpOX Ñ f˚ pOY qq, but this may fail to be quasi-coherent for a general morphism f .
Show however that if X and Y are Noetherian, then it is quasi-coherent.
Exercise 14.12.28. Let X be a scheme.
a) Show that the direct sum of two OX -modules of finite type sheaves is again of
finite type.
b) Show that the ‘2-out-of-3’-property holds for OX -module of finite type. That
is, if 0 Ñ F 1 Ñ F Ñ F 2 Ñ 0 is an exact sequence of OX -modules, and if
two of F, F 1 , F 2 are of finite type, then so is the third.
c) Let ϕ : F Ñ G be a map of OX -module of finite type. Show that Ker ϕ, Im ϕ
and Coker ϕ are all of finite type. H INT: Apply Exercise 14.12.28.
Exercise 14.12.29. Let F and G be OX -modules and let x P X be a point. Show that
pF bOX Gqx “ Fx bOX,x Gx .

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Exercise 14.12.30. Use Yoneda’s lemma for sheaves to prove the identity MČ
bA N “
M
Ă bO X N r . H INT: Proposition 14.6 may also be useful.

Exercise 14.12.31. Let f : X Ñ Y be a morphism of schemes, F an OX -module, and G an

OY -module.
a) Given a map of OX -modules OX Ñ F , show that f 7 induces a map of OY -
modules OY Ñ f˚ F .
b) Given a map of OY -modules OY Ñ f˚ F , show that there is an induced map
of OX -modules OX Ñ F .
c) Show that the constructions in a) and b) are inverse to each other and conclude
that f ˚ OY “ OX .
Exercise 14.12.32. Let f : X Ñ Y be a morphism and let G a be a quasi-coherent OY -
module. Show that the stalk of f ˚ G at x P X is given by Gf pxq bOY,f pxq OX,x .
Exercise 14.12.33. Show that the map in (14.38) is a bijection, and that the isomorphism is
functorial in M and N .
Exercise 14.12.34. Show directly that the pullback f ˚ G can be constructed by gluing together
sheaves of the form MČbA B .
Exercise 14.12.35. Let f : X Ñ Y be a morphism of schemes and let F be an OX -module
of finite type.
a) Show that f˚ F need not be of finite type in general.
b) Show that if f is finite, then f˚ F is an finite type OY -module. In particular,
this holds for closed embeddings.
Exercise 14.12.36. Let f : X Ñ Y be a morphism and let J be an ideal sheaf of OY .
a) Show that there is a map f ˚ J Ñ OX
b) Give an example to show that f ˚ J needs not be an ideal of OX in general.
c) Let I Ă OX denote the image of f ˚ J in OX . Show that I is an ideal sheaf
and that
SupppOX {Iq “ f ´1 SupppOY {J q
H INT: Reduce to the affine case.
Exercise 14.12.37. Consider the affine plane A2k “ Spec krx, ys over a field k and let
P P A2k be the closed point corresponding to the origin. Define the sheaf I by
#
0 P PU
IpU q “
OX pU q O R U
a) Show that I is an ideal sheaf and that SupppOX {Iq is not a closed subset of
X.
b) Show that I is not quasi-coherent. In fact, show that I is not even locally
generated by sections. H INT: The idea is that any section of I is 0 in any
neighbourhood of P , but F is not 0 when restrcted to any neighourhood of P .
c) Show directly that I is not quasi-coherent by showing that IpXq “ 0, but
I ‰ 0.

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d) Show that there is an exact sequence

0 ÝÝÑ I ÝÝÑ OX ÝÝÑ i˚ OA2k ,P ÝÝÑ 0
where i˚ OA2k ,P is the skyscraper sheaf on P .
e) Show that i˚ OA2k ,P is an OX -module of finite type, but not quasi-coherent.
Exercise 14.12.38. Let f : X Ñ Y and g : Y Ñ Z be morphisms and let G be a sheaf on
Z . Show that f ´1 pg ´1 Gq “ pg ˝ f q´1 G .
Exercise 14.12.39. Let X be a scheme and let F be a quasi-coherent sheaf of finite type.
n
Let s1 , . . . , sn be sections of F and consider the map of OX -modules ρ : OX Ñ F send-
ing ei to si . Show that ρ is surjective in a neighbourhood of x if and only if the images
s1 pxq, . . . , sn pxq generate Fpxq. H INT: Apply the previous exercise to the cokernel.
Exercise 14.12.40 (Torsion sheaves). Let X be an integral scheme, and let F be a quasi-
coherent sheaf on X . Define for each open set U Ă X , a subgroup T pU q Ă FpU q consisting
of all the elements m P FpU q such that the germ mx is torsion in Fx for all x P X , i.e.,
ax ¨ mx “ 0 for some non-zero ax P OX,x .
a) Show that T is a subsheaf of F . Also, show that T is quasi-coherent. T is called
the torsion subsheaf of F . Another notation for it is Ftors .
b) Let K denote the constant sheaf on K “ KpXq. Define a map of sheaves
ν : F Ñ F bO X K .
Show that T “ Ker ν .
c) A sheaf is called torsion free if Ftors “ 0. Show that the quotient F{T is always
torsion free, i.e., pF{T qtors “ 0.
d) Show that any locally free sheaf is torsion free.
À
Exercise 14.12.41. Let X “ Spec Z and consider the Z-modules M “ N Z and
N “ Z. À Show that ś Hom OX pM , N q is not quasi-coherent. H INT: Use the identity
Ă r
HomA p N A, Aq “ N A.
Exercise 14.12.42. Let F be a quasi-coherent sheaf on X and let U Ă X be an open set.
Show that the restriction F|U is also quasi-coherent.
Exercise
À 14.12.43. Let X be a Noetherian scheme. Show that the direct sum presheaf
iPI Fi of (??) is a sheaf.

Exercise 14.12.44. Let X be a scheme and let F be an OX -module on X . Show that F is

quasi-coherent if and only if for any pair V Ă U open affine subsets the natural map
FpU q bOX pU q OX pV q Ñ FpV q (14.49)
that sends s b g to gs|V , is an isomorphism.
Exercise 14.12.45. Show that a morphism f : X Ñ Y of schemes is a closed embedding if
and only if it is affine and map ι7 : OY Ñ ι˚ OX is surjective.
Exercise 14.12.46. Find a scheme X such that f˚ OX is not quasi-coherent sheaf on Spec Z,
where f : X Ñ Spec Z is the canonical map.

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Exercise 14.12.47. Let ϕ : F Ñ G be a map of quasi-coherent sheaves. Show that ϕ is

injective (resp. surjective) if and only if ϕU : FpU q Ñ GpU q is injective (resp. surjective)
for each affine open U Ă X .
Exercise 14.12.48. A morphism f : X Ñ Y is called quasi-compact if for every affine open
subset U Ă Y , the inverse image f ´1 pU q is quasi-compact. f is called quasi-separated if
for any pair of affine open subsets U, V Ă Y , the intersection f ´1 pU q X f ´1 pV q can be
covered by finitely many affine open subsets.
a) Show that the properties of being quasi-compact and quasi-separated are stable
under base change.
b) Let f : X Ñ Y be a morphism which is both quasi-compact and quasi-
separated. Show that f˚ F is quasi-coherent if F is.
Exercise 14.12.49. Let P be the property of affine subsets U Ă X : FpU q is a finitely gener-
ated OX pU q-module. Show that P is a distinguished property and hence prove Proposition
14.21. H INT: See the proof of Proposition 7.23.

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Locally free sheaves

The most important examples of quasi-coherent sheaves are the locally free sheaves. As the
name suggests, these are sheaves which are locally isomorphic to a direct sum of copies of
the structure sheaf of the scheme. Because of this ‘freeness’ property, these sheaves are in
many respects the nicest examples of sheaves on a scheme and the easiest to work with. They
are also the algebraic counterpart to the vector bundles in topology.

15.1 Basic definitions

Let X be a scheme. We say that an OX -module E is free if it is isomorphic to a direct sum
of copies of OX . E is said to be locally free if there is an open cover tUi uPI of X such that
the restriction E|Ui is free for every i. The cover tUi u is called a ‘trivializing cover’, and a
locally free sheaf which is globally free is sometimes said to be trivial.
If x is a point of X , say contained in Ui , the rank of E at x is the number of copies of
OUi needed to express E|Ui as a direct sum of OUi ’s. The rank may be finite or infinite, but
we shall almost exclusively concern ourself with the case of finite rank. The sheaf E is also
allowed to have different ranks at different points of X , but the rank will be constant on each
connected component of X . If E has rank equal to r at every point of X , we say that E is
locally free of rank r and write rankpEq “ r.
A locally free sheaf of rank 1 is called an invertible sheaf (these will be studied in more
detail in Section 16.1.)

Example 15.1. On the projective line X “ P1A , one has the sheaves OP1A pmq constructed
on page 102. These were constructed by gluing together two copies of OA1A , so they are
invertible sheaves. These sheaves are non-trivial for all m ‰ 0, as we computed the global
sections of OP1A pmq is not isomorphic to ΓpP1A , OP1A q “ A. △
Example 15.2. It is easy to give examples of locally free sheaves with varying rank. If X is
a disjoint union of two open sets U and V , we can simply define E by letting E|U “ OUn and
E|V “ OVm with n, m P N arbitrary. △
Example 15.3. If E and F are locally free, their direct sum will be locally free as well. Indeed,
if tUi u is a trivializing cover for E and tVi u one for F , the cover tUi XV
Àrj u will be trivializing
for E ‘ F . In particular, if m1 , . . . , mr are integers, the direct sum i“1 OP1A pmi q will be
locally free of rank r. △
If E is a locally free sheaf of rank r, and tUi u is a trivializing open cover of E , then by

297

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definition there are isomorphisms of OUi -modules

ϕi : OUr i » E|Ui . (15.1)

Over the intersections Uij “ Ui X Uj the maps τji “ ϕ´1

j ˝ ϕi are well-defined, and they
give isomorphisms
τji : OUr ij » OUr ij ,

which restricted to triple intersections Uijk “ Ui X Uj X Uk satisfy the cocycle condition

τki “ τkj ˝ τji . (15.2)
Indeed, we have ϕ´1
˝ ϕi “
k pϕ´1
˝ ϕj q ˝
k pϕ´1
˝ ϕi q .
j
Conversely, we know from the Gluing Lemma for sheaves that given isomorphisms τji as
above, satisfying (15.2) on the triple overlaps, the sheaves OUr ij may be glued together to a
sheaf E , which by definition is locally free of rank r.
Note that any isomorphism of OUij -modules OUr ij Ñ OUr ij is given by some r ˆ r-matrix
with entries in OX pUij q. Thus it is possible to define E by specifying a collection of matrices
τji satisfying the cocycle condition (15.2). These are sometimes referred to as the transition
functions of E .
The isomorphisms τji defining E are not unique. For instance, if we choose any isomor-
phisms ψi : OUr i Ñ OUr i , we can form new isomorphisms τji 1
“ ψj´1 ˝ τji ˝ ψi , satisfying
the cocycle conditions and which define the same sheaf E . Furthermore, E could admit iso-
morphisms defined on a different open cover. This is therefore not an ideal way of classifying
locally free sheaves, and in practice it is very difficult to decide when two constructions yield
isomorphic sheaves.

15.2 Examples
Example 15.4. Let A “ Z{2 ˆ Z{2 and X “ Spec A. Consider the ideal I “ Z{2 ˆ p0q
and the sheaf E “ Ir. Note that Spec A is the disjoint union of two copies of Spec Z{2, and E
restricts to the structure sheaf on one of these and to the zero sheaf on the other. E is therefore
locally free. However, E is not free, because EpXq “ Z{2, whereas any free A-module must
have at least four elements. △
Example 15.5 (The integers). Let X “ Spec Z. If E is a OX -module of finite type, then
E “M Ă for some finitely generated Z-module M , and by the structure theorem for finitely
generated abelian groups, we may write M “ Zr ‘ T , where T is a finite direct product of
groups of the form Z{nZ . If E in addition is required to be locally free, it must hold that
T “ 0 (otherwise, if p is a prime factor of an n appearing in one of the summands of T , the
r
stalk at ppq will not be free). Hence E “ Z
Ăr “ OX , and we conclude that every locally free
sheaf on Spec Z is trivial. △
Example 15.6 (The affine line). The argument of the previous example in fact applies over
any PID A: every OX -module of finite type on X “ Spec A must have the form M Ă for
r
M “ A ‘ T where T is a finitely generated torsion module, and if we require M to be
Ă

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locally free, the torsion part must vanish; i.e. we must have T “ 0. In particular, this applies
to locally free sheaves on the affine line A1k “ Spec krxs over any field k . △

Example 15.7 (The tangent bundle of the 2-sphere). Consider X “ Spec A where A is
the ring A “ Rrx, y, zs{px2 ` y 2 ` z 2 ´ 1q, and consider the A-module map ϕ : A3 Ñ A
given by multiplication by the vector V “ px, y, zq. Then M “ Ker ϕ gives rise to a
quasi-coherent sheaf T “ MĂ. We claim that T is locally free of rank 2.
Any element in M corresponds to a vector of polynomials pp, q, rq P A3 so that

xp ` yq ` zr “ 0

If we localize at x, we may solve this equation for p, so pp, q, rq is uniquely determined

by the elements q, r. Conversely, given any pair q, r of elements in A, we may define the
element p´x´1 pyq ` zrq, q, rq which lies in Mx . This implies that Mx » A2x and so
T |Dpxq “ M Ăx » O2 is trivial over Dpxq. A similar argument works for y and z , showing
Dpxq
that T is locally free of rank 2.
It is a non-trivial fact that M is not free, i.e. not isomorphic to A2 . Every element of A3
gives a vector field on the sphere S 2 . For instance, px, y, zq P A3 defines the vector field
normal to the sphere which points out from the origin to the point px, y, zq. Any element of
M therefore gives a tangent vector to S 2 . If M were free, elements of a basis would be non-
vanishing vector fields on S 2 , which is impossible (from topological reasons). Interestingly,
if we replace R by C, the corresponding sheaf is in fact trivial. △
? ?
Example 15.8. Consider the ring A “ Zr ´5s and the ideal a “ p2, 1 ` ´5q. Then one
can show that a is projective (see Exercise 15.7.16). However, a is not free: being an ideal in
A, it is free if and only if it is principal, but a requires two generators. Compare this with
Example 5.7. △

Example 15.9. Let X “ Spec Z and consider the Z-module ř M Ă Q consisting of all
rational numbers ab with b square-free. In other words, M “ p Z ¨ p´1 . Then Mppq » Zppq
for every prime number p. The module M is however not finitely generated. △

Example 15.10. Let k be a field and let R “ kru0 , . . . , un s. Let further An`1 “ Spec R
and U “ An`1 ´ t0u. Consider the exact sequence of R-modules

ϕ
0 Rn`1
R M 0 (15.3)
ř
where the map ϕ sends a polynomial p to i pxi ei where ei is the i-th standard basis vector.
We claim that the restriction E “ MĂ|U is a locally free sheaf of rank n. Taking tildes and
restricting to U we obatin the sequence

0 OU OUn`1 E 0 (15.4)

on U . Over the distinguished open set Dpxi q, this sequence splits since the map π
of sheaves ř
n
that sends j aj ej to ai x´1
i is section of ϕ. Consequently EDpxi q » ODpxi q . △

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15.3 Locally free sheaves and stalks

r
If E is a locally free OX -module of rank r, then the stalk Ex is clearly isomorphic to OX,x
for every x P X . In fact, the converse holds under some mild assumptions on X .

Theorem 15.11. Let A be a Noetherian ring and let M be a finitely generated A-module.
Then the following are equivalent:
(i) M
Ă is locally free.
(ii) Mp is free for every p P Spec A.
(iii) M is projective (that is, a direct summand of some free module An ).

Proof The direction (i)ñ(ii) is trivial. As projective modules will not play a big role in the
rest of the book, we will focus on the implication (ii)ñ(i).
Let p P Spec A be a point. Assume Mp » Arp , and pick elements m1 , . . . , mr P M so
that their classes generate Mp . Consider the map of A-modules ϕ : Ar Ñ M sending the
i-th basis vector to mi . Letting K “ Ker ϕ and C “ Coker ϕ, there is an exact sequence
ϕ
0 K Ar M C 0 (15.5)
As A is Noetherian and M is finitely generated, both K and C are also finitely generated.
This implies that the supports SupppKq and SupppCq are closed in SpecpAq (they are
equal to V pAnnpKqq and V pAnnpCqq respectively). These subsets are are moreover proper
closed subsets, because ϕp is an isomorphism. Therefore, we may pick an f P A so that
Dpf q Ă Spec A ´ V pAnnpCqq Y V pAnnpKqq is an open set containing p. This means that
Cf “ Kf “ 0 and hence that ϕf : Arf Ñ Mf is an isomorphism. Consequently, M Ă is free
over the open set Dpf q Ă Spec A.

Corollary 15.12. Let X be a Noetherian scheme and let E be a coherent sheaf. Then the
following are equivalent:
(i) E is locally free of rank r
(ii) Ex is a free OX,x -module of rank r for every x P X .

Example 15.13. Let A be DVR with fraction field K , and let x and η be respectively the
closed and the open point of X “ Spec A. Let E be the OX -module with ΓpX, Eq “ A and
Γptηu, Eq “ K , and with the restriction map A Ñ K equal to the zero map. Then E is an
OX -module with exactly the same stalks as the structure sheaf OX , but it is not locally free
(in fact, it is not even quasi-coherent). △
Let E be a coherent sheaf and let x P X be a point. The fiber of E at x is defined as the
κpxq-vector space
Epxq “ Ex {mx Ex .
Geometrically, Epxq » Ex bOX,x κpxq is the κpxq-vector space which corresponds to the
pullback of E via the map ι : Spec κpxq Ñ X .
If U Ă X is an open subset containing x and s P ΓpU, Eq is a section of E over U , we
denote by spxq the image of the germ sx P Ex in the fiber Epxq. This is in close analogy with
what we called the ‘value’ of a regular function in Chapter 4.

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If E is locally free of rank r, then Epxq is a κpxq-vector space of rank r at every point x.
The next result states that the converse holds, that is, a coherent sheaf with the property have
the fibers have the same dimension at every point is locally free.

Proposition 15.14. Let X be an integral Noetherian scheme with generic point η and let
E be a coherent sheaf. Then for all x P X ,
dimκpxq Epxq ě dimκpηq Epηq
If equality holds for every x P X , then E is locally free of rank r “ dimκpηq Epηq.

To prove this, we immediately reduce to the affine case X “ Spec A, where E “ M

Ă for a
finitely generated A-module M . Using Theorem 15.11, we reduce to proving the following:

Lemma 15.15. Let A be a local Noetherian ring with maximal ideal m and fraction field
K . Let M be a finitely generated A-module. Then
dimA{m pM bA A{mq ě dimK pM bA Kq (15.6)
with equality if and only if M is free.

Proof Let x1 , . . . , xs be elements of M so that x1 , . . . , xs form a basis for M {mM “

M bA A{m as a A{m-vector space. By Nakayama’s Lemma, the x1 , . . . , xs generate M as
an A-module. Then the elements x1 b 1, . . . , xr b 1 also generate M bA K as a K -vector
space, and hence (15.6) holds.
We next need to show that if dimκ M bA A{m “ dimK M bA K “ r, then M is free of
rank r. Choose generators x1 , . . . , xr P M so that their classes define a basis
řr for M bA A{m
r
as before. Consider the map ϕ : A Ñ M defined by pa1 , . . . , ar q ÞÑ i“1 ai xi . There is
an exact sequence
ϕ
0 Ker ϕ Ar M 0

Tensoring the sequence by K (which corresponds to localizing at p0q, and hence preserves
exactness), gives a sequence of finite-dimensional K -vector spaces

0 Ker ϕ bA K Kr M bA K 0

As K r and dimK pM bA Kq “ r have the same dimension, and the sequence is exact, we
must have Kerpϕq bA K “ 0. Hence Ker ϕ is torsion module. But being a submodule of
Ar , where A is an integral domain, it must be torsion free, and hence Ker ϕ “ 0. Therefore,
ϕ is an isomorphism and hence M is free.
Finally, if M » Ar then M bA A{m » pA{mqr and M bA K » K r , so equality holds
in (15.6).

If A is a Noetherian integral domain and M is a finitely generated A-module then there is

an open set U Ă Spec A for which M Ă is locally free. To see this, choose a presentation
ϕ
Am ÝÝÑ An ÝÝÑ M ÝÝÑ 0. (15.7)

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For r P N, define the ideal Ir pM q generated by the pn ´ rq ˆ pn ´ rq-minors of the

matrix representing ϕ. By (?, Section 20.3), the ideal Ir pM q is independent of the chosen
presentation of M , and

V pIr pM qq “ tp P Spec A | rankAp pMp q ě r ` 1u.

In particular, setting r “ dimK pM bA Kq, then M

Ă will be locally free over the open subset

U “ Spec A ´ V pIr q.

In particular, M
Ă is locally free of rank r if and only if Ir´1 pM q “ 0 and Ir pM q “ A.
This criterion is often useful for proving local freeness. Here’s an example:

Example 15.16 (A four-dimensional quadric). Let k be a field and let R “ krx0 , x1 , x2 , x3 , x4 , x5 s.

Consider the matrix
¨ ˛
0 x0 x1 x2
˚´x0 0 x3 x4 ‹
ϕ“˚ ‹.
˝´x1 ´x3 0 x5 ‚
´x2 ´x4 ´x5 0
The determinant of ϕ is the square of the quadratic polynomial

q “ x0 x5 ´ x1 x4 ` x2 x3

This means that the locus of points where ϕ has rank at most 3 is given by the hypersurface
X “ V pqq Ă P5k .
The locus of points where ϕ has rank at most 2 is defined by the ideal generated by the
2 ˆ 2-minors, which by direct calculation has radical equal to the irrelevant ideal R` .
The matrix ϕ defines an exact sequence of graded R-modules
ϕ
0 ÝÝÑ Rp´1q4 ÝÝÑ R4 ÝÝÑ Coker ϕ ÝÝÑ 0.

Applying the tilde-functor we obtain an exact sequence of sheaves

0 ÝÝÑ OP5 p´1q4 ÝÝÑ OP45 ÝÝÑ F ÝÝÑ 0 (15.8)

where F “ Coker
Čϕ.
If ι : X Ñ P5 denotes the closed embedding of X , applying ι˚ gives an exact sequence of
sheaves on X :
OX p´1q4 ÝÝÑ OX
4
ÝÝÑ E ÝÝÑ 0

where E “ ι˚ F (recall that ι˚ is only right-exact). Proposition 15.14 then shows that E is
locally free of rank 2 on X . △

15.4 Operations on locally free sheaves

The next proposition summarises some of the basic properties of locally free sheaves.

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Proposition 15.17. Let X be a scheme and let E and F be two locally free OX -modules
of rank e and f respectively. Then
(i) The direct sum E ‘ F is locally free of rank e ` f .
(ii) The tensor product E bOX F is locally free of rank e ¨ f .
(iii) The Hom sheaf Hom OX pE, Fq is locally free of rank e ¨ f .
(iv) The dual sheaf E _ “ Hom OX pE, Oq is locally free of rank e, and there
is a canonical isomorphism pE _ q_ “ E . Moreover, there is a canonical
isomorphism
E _ bOX F Ñ Hom OX pE, Fq. (15.9)

Proof Let tUi uiPI be an open cover of X which trivializes both E and F . Refining the
cover if necessary, we may assume that the Ui ’s are affine. Let U “ Spec A be one such
open affine. Then the restrictions of the sheaves in question are given by
E ‘ F|U “ pE ‘ F qr
E bOX F|U “ pE bA F qr
Hom OX pE, Fq|U “ HomA pE, F qr.
where E and F are free A-modules. The modules on the right-hand side are all free the ranks
indicated in the proposition, so we get (i), (ii) and (iii).
For (iv): if M is free, then the dual module M _ “ HomA pM, Aq is also free of the
same rank. The natural evaluation map M Ñ pM _ q_ defined by m ÞÑ pϕ ÞÑ ϕpmqq is
an isomorphism when M is free of finite rank. Moreover, for any A-module N , there is a
canonical map
M _ bA N Ñ HomA pM, N q, (15.10)
defined by ϕ b n ÞÑ pm ÞÑ ϕpmqnq. This map is an isomorphism if M and N are both free
of finite rank. (This is clearly the case when N “ A, and as both sides are additive, it holds
in general.) Finally, since the isomorphisms (15.10) are compatible with localization, they
glue together to a global isomorphism (15.9).
Example 15.18. Suppose E is locally free of rank r. Let Ui be a trivializing cover, and let τji
denote the gluing functions for E . As before, we interpret τji as an r ˆ r matrix with entries
t ´1
in OX pUi X Uj q. Then E _ is obtained by the transition matrices νji “ pτji q . △
Example 15.19. Suppose E and F are locally free of ranks r and s respectively. After
refining, we may assume that they admit the same trivializing cover. Suppose that the gluing
functions are given by τji and νji respectively. Then E ‘ F is obtained by gluing together
the different OUr i ‘ OUs i with help of the matrices
ˆ ˙
τji 0
Φji “
0 νji
For instance, the sheaf OP1A ‘ OP1A p´1q on the projective line is obtained using the gluing
matrix ˆ ˙
1 0
τ01 “
0 u

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over U0 X U1 “ Spec Aru, u´1 s with U0 “ Spec Arus and U1 “ Spec Aru´1 s. △

Locally free sheaves behave well with respect to pullbacks:

Proposition 15.20. Let f : X Ñ Y be a morphism of schemes. If E is a locally free

OY -module of rank r, then f ˚ E is a locally free OX -module of rank r.

Proof Let U Ă Y be an open over which E is trivial; that is E|U » OUr . Then, since
pullbacks commute with restriction and direct sums, and f ˚ OY “ OX , we see that
f ˚ E|f ´1 U » Ofr ´1 U , and hence f ˚ E is free over f ´1 pU q.
In fact, if tτij u are the transition functions for E over the trivializing cover tUi u of Y ,
then the transition functions for f ˚ E over the cover tf ´1 pUi qu of X are given by fU7 ij pτij q,
where fU7 ij : OY pUij q Ñ OX pf ´1 pUij qq is the usual pullback map.

Example 15.21 (Pushforwards). The pushforward of a locally free sheaf is not locally free
in general. For instance, if ι : Y Ñ X is a closed subscheme, then ι˚ OY is a sheaf with zero
stalks at points x R Y , and nonzero stalks for x P Y . △

15.5 Locally free sheaves on P1

In this section, we prove that every locally free sheaf on the projective line over a field
decomposes into a direct sum of invertible sheaves. This result was first established by
Dedekind and Weber in 1882 and has since been rediscovered multiple times, most notably
by Birkhoff in 1913 and Grothendieck in 1955.

Theorem 15.22. Let k be a field. If E is a locally free sheaf of rank r on P1k , then there
are integers a1 , . . . , ar such that
E » OP1k pa1 q ‘ ¨ ¨ ¨ ‘ OP1k par q. (15.11)
This decomposition is unique up to ordering of the factors.

Proof The result follows from the classification of quasi-coherent sheaves on P1k of Example
14.13. In the notation there, we have M0 “ krxsr , M1 “ krx´1 sr and

τ : krx˘1 sr ÝÝÑ krx˘1 sr .

As krx, x´1 s is a PID, using the Smith Normal Form (Theorem A.59), one can find invertible
matrices P and Q with entries in krx, x´1 s, so that D “ P AQ is a diagonal matrix. More-
over, as τ is an isomorphism, the diagonal entries must be units in krx, x´1 s, hence powers
of x. Therefore, D “ diagpxe1 , . . . , xer q for some integers e1 , . . . , er . By multiplying P by
a suitable power of x, and Q by the same power of x´1 , we can even assume that their entries
lie in krxs and krx´1 s respectively. This means that the data defining E are equivalent to the
data defining the locally free sheaf OP1k p´e1 q ‘ ¨ ¨ ¨ ‘ OP1k p´er q and hence E is isomorphic
as a direct sum as above. For the uniqueness part, see Exercise 15.7.14.

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Proposition 15.23. Let f : X Ñ Y be a morphism of schemes and let E be a locally

free sheaf on Y . Then
f ´1 pV psqq “ V pf ˚ sq and f ´1 pDpsqq “ Dpf ˚ sq.

Proof For each of these statements, we may reduce to the case X “ Spec B , Y “ Spec A
and L “ OYr . Then the first part follows because f ´1 pV paqq “ V pϕpaqq for a P A, which
we have seen several times before.

15.6 Zero sets of sections

Let E be an locally free sheaf on the scheme X , and suppose s P ΓpX, Eq is a global section.
We define the zero set of s by
V psq “ tx P X | spxq “ 0 in Epxqu.
Equivalently, V psq consists of the points x P X such that sx P mx Ex .
The set V psq is a closed subset of X . In fact, it carries a canonical structure as a closed
subscheme of X , which we will also denote by V psq. To see this, consider an affine open
r
set U “ Spec A such that E|U » OX |U . Under this isomorphism, the restruction of s
corresponds to an r-tuple of elements pf1 , . . . , fr q P Ar . This r-tuple generates an ideal
I “ pf1 , . . . , fr q, which defines a closed subscheme SpecpA{Iq Ñ Spec A. One can check
that that as U runs over all open affines where E is trivial, these closed subschemes glue
together to form a closed subscheme of X .
The ideal sheaf describing V psq has the following description. Viewing s P ΓpX, Eq “
HompOX , Eq, as a map of OX -modules s : OX Ñ E , we get, after applying HomOX p´, OX q,
a map of OX -modules
s_ : E _ ÝÝÑ OX . (15.12)
The image of s_ is a quasi-coherent ideal sheaf I of OX , and this is the ideal sheaf that defines
V psq. Indeed, over an affine open U “ Spec A, after identifying s with pf1 , . . . , fr q P Ar ,
the map s_ is simply the tilde of the map Ar Ñ A that sends the i-th basis vector ei to fi .
Hence I|U corresponds to the ideal generated by the fi ’s.

Let us now specalize to the case where E “ L is an invertible sheaf. If s P ΓpX, Lq is a

section of L, and Y “ V psq is the zero scheme of s, then the map s_ : L_ Ñ OX is given
by multiplication by s. In particular, if X is integral, it is injective and identifies L_ with the
ideal sheaf I of Y . Therefore, the ideal sheaf sequence takes the form
0 ÝÝÑ L_ ÝÝÑ OX ÝÝÑ ι˚ OY ÝÝÑ 0 (15.13)
Tensoring by L, we also get
0 ÝÝÑ OX ÝÝÑ L ÝÝÑ L|Y ÝÝÑ 0 (15.14)
where we for simplicity write L|Y “ ι˚ ι˚ L.
Both of these sequences are useful in computations.
There is a similar sequence involving the zero set of two sections. Let us start by the

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affine setting. If x, y P A are ring elements, then there are two maps σ : A2 Ñ A, sending
pa1 , a2 q ÞÑ a1 x ` a2 y and ρ : A Ñ A2 which sends a to p´ay, axq. If we assume that
x, y form a regular sequence, that is, x and y are nonzerodivisors, and x is a nonzerodivisor
modulo y , then the sequence
ρ σ
0 ÝÝÑ A ÝÑ A ‘ A ÝÑ I ÝÝÑ 0 (15.15)
is exact, where I “ px, yq in A. (See Exercise 15.7.15.)
If we connect the sequence above with the sequence 0 Ñ I Ñ A Ñ A{I Ñ 0, we get an
exact sequence
ρ σ
0 ÝÝÑ A ÝÑ A ‘ A ÝÑ A ÝÝÑ A{I ÝÝÑ 0.
This generalizes to sections of invertible sheaves as follows. For two sections s, t P ΓpX, Lq,
and Y “ V ps, tq, there are exact sequences
ρ σ
0 ÝÝÑ L´2 ÝÑ L´1 ‘ L´1 ÝÑ I ÝÝÑ 0 (15.16)
and
ρ σ
0 ÝÝÑ L´2 ÝÑ L´1 ‘ L´1 ÝÑ OX ÝÝÑ ι˚ OY ÝÝÑ 0. (15.17)
where I is the ideal sheaf of the zero scheme Y “ V ps, tq of s and t.
In the special case when Y is the empty scheme, the sequence reduces to
0 Ñ OX Ñ L ‘ L Ñ Lb2 Ñ 0
This has the consequence that if H 1 pX, OX q “ 0, the multiplication map
ΓpX, Lq ‘ ΓpX, Lq ÝÝÑ ΓpX, Lb2 q; pa, bq ÞÑ a b s ` b b t (15.18)
is surjective.
Example 15.24. For X “ P1k , then the two sections x0 and x1 of L “ OP1k p1q have an
empty zero scheme. In this case we know that (15.18) is surjective, as sections of Op2q are
degree 2 polynomials in x0 , x1 . △

15.7 Exercises
Exercise 15.7.1. Let ϕ : L Ñ M be a map of invertible sheaves.
a) Show that if ϕ is surjective, then it is an isomorphism.
b) Give an example of an injective map ϕ which is not an isomorphism.
Exercise 15.7.2. Let A be a ring, let M be a finitely generated A-module and let S Ă A is a
multiplicative set. Show that there are isomorphisms of S ´1 A-modules
n
a) S ´1 pSym Mq “ Symn pS ´1 M q.
Ź n Ź n
b) S ´1 p Mq “ pS ´1 M q.
Exercise 15.7.3. Let E be a quasi-coherent
Źn sheaf. Define the n-th symmetric power Symn pEq
and the n-th exterior power, E , to be the sheaves associated to the presheaves
n
ľ
U ÞÑ Symn pEpU qq and U ÞÑ pEpU qq.
respectively.

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e) Let E be a locally free of rank r. The sheaf detpEq “ ^r E is called the

determinant of E . Show that det E is an invertible sheaf.
f) Show that if f : X Ñ Y is a morphism, then for each locally free sheaf E on
Y,
f ˚ detpEq “ detpf ˚ Eq
g) Show that if E “ OP1k pa1 q ‘ ¨ ¨ ¨ ‘ OP1k par q for ai P Z, then
detpEq “ OP1k pa1 ` ¨ ¨ ¨ ` ar q.
h) Show that the determinant is additive on exact sequences: if
0 Ñ E1 Ñ E Ñ E2 Ñ 0
is exact, then
detpEq “ detpE 1 q bOX detpE 2 q. (15.19)
Exercise 15.7.4. Let A be a ring. Let 0 Ñ P Ñ N Ñ M Ñ 0 be an exact sequence of
A-modules. Show that there are exact sequences
P bA Symn´1 pN q Ñ Symn pN q Ñ Symn pM q Ñ 0
and
n´1
ľ n
ľ n
ľ
P bA NÑ NÑ M Ñ 0.
Generalize these to sequences of locally free sheaves.
Exercise 15.7.5. Let f : X Ñ Y and let E be a locally free sheaf of finite rank on Y .
Find examples where the pullback map f ˚ : ΓpY, Eq Ñ ΓpX, f ˚ Eq may fail to be injec-
tive/surjective.
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Exercise 15.7.6. Let X “ Spec A, where A “ i“0 Z. Show that M “ Z admits the
structure of an A-module which is projective, but not free.

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Exercise 15.7.7. If Y Ă X is a closed subscheme and F is an OX -module of finite type

with support contained in Y , show that F admits a finite filtration of OX -modules of finite
type
0 “ F0 Ă F1 Ă ¨ ¨ ¨ Ă Fn “ F
such that each Fi {Fi´1 is the pushforward of an OY -module of finite type.
Exercise 15.7.8. Let X be a Noetherian scheme. We define the Grothendieck group K0 pXq
as the quotient of the free abelian group on symbols rFs for F an OX -module of finite type,
by the subgroup generated by the expressions rFs ´ rF 1 s ´ rF 2 s whenever there is an exact
sequence
0 Ñ F1 Ñ F Ñ F2 Ñ 0
of finite type OX -modules.
a) If X is integral and F is an OX -module of finite type, we define the rank of F
as the number
rankpFq “ dimOX,η pFη q
where η is the generic point of X . Show that the rank defines a surjective map
of groups K0 pXq Ñ Z.
b) If Y Ă X is a closed subscheme, show that there is an exact sequence of groups
α β
K0 pY q Ý
Ñ K0 pXq Ý
Ñ K0 pX ´ Y q Ñ 0
where α is induced by i˚ and β is induced by restriction. H INT: Compute that
β ˝ α “ 0, so that β induces a map β̄ : K0 pXq{αpK0 pY qq Ñ K0 pX ´ Y q.
Use Exercise 15.7.7 and Exercise ??.
c) Let k be a field. Compute K0 pSpec kq and K0 pA1k q.
d) Compute K0 pP1k q.
Exercise 15.7.9 (The projection formula). Let f : X Ñ Y be a morphism of schemes, and
let E and F be OX -modules. Use the adjunction between f˚ and f ˚ to define a map
f˚ pF b f ˚ Eq ÝÝÑ f˚ pFq b E. (15.20)
Show that if E is locally free of finite rank then (15.20) is an isomorphism.
Exercise 15.7.10. Show that the subscheme Zpsq satisfies the following universal property:
A morphism f : T Ñ X satisfies f ˚ s “ 0 if and only if it factors through Zpsq. (Hint:
Understand the subscheme on each open affine Spec A Ă X first. Reduce to the case
E “ OX .)
Exercise 15.7.11 (Projective modules). An A-module M is called projective if it is a direct
summand in a free module; that is, if there is another module N so that M ‘ N » AI . Show
that a module M is projective if and only if the functor N ÞÑ HomA pM, N q is exact.
Exercise 15.7.12. Let A be a Noetherian ring. Show that a finitely generated module
is projective if and only if Mp is projective for all p P Spec A. H INT: Show that
HomA pM, N qp “ HomAp pMp , Np q.

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15.7 Exercises 309
ś8
Exercise 15.7.13. Let R “ i“1 F2 beÀa direct product of countably many copies of the
8
field F2 with two elements, and let I “ i“1 F2 .
a) Show that I is an ideal which is not finitely generated.
b) Show that R{I is not projective. H INT: I is generated by the elements ei P R
with a 1 in the i-th entry and 0 in all other entries. No non-zero element in R is
killed by all the ei ’s.
c) For a prime p P Spec R, show that pR{Iqp is free. H INT: If I Ă p and α P I ,
let n P N be such that αi “ 0 for every i ą n and define řn β by βi “ 0 for
i ă n and βi “ 1 for i ą n. Then β ¨ α “ 0 and 1 “ i“1 ei ` β . Show that
β R p and deduce that α maps to 0 in Ip and hence that Ip “ 0.
Exercise 15.7.14. Show that the integers appearing in Theorem 15.22 are unique. H INT:
Consider the global sections of a suitable twists Epmq.
Exercise 15.7.15. Let A be a ring and let x, y P A be two elements forming a regular
sequence. Show that the Koszul complex (15.15) is an exact sequence.
? ?
Exercise 15.7.16. Let R “ Zr ´5s and let a “ p2, 1` ´5q. Define two? maps p : R2 Ñ a
?
and s : a Ñ R2 by ppa, bq “ 2a ` p1 ` ´5qb, and spxq “ p´x, x 1´ 2 ´5 q.
a) Show that p ˝ s “ id and deduce that
R2 » a ‘ Ker p.
b) Show that a is projective, but not free.
Exercise 15.7.17. Let X be a scheme and let F be a quasi-coherent sheaf of finite type.
a) Show that the function x ÞÑ dimκpxq Fpxq is upper-semicontinuous, that is,
the subsets t x P X | dimκpxq Fpxq ě r u are closed in X . H INT: Apply the
previous exercise to a set of generators for F .
b) Show that if X satisfies ???, then this function is the constant function r if and
only if F is locally free of r.
Exercise 15.7.18. Find an example showing that if E is locally free, then E|U needs not be
free for every affine U Ă X .
Exercise 15.7.19. If X is an integral Noetherian scheme and ϕ : E Ñ F is map of locally
free sheaves. Assume that the induced map Ex bOX,x kpxq Ñ F bOX,x kpxq has constant
rank for all x P X . Then Coker ϕ is locally free.
Exercise 15.7.20 (Vector bundles). A vector bundle of rank r over a scheme X is a scheme
V together with a morphism π : V Ñ X such that there is an open cover tUi uiPI such
that for each i P I , there is an isomorphism ϕi : π ´1 pUi q » Ui ˆ Ar so that the following
diagram commutes:
ϕi
π ´1 pUi q Ui ˆ Ar
π pr1

“
Ui Ui
and for any pair i, j P I , and affine V “ Spec A Ă Ui X Uj the automorphism ϕji “

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ϕj ˝ ϕ´1
i induces an isomorphism V ˆ Ar Ñ V ˆ Ar which is A-linear, i.e., given ř by
θ : Arx1 , . . . , xr s Ñ Arx1 , . . . , xr s where θpaq “ a for all a P A and θpxi q “ aij xj
for every i “ 1, . . . , r.
a) Given a vector bundle π : V Ñ X , show that π˚ OV is a locally free sheaf of
rank r.
b) Conversely, given a locally free sheaf E of rank r on X , construct a vector
bundle π : V Ñ X such that π˚ OV “ E . H INT: Define V by gluing, using
the transition functions of E .
Exercise 15.7.21. Let E and F be locally free sheaves of ranks e and f respectively, and
let tUi u be an affine cover trivializing both E and F . Suppose the gluing isomorphisms
for E be given by τji : OUr i Ñ OUr j and the gluing isomorphisms for F are given by
νji : OUs i Ñ OUs j . Show that the gluing isomorphisms for the tensor product E bOX F
are given by τji b νji . Show that the matrix representing E b F is given by the Kronecker
product of the matrices representing τji and νji .
Exercise 15.7.22. Let ι : Y Ñ X be a closed subscheme. Show that ι˚ OY is globally
generated.
Exercise 15.7.23. Consider the scheme U “ P2k ´ tp0 : 0 : 1qu, covered by the two affine
open sets
U0 “ Spec krx, ys and U1 “ Spec krx´1 , x´1 ys.
with U0 X U1 “ Dpxq “ Dpx´1 q. Consider the matrix τ01 “ p 10 x1 q. Show that τ01 defines
a locally free sheaf E on U which fits into an exact sequence
0 ÝÝÑ OU ÝÝÑ E ÝÝÑ OU ÝÝÑ 0.
Compute ΓpX, Eq. Is E globally generated?
Exercise 15.7.24. Compute AutpA1k q by showing that every automorphism ϕ : A1k Ñ A1k
extends uniquely to an automorphism ϕ̄ : P1k Ñ P1k which fixes the point at infinity p0 : 1q.
Exercise 15.7.25. Show that the map px, yq ÞÑ px, y ` ppxqq defines an automorphism of
A2k . Conclude that AutpAnk q is enormous for n ě 2.

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Invertible sheaves and projective space

16.1 Invertible sheaves and the Picard group

Among the locally free sheaves, the most important ones are the invertible sheaves. By
definition, an OX -module L is invertible whenever there exists a covering U “ tUi u and
isomorphisms
ϕi : OUi ÝÝÑ L|Ui .
We say that gi “ ϕi p1q P LpUi q is a local generator for L.

Proposition 16.1. Let X be a scheme and L and M two invertible sheaves on X . Then:
(i) L bOX M is also an invertible sheaf. If g and h are local generators for L
and M respectively, then g b h is a local generator for L bOX M .
(ii) Hom OX pL, M q is also invertible. In particular, Hom OX pL, OX q is invert-
ible, and
Hom OX pL, OX q bOX M » Hom OX pL, M q. (16.1)

Proof This is a special case of Proposition 15.17.

Note that the tensor product acts as a sort of binary operation on the set of invertible
sheaves; L b M is invertible if L and M are, and the tensor product is associative. Tensoring
an invertible sheaf by OX gives an isomorphism L bOX OX » L, so OX serves as the
identity. For an invertible sheaf L, we define L´1 “ Hom OX pL, OX q. By the proposition,
we see that L´1 is again invertible, and serves as a multiplicative inverse of L under b. In
particular, this explains the term ‘invertible’.
This discussion leads to the following definition:

Definition 16.2. For a scheme X , the Picard group PicpXq of X is the group of
isomorphism classes of invertible sheaves on X under the tensor product.

Note that it is the set of isomorphism classes of invertible sheaves that form a group, not the
invertible sheaves themselves. That is, L bOX L´1 is isomorphic, but strictly speaking, not
equal to OX . Note also that PicpXq is an abelian group, because L bOX M is canonically
isomorphic to M bOX L.
Invertible sheaves behave well with respect to pullbacks. If f : X Ñ Y be a morphism of
schemes and L be an invertible sheaf on Y , then f ˚ L is invertible on X . If g P LpU q is a
local generator for L over U Ă Y , then f ˚ pgq is a local generator for f ˚ L over the open set

311

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f ´1 pU q. Moreover, if gij P OX pUi X Uj qˆ are the transition elements of L, then f ˚ L is

defined by the elements f ˚ pgij q “ f 7 pgij q P OX pUi X Uj qˆ .
Moreover, if L and M are two invertible sheaves on X , then
f ˚ pL bOY M q “ f ˚ pLq bOX f ˚ pM q
Therefore, the assignment L ÞÑ f ˚ L induces a morphism of groups
f ˚ : PicpY q Ñ PicpXq.
Example 16.3 (Spec Z). By Example 15.5, any invertible sheaf on X “ Spec Z is trivial,
so that
PicpSpec Zq “ 0.
?
On the other hand, PicpZr ´5sq ‰ 0, as we will show in Proposition 17.23. △
Example 16.4 (The affine line). By Example 14.24, any invertible sheaf on A1k is trivial. In
particular,
PicpA1k q “ 0.
We will prove more generally that PicpAnk q “ 0 for any n ě 0 in Chapter 17. △
Example 16.5 (The projective line). By Theorem 15.22, every invertible sheaf on P1k is
isomorphic to OP1k pmq for some m P Z. Hence we have an isomorphism
Pic P1k » Z.
△
The construction of the projective spectrum Proj R is similar to that of the affine spectrum
Spec R: the underlying topological space is defined with the help of prime ideals and the
structure sheaf from localizations of R. However, there are some fundamental differences
between the two: in the proj-construction one only considers graded rings R, and only
homogeneous prime ideals that do not contain the irrelevant ideal R` .
Another important feature of Proj R is that it comes equipped with a distighuished
invertible sheaf which is denoted by OProj R p1q. This is the geometric manifestation of the
fact that R is graded. This sheaf, along with its tensor powers OProj R pdq “ OProj R p1qbd ,
will play a cruicial role in classifying all quasi-coherent sheaves on Proj R in terms of graded
R-modules.

16.2 The graded tilde-functor

Let R be a graded ring and let M be a graded module. To M we can define a sheaf M Ă on
Proj R as follows. Let B be the collection of distinguished open sets D` pf q where f P R
is homogeneous of positive degree. For an inclusion D` pgq Ă D` pf q, we have a relation
of the form g r “ af for some homogeneous a P R and some r P N. There is a canonical
localization map Mf Ñ Mg which respects the grading when f and g are homogeneous.
Taking degree 0 parts, we have a canonical map
pMf q0 ÝÝÑ pMg q0 , (16.2)

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which sends an element m{f n with m homogeneous and deg m “ n deg f to the element
an mg ´nr .
We define a B -presheaf M
Ă by setting

M
ĂpD` pf qq “ pMf q0 ,

and for D` pgq Ă D` pf q, we define the restrictions to be the localization maps (16.2). The
two sheaf axioms are easily verified, for instance by taking the degree zero part of the sheaf
sequence (4.9) for the sheaf M
Ă on Spec R. We continue to denote this sheaf by M Ă.

Proposition 16.6. The sheaf M

Ă is a quasi-coherent sheaf on Proj R.

Proof The argument is very similar to the proof that OProj R |D` pf q » OSpecpRf q0 on page
??. The main thing to note is that Lemma ?? holds by replacing R by an R-module M ,
which implies that the restriction of M
Ă to D` pf q and pM
Č f q0 on takes the same values on
distinguished opens D` pf gq contained in D` pf q.
Note that a map of graded modules M Ñ N induces maps between the degree 0 part
of the localizations pMf q0 Ñ pNf q0 . These are compatible with further localizations, so
we get induced maps of OProj R -modules MĂÑN r . It follows that the grade tilde operation
M ÞÑ M defines a functor GrModR Ñ QCohProj R .
Ă

Each element m P M0 determines a section of ΓpD` pf q, M Ăq “ pMf q0 , by taking the

image of m in pMf q0 . If g is another homogenous element, then the two sections clearly
agree when restricted to the overlaps D` pf gq, (they both map to the image of m in pMf g q0 ).
Therefore, they glue together to a global section of M
Ă. This observation gives the following:

Lemma 16.7. There is a canonical map of R0 -modules

M0 Ñ ΓpProj R, M
Ăq,

which is functorial in M .

Basic properties of the tilde-functor

The following proposition summarizes the basic properties of the tilde-functor.

Proposition 16.8. Let R be a graded ring. The graded tilde functor M ÞÑ M

Ă has the
following properties:
(i) It is additive and exact.
(ii) For each p P Proj R, we have pM Ăqp “ pMp q0 .
(iii) If M is finitely generated, then M
Ă is of finite type.

Proof The item (i) holds because localization and taking degree zero parts are exact op-
erations that commute with forming direct limits and direct sums. The second item is a

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consequence of M Ă|D` pf q “ pM
Č f q0 ; that a prime p P D` pf q Ă Proj R corresponds to
q “ ppRf q0 ; and that we have the equality M Ăp “ ppMf q0 qq “ pMq q0 “ pMp q0 .
If M is finitely generated, then choosing a graded surjection Rn Ñ M Ñ 0, we get
n
OX ÑM Ă Ñ 0, showing M Ă is of finite type.
In many aspects the projective tilde-functor behaves as the one of Spec, but there are also
many important differences. The most striking is that different non-isomorphic modules can
give rise to the same sheaf. This can be explained from the fact that primes contained in
V pR` q are disregarded in the Proj-construction – this has the effect that graded modules
supported in V pR` q must give the zero sheaf after applyingÀ the tilde-functor.
For any integer d, we let Mąd be the R-module Mąd “ iąd Mi .

Lemma 16.9. Assume that R is a graded ring and let M and N be two graded R-
modules,
(i) If Supp M Ă V pR` q, then M Ă “ 0.
(ii) Assume that Mąd » Nąd for some d. Then M Ă»N r.
If R is generated in degree 1, the converse of (i) also holds.

The converse of (ii) holds when R is finitely generated in degree 1 and M and N are
finitely generated (Exercise 16.9.2).
Proof To prove (i), suppose that Supp M Ă V pR` q. Then item (ii) of Proposition 16.8
implies that the stalks of M
Ă are zero for every p P ProjpRq, and hence M Ă “ 0.
To prove (ii), note that the quotient N “ M {Mąd is a graded module which is killed
by the power pR` qd and consequently has support in V pR` q. By (i), we have N r “ 0, and
hence Mąd “ M . As this holds for N as well, we get M » Mąd » Nąd » N .
Ć Ă Ă Ć Ą r
For the converse of (i): if the support of M is not contained in V pR` q, there is a homoge-
neous prime ideal p P Proj R such that Mp ‰ 0. When R is generated in degree 1, this in
turn implies that pMp q0 ‰ 0. Indeed, Proj R is covered by distinguished open sets D` pf q
where f has degree 1, and so there is an f of degree 1 not lying in p. Then for an non-zero
homogeneous element x P Mp , the element x{f deg x yields a non-zero element of degree
zero in Mp .
Example 16.10. On P1k “ Proj krx0 , x1 s, the module M “ krx0 , x1 s{px20 , x21 q satisfies
M
Ă “ 0, but it is non-zero. △
The Proj-construction behaves best when the ring R is generated in degree 1. Here is a
simple example of what can happen if it is not.

Tensor products
Let M and N be two graded modules over the graded ring R. There is a natural way of
defining a grading on the tensor product, by defining the degree d piece pM bR N qd to be
the additive subgroup of M bR N generated by elements x b y where x P Mp and y P Nq
where p ` q “ d. One checks that M bR N is the direct sum of these graded parts (as an
R0 -module).

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Example 16.11. Let R “ krxs with the standard grading, and let M “ N “ R. Then
À bR N qd “ krxsd . Note that the graded pieces of pM bR N qd are not equal to
pM
p`q“d pMp bR0 Nq q: in the example that would give a vector space isomorphic to krx, ysd .
△
Recall that the tilde-functor for affine schemes is well-behaved when it comes to tensor
products in that M
Ă bOSpec A Nr “ MČ bA N . For Proj R, this identity does not always hold,
unless say R is generated in degree 1 (see Example 16.13).
Let us proceed to compare MČ bR N with M Ă bOX N r . For each homogeneous element
f P R of positive degree, there is a canonical map
Mf bpRf q0 Nf Ñ Mf bRf Nf » pM bR N qf
induced by sending x{f n b y{f m to px b yq{f n`m . When restricted to elements of degree
zero, we get a map
pMf q0 bpRf q0 pNf q0 Ñ ppM bR N qf q0 , (16.3)

which one checks is compatible with the restriction maps induced from inclusions D` pgq Ă
D` pf q, and so it is a map of B -sheaves with B being the basis of distinguished open subsets.
Therefore, we get a natural map of sheaves

M
Ă bOProj R N
r ÝÝÑ MČ
bR N . (16.4)

It is, as the Example ?? above shows, not always an isomorphism, but when R is generated
ine degree 1, it is well behaved:

Proposition 16.12. Let R be a graded ring and suppose that R is generated in degree 1.
For every graded R-modules M and N , the natural map
M
Ă bOProj R N
r ÝÝÑ MČ
bR N
is an isomorphism.

Example 16.13. Consider the polynomial ring R “ krx0 , x1 , x2 s with a grading defined by
degpx0 q “ 1, degpx1 q “ 2, and degpx2 q “ 3. After localizing at x2 , the degree zero part
of Rx2 is given by
„ 3 ȷ
x0 x0 x1 x31
pRx2 q0 “ k , , .
x2 x2 x22

We claim that M
Ăb Nr ‰M Č b N for the two graded R-modules M “ Rp1q and N “ Rp1q.
Of course, we have Rp1q b Rp1q “ Rp2q. However, we have the following equalities of
pRx2 q0 -modules:
pRp1qx2 q0 “ pRx2 q0 ¨ x0 ` pRx2 q0 ¨ x21 {x2
pRp2qx2 q0 “ pRx2 q0 ¨ x1 ` pRx2 q0 ¨ x20
Next, if we compute the tensor product of pRp1qx2 q0 with itself over pRx2 q0 we get an
pRx2 q0 -module which does not contain the monomial x1 . However, x1 clearly belongs to

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pRp2qx2 q0 . This shows that Rp1q

Ć b Rp1q
Ć fi Rp2q
Ć , as the two sheaves are different over the
open set U “ D` px2 q. △

16.3 Serre’s twisting sheaf Op1q

The most important sheaves on Proj R are the so-called twisting sheaves, denoted by
OProj R pdq. These generalize the sheaves OP1A pdq on P1A which were introduced in Sec-
tion 5.1.
Just as elements of R0 define sections of the structure sheaf on Proj R, the homogeneous
elements of degree d give sections of the sheaf OProj R pdq. As we shall see, in good cases, we
can in fact recover the entire ring R by knowing the sections of all the sheaves OProj R pdq.
The sheaves Opdq are consequences of the grading on R. If M is a graded R-module, and
n is an integer, we define an R-module M pnq as follows: the underlying R-module of M pnq
is just M , but the grading is shifted:
M pnqd “ Md`n . (16.5)
Hence N “ M pnq is a graded R-module with N0 “ Mn , N1 “ Mn`1 and so on. So
elements from Md considered as elements in M pnq will be of degree d ´ n.
In the particular case when M “ R, this gives a graded and free R-module Rpnq, which is
generated by the element 1 P R´n . Note the canonical isomorphism M pnq “ M bR Rpnq:
both have M as underlying module, and the gradings coincide by the way we have defined
the grading on the tensor product.
Example 16.14. For each m, n P Z, we have Rpnq bR Rpmq “ Rpn ` mq. △
Applying the tilde-functor to Rpnq gives us a quasi-coherent OProj R -module on Proj R:

Definition 16.15. Let R be a graded ring. For each integer n P Z, and for X “ Proj R,
we define
OX pnq “ Rpnq.
Ć

For an OX -module F , we define the twist of F by n to be the OX -module

Fpnq “ F bOX OX pnq.

If h P R is homogeneous of degree d, then it defines a section of OX pdq via the map in

Lemma 16.7. Abusing notation slightly, we will continue to write h for this section.
For an OX -module F , the tensor product gives us natural multiplication maps
OX pdq bOX Fpeq ÝÝÑ Fpd ` eq
In the case F “ M
Ă, this is simply the map of sheaves induced by the multiplication maps

pRf qd bpRf q0 pMf qe ÝÝÑ pMf qd`e ; h{f a b m{f b ÞÑ ph ¨ mq{f a`b
over each distinguished open.

Consider a homogeneous element f P R of degree 1. As f is invertible in Rf , we see that

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f n Rf “ Rf for each n P Z. Since f is of degree 1, we find that the degree 0 part of the
localization of Rpnq at f is given by

pRpnqf q0 “ pRf qn “ f n ¨ pRf q0 .

Therefore, on the distinguished affine open set D` pf q we see that

OX pnq|D` pf q “ f n ¨ OX |D` pf q .

This means that if R is generated by degree 1 elements, the OX -module OX pnq is an

invertible sheaf.

Proposition 16.16. When R is generated in degree 1, the sheaf OX pnq is invertible for
every n P Z. Moreover, there are canonical isomorphisms
OX pm ` nq » OX pmq bOX OX pnq.

Proof If R is generated in degree 1, Proposition 16.12 shows that OX pmqbOX OX pnq is the
sheaf associated to RpmqbR Rpnq » Rpn`mq; that is, it is isomorphic to OX pn`mq.

Example 16.17 (The twisting sheaves on P1A ). Let us show that the sheaves OP1A pnq agree
with the ones defined earlier in Section 5.1. Let P1A “ Proj Aru0 , u1 s. Over the distinguished
open sets D` pu0 q and D` pu1 q, we have

OP1A pnq|D` pu0 q “ un0 ¨ OP1A |D` pu0 q and OP1A pnq|D` pu1 q “ un1 ¨ OP1A |D` pu1 q .

Over D` pu0 q X D` pu1 q, the gluing maps are given by multiplication by pu0 {u1 qn , which
is precisely the gluing maps used in Section 5.1. △

The sheaves Opdq exhibit another notable difference between affine schemes and projective
schemes: Proj R typically comes equipped with a collection of non-isomorphic invertible
sheaves.

16.4 The associated graded module

For a graded R-module M , we have defined a sheaf M Ă on X “ Proj R, which is a quasi-
coherent sheaf on Proj R. The aim of this section is to show that any quasi-coherent sheaf
arises in this manner.
The situation is similar, but not identical, to the case of affine schemes. Recall that if F is a
quasi-coherent sheaf on X “ Spec A, then we can recover the unique module corresponding
to F by M “ ΓpX, Fq. For X “ Proj R, simply taking the global sections of F does not
work. For instance, for F “ OP1k on P1k , we have ΓpP1k , Fq “ k , from which we certainly
cannot recover F . The remedy is to look at the various Serre twists Fpdq of F , in fact, all of
them at once:

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Definition 16.18. Let R be a graded ring and let F be an OX -module on X “ Proj R.

We define the graded R-module associated to F , denoted Γ˚ pFq as
à
Γ˚ pFq “ ΓpX, Fpdqq.
dPZ

In particular, from X alone we get the associated graded ring

à
Γ˚ pOX q “ ΓpX, OX pdqq.
dPZ

The associated graded module Γ˚ pFq is naturally a graded module over R, with the
multiplication defined using the multiplication maps OX pdq b Fpeq Ñ Fpd ` eq. In other
words, if h P Rd , and s P Γ˚ pFqe , we regard h as a section of OX pdq and define the product
h ¨ s to be the image of h b s in ΓpX, Fpd ` eqq “ Γ˚ pFqd`e . The multiplication maps
also show that Γ˚ pOX q is a ring.
In the case F “ M Ă, there is a canonical map of graded R-modules

αM : M ÝÝÑ Γ˚ pM
Ăq, (16.6)
defined as follows. For each integer d, we have pM pdqq0 “ Md , and the map in Lemma 16.7
gives a map Md Ñ ΓpX, M Ăpdqq. Taking the direct sum for all d we get the map αM .
In particular, when M “ R, there is a natural map of graded rings
α : R ÝÝÑ Γ˚ pOX q.
In good cases, this map is an isomorphism.

Proposition 16.19. Let X “ Proj R, where R is a graded ring, finitely generated over
R0 by degree 1 elements x0 , . . . , xn which are nonzerodivisors in R. Then
n
č
Γ˚ pOX q “ Rxi Ă R0 rx0 , x´1 ´1
0 , . . . , xn , xn s. (16.7)
i“0

Moreover, if each xi is a prime element, then R “ Γ˚ pOX q.

Proof Cover X by the distinguished open sets Ui “ D` pxi q. As ΓpD` pxi q, OX pdqq »
pRxi qd , the sheaf sequence for OX pdq takes the form
n
à à
0 Ñ ΓpX, Opdqq Ñ pRxi qd Ñ pRxi xj qd .
i“0 i,j

Taking the direct sum over all integers d gives

n
à à
0 Ñ Γ˚ pOX q Ñ Rxi Ñ Rxi xj .
i“0 i,j

Àn this we see that a section of Γ˚ pOX q corresponds to an pn ` 1q-tuple pt0 , . . . , tn q P

From
i“0 pRxi q such that ti and tj coincide in Rxi xj for each i ‰ j . Now, the xi are not
zerodivisors in R, so the localization maps R Ñ Rxi are injective. It follows that we can
view all the localizations Rxi as subrings of Rx0 ...xn , and then Γ˚ pOX q coincides with the

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intersection in (16.7). In the case that the xi ’s are prime elements, this intersection is simply
R.

Corollary 16.20. Let X “ PnA “ Proj Arx0 , . . . , xn s for a ring A. Then

Γ˚ pOX q » Arx0 , . . . , xn s
In particular we can identify ΓpPnA , Opdqq with the A-module generated by homogeneous
degree d polynomials.

When R is not a polynomial ring, it can easily happen that Γ˚ pOX q is different than R.
Here is a concrete example:

Example 16.21. Let R be Crx0 , x1 s, but with with the degree 0 piece replaced by R0 “ Z.
Then X “ Proj R is isomorphic to P1C , because pRx0 q0 “ Crx1 {x0 s and pRx1 q0 “
Crx0 {x1 s. However, Γ˚ pXq “ Crx0 x1 s, which is not isomorphic to R. Note that R is not
finitely generated by over R0 in this example, so the proposition above does not apply. △

Example 16.22 (A quartic rational space curve). A systematic way of producing examples of
rings so that Γ˚ pOX q ‰ R, is to start with a closed subscheme X Ă Pnk and project it into
Pn´1
k . In some cases this will again be a closed embedding of X , but in this new embedding
X will be equal to Proj S with a different graded ring S .
The simplest example of this set-up is the rational normal quartic curve in P4k . This scheme
is defined as X “ Proj R where R is the ring

R “ kru4 , u3 v, u2 v 2 , uv 3 , v 4 s Ă kru, vs,

where all of the indicated monomials are given degree 1. Note that this equals the Veronese
ring kru, vsp4q from Section ??.
Projecting into a lower projective space corresponds to discarding some of the generators,
and in our example we discard the monomial u2 v 2 and work with X “ Proj S where

S “ kru4 , u3 v, uv 3 , v 4 s.
Evidently, S1 is of dimension 4, and we shall se that the monomial u2 v 2 reappears in
ΓpX, OX p1qq, and so ΓpX, OX p1qq will be of dimension 5 as a k -vector space.
Let us compute ΓpX, OX p1qq using the sheaf sequence, using the open affine cover
consisting of U0 “ D` pu4 q and U1 “ D` pv 4 q (these two cover X because S`
4
Ă pu4 , v 4 q).
Moreover, we have equalities OX pU0 q “ pSu4 q0 “ krv{us and OX pU1 q “ kru{vs. We
have isomorphisms OX p1q|U0 » OU0 ¨ u4 and OX p1q|U1 » OU1 ¨ v 4 . The sheaf sequence
then takes the form

0 Ñ ΓpX, OX p1qq Ñ krv{usu4 ‘ kru{vsv 4 Ñ kru{v, v{usu4 .

Note that u2 v 2 “ pv{uq2 u4 “ pu{vq2 v 4 , so the monomial u2 v 2 belongs to both the rings
kru{vsv 4 and krv{usu4 and hence defines a global section in ΓpX, OX p1qq. In fact,
ΓpX, OX p1qq “ ku4 ‘ ku3 v ‘ ku2 v 2 ‘ kuv 3 ‘ v 4 .
Therefore ΓpX, OX p1qq contains all 5 monomials, while u2 v 2 is missing from S1 . In this

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example, the graded ring Γ˚ pOX q is isomorphic to R, and R is the integral closure of S .
Exercise 16.9.5 shows that this is not a coincidence. △
Exercise 16.4.1. Let X “ P1k “ Proj krx0 , x1 s. Show that the functor F ÞÑ Γ˚ pFq is not
right exact, by considering the exact sequence

0 ÝÝÑ Rp´nq ÝÝÑ R ÝÝÑ R{pxn0 q ÝÝÑ 0.

Here the main result of this section.

Proposition 16.23. Let R be a graded ring, finitely generated in degree 1 over R0 .

Suppose F is a quasi-coherent sheaf on Proj R. Then there is a canonical isomorphism
Ă ÝÝÑ F
β: M (16.8)
where M “ Γ˚ pFq.

Proof Let X “ Proj R. We will define the map (16.8) over each distinguished subset
D` pf q “ SpecpRf q0 for f homogeneous of degree 1. As D` pf q is affine, the isomorphism
Fpdq b OX p´dq » F induces an isomorphism
ιd : ΓpD` pf q, Fpdqq bpRf q0 ΓpD` pf q, OX p´dqq Ñ ΓpD` pf q, Fq (16.9)

Therefore, if we regard f ´d as a section in OX p´dqpD` pf qq, then the tensor product

m|D` pf q b f ´d can be regarded as a section of F . This gives a map of pRf q0 -modules
βf : pMf q0 ÝÝÑ ΓpD` pf q, Fq (16.10)

sending m{f d to ιd pm|D` pf q b f ´d q. This assignment is compatible with localizations and

restrictions, because if f and g of degree 1, the following diagram commutes:
βf
pMf q0 ΓpD` pf q, Fq

βf g
pMf g q0 ΓpD` pf gq, Fq

Therefore, the βf glue to a map of OX -modules (16.8). To prove the proposition, it suffices
to show that β is an isomorphism over each D` pf q where f has degree 1.
Injectivity of (16.8): Suppose that m{f d maps to zero via the map βf . This means that
m P ΓpX, Fpdqq is a section such that m|D` pf q b f ´d “ 0. Then m|D` pf q “ 0, as f is
invertible in pRf q0 . We want to infer from this that f N m “ 0 for some N P N, as a section
of ΓpX, Fpd ` N qq. If this holds, then m{f d “ pf N mq{f d`N “ 0 in pMf q0 , and βf is
injective.
Note that the distinguished open D` pf q is covered by the open sets D` pf q X D` pxi q
for i “ 0, . . . , n. We will identify the latter with the distinguished open set D` pfi q Ă
SpecpRxi q0 , where fi “ f {xi . Consider the section m|D` pxi q , which is an element of
ΓpD` pxi q, Fq. As m|D` pxi q vanishes when restricted to the distinguished open set Dpfi q Ă
SpecpRxi q0 , we must have fiNi m|D` pxi q “ 0 for some integer Ni P N (by Exercise
14.12.22). As this happens for every i “ 0, . . . , n, we may choose N so that fiN m|D` pxi q “

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0 for every i. But then f N m defines a global section of FpN ` dq which vanishes when
restricted to every D` pxi q. Therefore f N m “ 0 by Locality, and so injectivity follows.
Surjectivity of (16.10): We keep the notation from the previous paragraph. Let t P
ΓpD` pf q, Fq be any section and consider the restrictions ti “ t|Dpfi q for i “ 0, . . . , n, to
the open set Dpfi q Ă D` pxi q. Since D` pxi q is affine, we know from Exercise 14.12.22 that
some product fiN ti extends to a section mi in ΓpD` pxi q, Fq (as before we may choose an
N that works for all i).
In view of the isomorphism FpN q|D` pxi q “ xN i F|D` pxi q , we find

f N ti “ xN N
i fi ti “ mi P ΓpD` pxi q, FpN qq.

One issue is that mi and mj may potentially not agree over the overlaps D` pxi q X D` pxj q,
preventing us from beging able to glue them together to a section of F . However, we have
mi |Dpfi q “ f N t|Dpxi q |Dpfi q ,
so at least mi “ mj when restricted to D` pxi q X D` pxj q X D` pf q. Now, D` pxi q X
D` pxj q “ D` pxi xj q is also affine, and D` pxi q X D` pxj q X D` pf q is a distinguished
open subset of D` pxi xj q. Arguing as in the injectivity part above shows that there is a large
integer l ą 0 such that
f l ¨ pmi |D` pxi qXD` pxj q ´ mj |D` pxi qXD` pxj q q “ 0
in ΓpD` pxi xj q, FpN `lqq. But this means that the sections f N `l ti can be glued to a section
m P ΓpX, FpN ` lqq. By construction, this section restricts to tf N `l |D` pf q over D` pf q,
and hence m{f N `l maps to t via the map in (16.10).

We have now defined two functors

„ : GrModR ÝÝÑ QCohX
Γ˚ : QCohX ÝÝÑ GrModR

Since β : ΓČ ˚ pFq Ñ F is an isomorphism, it follows that the tilde functor is essentially

surjective; that is, every quasi-coherent sheaf on X is the tilde of a graded module. However,
unlike the affine case, the functors do not give mutual inverses. The functor „ is not faithful
as the tilde of any module M with support in V pR` q is the zero sheaf. By Lemma 16.9
however, this is the only source of ambiguity.
Putting everything together, we find

Theorem 16.24 (Quasi-coherent sheaves on Proj). Let R be a graded ring, finitely

generated in degree 1 over R0 and let X “ Proj R. Then every quasi-coherent sheaf F
is of the form M
Ă for some graded R-module. More precisely,

pX, Fq » F.
Γ˚Č
For a graded R-module M , we have M
Ă “ 0 if and only if Mp “ 0 for all p R V pR` q.

We get a similar correspondence if we restrict our attention to finitely generated R-modules

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and OX -modules of finite type. For finitely generated graded modules, the converse of claim
(ii) in Lemma ?? holds and gives another criterion for when two modules have isomorphic
Àeach graded module M and each integer d, we defined the graded
tildes. Recall that to
R-module Mąd “ iąd Md . We then have:

Theorem 16.25 (Coherent sheaves on Proj). Assume R be generated over R0 by

finitely many elements of degree 1. Then every quasi-coherent sheaf F of finite type on
X “ Proj R is of the form M Ă for some finitely generated R-module M . In fact, we can
take
M “ Γ˚Č pX, Fq. (16.11)
Two finitely generated modules M and N give rise to the same sheaf F if and only if
Mąd » Nąd for some d.

Proof See Exercise 16.9.10 and Exercise 16.9.11.

16.5 Closed subschemes of Proj R

Having discussed quasi-coherent sheaves on projective spectra, we will now use this to
study closed subschemes. We saw in Chapter ??, that each graded ideal I Ă R gives a
closed subscheme ProjpR{Iq Ñ Proj R. The main result of this section is that every closed
subscheme is of this form, at least when R is finitely generated by degree 1 elements.

Proposition 16.26. Let R be a graded ring generated by finitely many elements of degree
1, and let X “ Proj R. If Y Ñ Proj R is a closed subscheme of Proj R, then there is a
homogeneous ideal I Ă R so that Y is equal to ProjpR{Iq.

Proof Let I denote the ideal sheaf of Y . Note that Γ˚ pIq Ă Γ˚ pOProj R q is a homogeneous
ideal of the ring Γ˚ pOProj R q. The preimage I “ α´1 pΓ˚ pIqq Ă R under the natural map
α : R Ñ Γ˚ pOProj R q is therefore also a homogeneous ideal of R. Applying tilde to the
diagram on the left, gives the right diagram, which shows that Ir “ I .

I R Ir OProj R
α α

Γ˚ pIq Γ˚ pOProj R q Γ˚ pIq “ I OProj R

Finally, note that ideal sheaf Ir induces the subscheme ProjpR{Iq Ñ Proj R, so we have an
equality of subschemes Y “ ProjpR{Iq.

As in the discussion on quasi-coherent sheaves on Proj R, it can happen that several ideals
correspond to the same subscheme. There is however, a ‘canonical representative’, defined as
follows.
We will for simplicity consider the case R “ Arx0 , . . . , xn s, so that Proj R “ PnA . Let
ι : Y Ñ Proj R be a closed subscheme. If we restrict to Ui “ D` pxi q Ă PnA , ι´1 pUi q

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defines a closed subscheme of Ui , hence an ideal ai Ă pRxi q0 . Here, since Ui is affine,

ai “ IpUi q for each i. Then Y is the closed subscheme defined by the homogeneous ideal
à␣ (
I“ r P Rd | r{xdi P ai for all i . (16.12)
dě0

If Y “ ProjpR{Jq then the ideal (16.12) is equal to the saturation of J with respect to the
irrelevant ideal px0 , . . . , xn q. More precisely, for an ideal J Ă Arx0 , . . . , xn s, the ideal J sat
is defined by

J sat :“ tr P R | DN P N such that xN

i ¨ r P J for every i “ 0, . . . , nu.

Note that there is always an inclusion J Ă J sat . We say that J is saturated if J “ J sat . It is
not hard to check that J sat is homogeneous if J is.

Example 16.27. In R “ krx0 , x1 s, the saturation of px20 , x0 x1 q is the ideal px0 q. Note that
both px0 q and px20 , x0 x1 q define the same subscheme of P1k , but in some sense the latter ideal
is inferior, as it has a component in the irrelevant ideal px0 , x1 q. This example is typical – the
saturation is a process which throws away components of I supported in the irrelevant ideal.
△

Example 16.28. The saturation of the irrelevant ideal R` “ px0 , . . . , xn q is the whole ring
R. △

Example 16.29. If I Ă R “ krx0 , . . . , xn s is a radical homogeneous ideal, and I is not

equal to px0 , . . . , xn q, then I is saturated. Indeed, as I is radical, the primary decomposition
takes the form
I “ p1 X ¨ ¨ ¨ X pr

where the pi are the minimal primes of I . By assumption, px0 , . . . , xn q is not among these
pi . Now, if r P I sat satisfies r ¨ xNi P I for all i, but r R I , then r is not contained in one of
the pi , say r R p1 . But then since p1 is prime, we must have xi P p1 for all i, which implies
that p1 “ px0 , . . . , xn q, a contradiction. △

For projective space PnA , the previous proposition can be extended as follows.

Proposition 16.30. Let A be a ring and PnA “ Proj R where R “ Arx0 , . . . , xn s.

(i) If Y is a closed subscheme of PnA defined by an ideal sheaf I , then the ideal
I “ Γ˚ pIq Ă R
is a homogeneous saturated ideal. Moreover, Y corresponds to the subscheme
ProjpR{Iq Ñ Proj R.
(ii) Two ideals I, J in R define the same subscheme if and only if they have the
same saturation.
In particular, there is a one-to-one correspondence between closed subschemes i : Y Ñ
PnA and saturated homogeneous ideals I Ă R.

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Example 16.31. Let X be an integral, projective scheme over a field k . Then ΓpX, OX q is a
finite extension of k . In particular, if k is algebraically closed, then
ΓpX, OX q “ k (16.13)
This follows because X “ Proj R where R “ krx1 , . . . , xn s{I where I is a homogeneous
prime ideal. In particular, R is △

16.6 Sheaves on projective space

In this section, we write Pnk “ Proj R where R “ krx0 , . . . , xn s with the standard grading.
We recall the following fundamental theorem from commutative algebra, which gives the
structure of graded modules over R in terms of free modules.

Theorem 16.32 (Hilbert’s syzygy theorem). Let k be a field and let R “ krx0 , . . . , xn s.
Then if M is a finitely generated graded R-module, then there is a finite free resolution
(that is, an exact sequence)
0 ÝÝÑ Fn ÝÝÑ . . . ÝÝÑ F1 ÝÝÑ F0 ÝÝÑ M ÝÝÑ 0, (16.14)
Àbk
where Fj “ i“1 Rp´dij q is a free graded R-module. Fi is called the i-th syzygy
module of the resolution.

By the theorem, M is the cokernel of a map F1 Ñ F0 of free R-modules of finite rank.

Any such map is defined by a matrix with entries being homogenous elements of R, so this is
a convenient way to present M : the generators come from F0 and the relations come from F1 .
Furthermore, the relations among the relations, and so on, are defined by matrices between
free modules. This sort of resolution is immensely useful in the study of M , as it allows us to
apply linear algebra techniques which can simplify computations and define invariants of M .
If we apply the „-functor to the sequence (16.14), we obtain an exact sequence of sheaves
on Pnk
0 ÝÝÑ En ÝÝÑ . . . ÝÝÑ E1 ÝÝÑ E0 ÝÝÑ M Ă ÝÝÑ 0
Àbk
where Ej “ i“1 OPk p´dij q is a direct sum of sheaves of the form OPk pdq. Hence any
n n

finite type OX -module admits a resolution where the terms are direct sums of invertible
sheaves. This shows very clearly why the invertible sheaves Opdq are so important: they are
the building blocks of all finite type sheaves on Pnk .
Here are a few important special cases:
Example 16.33 (Hypersurfaces). Let F P R denote an homogeneous polynomial of degree
d ą 0. Then F determines a projective hypersurface X “ ProjpR{F q Ă Pnk . Write
ι : X Ñ Pnk for the closed embedding.
Consider the map Rp´dq Ñ R given by multiplication by F . Note the shift in degrees
here, in order to make this into a map of graded modules (the constant ‘1’ gets sent to F ,
which should have degree d on both sides). We obtain a sequence
0 ÝÝÑ Rp´dq ÝÝÑ R ÝÝÑ R{pF q ÝÝÑ 0.
which is a resolution of the graded module R{pF q.

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Applying the tilde to this sequence, we obtain

0 ÝÝÑ OPnk p´dq ÝÝÑ OPnk ÝÝÑ R{F

Ć ÝÝÑ 0. (16.15)
Ć is isomorphic to the sheaf ι˚ OX ; both sheaves take the value pRf {pF qq0
Note that R{F
over each distinguished open set D` pf q. Therefore, comparing (16.15) to the ideal sheaf
sequence
0 ÝÝÑ I ÝÝÑ OPnk ÝÝÑ ι˚ OX ÝÝÑ 0,
we see that there is an isomorphism of OPnk -modules I » OPnk p´dq. △
Example 16.34 (Complete intersections). Let F, G be two homogeneous polynomials of
degrees d, e respectively without common factors. Let I “ pF, Gq and X “ ProjpR{Iq Ă
Pnk . The subscheme X then has codimension 2. X is called a complete intersection; it is the
scheme-theoretic intersection of the two hypersurfaces V pF q and V pGq. To study X , we
use the exact sequence
α β
0 ÝÝÑ Rp´d ´ eq ÝÝÑ Rp´dq ‘ Rp´eq ÝÑ I ÝÝÑ 0.
where the maps are defined by αphq “ p´hG, hF q and βph1 , h2 q “ h1 F ` h2 G. With the
shifts indicated, these maps preserve the grading.
To prove exactness, we first note that α is injective (because R is an integral domain) and
β is surjective (by the definition of I ). Then if ph1 , h2 q P Ker β , we have h1 F “ ´h2 G,
which, as F, G have no common factors, means that there is an element h so that h1 “ ´hG,
h2 “ hF .
Applying „, we obtain the following exact sequence
0 ÝÝÑ OPnk p´d ´ eq ÝÝÑ OPnk p´dq ‘ OPnk p´eq ÝÝÑ IX ÝÝÑ 0. (16.16)
This sequence, along with the ideal sheaf sequence, allows us to compute invariants of X
(see Section 18.14). △
Example 16.35 (The twisted cubic curve). Let k be a field and consider P3 “ Proj R where
R “ krx0 , x1 , x2 , x3 s. We will consider the twisted cubic curve C “ V pIq where I Ă R is
the ideal generated by the 2 ˆ 2-minors of the matrix
ˆ ˙
x0 x1 x2
A“
x1 x2 x3
i.e., I “ pq0 , q1 , q2 q “ px21 ´ x0 x2 , x0 x3 ´ x1 x2 , ´x22 ` x1 x3 q.
Consider the map of R-modules R3 Ñ I sending ei ÞÑ qi . This is clearly surjective, since
the qi generate I . Let us consider the kernel of this map, that is, the module of relations of
the form a0 q0 ` a1 q1 ` a2 q2 “ 0 for ai P R. There are two obvious relations of this form,
i.e., the ones we get from expanding the determinants of the two matrices
¨ ˛ ¨ ˛
x0 x1 x2 x0 x1 x2
˝x0 x1 x2 ‚ ˝x1 x2 x3 ‚
x1 x2 x3 x1 x2 x3
A¨
(So the first matrix gives x0 q2 ´ x1 q1 ` x2 q2 “ 0 for instance). These give a map R2 ÝÑ R3 ,

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where A is the matrix above. This map is injective, and it turns out that there is an exact
sequence of R-modules
A
0 ÝÝÑ R2 ÝÝÝÝÑ R3 ÝÝÑ I ÝÝÑ 0.
Again, we should consider these as graded modules, so we must shift the degrees according
to the degrees of the maps above
A
0 ÝÝÑ Rp´3q2 ÝÝÝÝÑ Rp´2q3 ÝÝÑ I ÝÝÑ 0.
This gives the resolution of the ideal I of C . Then applying „, and using the fact that I “ Ir,
we get a resolution of the ideal sheaf of C :
A
0 ÝÝÑ OP3k p´3q2 ÝÝÝÝÑ OP3k p´2q3 ÝÝÑ I ÝÝÑ 0.
We will see later in Chapter 18 how to use sequences like this to extract geometric information
about C . △

16.7 Globally generated sheaves

We say that an OX -module F is generated by global sections, or that F is globally generated,
if there exist sections si P FpXq, i P I , and the map of OX -modules
à
OX ÝÝÑ F
iPI

sending ei ÞÑ si is surjective. Equivalently, there is a set of sections si P FpXq, i P I , such

that for each x P X , the germs of si generate Fx as an OX,x -module.
From the definition, we see that any quotient of a globally generated sheaf is also globally
generated.
Example 16.36. On an affine scheme any quasi-coherent sheaf is globally generated. Indeed,
if X “ Spec A, and F “ M Ă, for some A-module M , then any set of generators of M will
define global sections which globally generate F . △
Example 16.37. The sheaf OPnA p1q on PnA “ Proj Arx0 , . . . , xn s is generated by the
sections x0 , . . . , xn . Indeed, for i “ 0, . . . , n, the section xi generates OPnn p1q over the open
set D` pxi q. The same argument shows that OPnA p1q is globally generated on Proj R for any
graded ring which is generated in degree 1.
On the other hand, if R is not generated in degree 1, then it can happen that the sheaf Op1q
has no global sections at all. This happens for instance for the weighted projective space
Pp2, 3, 4q “ Proj krx2 , x3 , x4 s (with deg xi “ i).
The invertible sheaf OPnA p´1q is never globally generated when n ą 0. △
We will now focus on the case where F is an invertible sheaf.
If s P ΓpX, Lq is a global section, we define the open set Dpsq by
Dpsq “ t x P X | spxq ‰ 0 in Lpxq u.
Equivalently, Dpsq is the set of points x P X where sx R mx Lx .
The set Dpsq is indeed open in X . If U Ă X denotes an open affine so that L|U » OU ,

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then s|U corresponds to an element f P OX pU q. For x P U , we have sx P mx Lx if and only

if fx P mx . This means that U X Dpsq “ Dpf q, and hence Dpsq is open.

Lemma 16.38. Let X be a scheme and let L be an invertible sheaf on X . If s P ΓpX, Lq

is a global section, then there is an isomorphism of OX -modules
ϕ : OX |Dpsq ÝÝÑ L|Dpsq
which sends 1 to s.

Proof We define ϕ over an open set U Ă Dpsq, by sending 1 P OX pU q to s|U P LpU q

and extend OX -linearly. This is an isomorphism if and only if it is an isomorphism locally, so
we may reduce to the case where X “ Spec A and L “ OX . In that case, Dpsq “ Spec As ,
s is invertible in As and so multiplication by s is an isomorphism As Ñ As .

Proposition 16.39. Let f : X Ñ Y be a morphism of schemes and let L be an invertible

sheaf on Y . Then if L is an invertible sheaf which is generated by global sections
s0 , . . . , sn , then f ˚ L is generated by the sections t0 “ f ˚ s0 , . . . , tn “ f ˚ sn , and X is
covered by the open sets Dpt0 q, . . . , Dptn q.

Proof As L is invertible, we may reduce to the case X “ Spec B , Y “ Spec A, L “ OY

and f is induced by a ring map ϕ : A Ñ B . Then we can view the sections s0 , . . . , sn as
elements of A, and the pullback f ˚ psq of a section s is simply given by ϕpsq.
Let x P X be a point. Then there is an index j so that sj pyq ‰ 0 P kpyq is nonzero at
y “ f pxq. Then f ˚ psj q “ ϕpsj q is then also nonzero at x, as its class in OX,x coincides
with the image of sj via the map of local rings fx : OY,y Ñ OX,x .
As the elements s0 , . . . , sn generate L, they generate the unit ideal in A, and so the
pullbacks ϕps0 q, . . . , ϕpsn q generate the unit ideal in B . Hence X “ Spec B is covered by
the open sets Dpf ˚ ps0 qq, . . . , Dpf ˚ psn qq.

Example 16.40. For a morphism f : X Ñ Y , a pushforward f˚ OX may fail to be globally

generated. For example, if f : P1 Ñ P1 is the ‘squaring map’ of Example 14.34, then
f˚ OP1 » OP1 ‘ OP1 p´1q, which is not globally generated (the space of sections is only
1-dimensinal). △

16.8 Morphisms to projective space

Given a scheme X , it is natural to ask when there is a morphism to a projective space

f : X ÝÝÑ Pn . (16.17)

Note that the corresponding question for AnZ has already been answered. Morphisms X Ñ AnZ
are in one-to-one correspondence with elements of ΓpX, OX qn , i.e., an n-tuple of regular
functions on X .
For a morphism into projective space it will not be the global sections ΓpX, OX q, but
rather sections of an invertible sheaf that will be the analogous data we need in order to
specify a morphism. Namely, given a morphism f : X Ñ PnZ , we get an invertible sheaf

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L “ f ˚ OPnZ p1q on X , as well as n ` 1 distinguished global sections si “ f ˚ xi by pulling

back the sections x0 , . . . , xn of Op1q. As Op1q is globally generated by x0 , . . . , xn , L will
be globally generated by the pullbacks s0 , . . . , sn .
The main result in this section is that there is a way to reverse this process. In other words,
from a given invertible sheaf L and n ` 1 global sections si P ΓpX, Lq that generate L, we
can uniquely construct a morphism f : X Ñ Pn so that f ˚ OPn p1q “ L and f ˚ xi “ si .
Hence pL, s0 , . . . , sn q is the data we are after.

Theorem 16.41. Let X be a scheme over a ring A, and let L be an invertible sheaf on
X with global sections s0 , . . . , sn P ΓpX, Lq which generate L. Then there is a unique
morphism over A
f : X ÝÝÑ PnA “ Proj Arx0 , . . . , xn s
and an isomorphism f ˚ Op1q » L which maps f ˚ xi to si for i “ 0, . . . , n.

Proof of Theorem 16.41 Write PnA “ Proj R, where R “ Arx0 , . . . , xn s and Op1q for
OP1A p1q.
Suppose that we are given n ` 1 sections s0 , . . . , sn which globally generate an invertible
sheaf L. By Lemma 16.38, the open sets Dpsi q provide a local trivializing cover of L, with
isomorphisms ψi : OX |Dpsi q Ñ L|Dpsi q which sends 1 to the section si . We write t ÞÑ t{si
for the inverse of this isomorphism. Concretely, if we restrict the section sj to Dpsi q, we
have sj “ rij si for some rij P ΓpDpsi q, OX q; then sj {si “ rij . These sections define a
map of A-algebras

pRxi q0 ÝÝÑ ΓpDpsi q, OX q (16.18)

xj sj
ÞÑ
xi si
By the correspondence between ring homomorphisms and maps into affine schemes, we
obtain a morphism of A-schemes fi : Dpsi q Ñ D` pxi q. Over Dpsi q X Dpsj q “ Dpsi sj q,
the map sends xxkj “ xxkj {x
{xi
i
to sskj “ sskj {s
{si
i
. In other words, the following diagram commutes:

pRxi q0 ΓpDpsi q, OX q

` ˘
Rxi xj 0
ΓpDpsi sj q, OX q

` ˘
Rxj 0
ΓpDpsi q, OX q

That means that the morphisms glue to a morphism f : X Ñ Pn .

Next, we show that there is an isomorphism f ˚ Op1q » L. Recall that Op1q|D` pxi q “
OD` pxi q xi , so xi serves as the local generator. Over the open sets Dpsi q “ f ´1 pDpxi qq,
we therefore have f ˚ Op1q|Dpsi q » ODpsi q with local generator f ˚ xi . Likewise, L|Dpsi q »
ODpsi q with local generator si . As both L and f ˚ Op1q are trivial over Dpsi q, we may
define an isomorphism f ˚ Op1q|Dpsi q Ñ L|Dpsi q by sending f ˚ xi to si . To check that these
isomorphisms glue, we need to check that they are compatible with the gluing maps. Recall

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that Op1q is obtained by gluing together the ODpxi q using the gluing maps τij “ xj {xi .
Therefore f ˚ Op1q is obtained by gluing together ODpsi q using multiplication by f ˚ τij “
f 7 pxj {xi q as gluing maps. However, f 7 pxj {xi q “ sj {si . Hence f ˚ Op1q and L are defined
by the same gluing data, and the local isomorphisms define an isomorphism f ˚ Op1q » L. It
is clear that the generator f ˚ xi maps to si via this identification, because this happens over
each open set Dpsi q.
The uniqueness part is clear, because the sections s0 , . . . , sn determine the ratios sj {si ,
which in turn determine f .
We will refer to a morphism ϕ : X Ñ PnA as given by the data pL, s0 , . . . , sn q and
informally write
X ÝÝÑ PnA
x ÞÑ ps0 pxq : ¨ ¨ ¨ : sn pxqq
One should still keep in mind that the sections si are sections of L, not regular functions. In
light of the above proof, we see that it is the ratios sj {si which can be interpretated as regular
functions, locally on Dpsi q “ tx P X | si pxq ‰ 0u.
We also see that two sets of data pL, s0 , . . . , sn q, pL, t0 , . . . , tn q give rise to the same
morphism f : X Ñ PnA if and only there is a unit λ P OX ˆ
pXq so that ti “ λsi for each i.
n
Thus morphisms f : X Ñ PA are in bijective correspondence with the data pL, s0 , . . . , sn q
modulo this equivalence relation. One can compare this with the description of the A-valued
points of PnA from Chapter 5:
Example 16.42. Let A “ Z and consider X “ Spec Z. Since any invertible sheaf on
Spec Z is isomorphic to the structure sheaf, we see that Z-morphisms f : X Ñ PnZ are in
one-to-one correspondance with pn ` 1q-tuples of elements in ΓpX, OX q “ Z. In other
words, any Z-point Spec Z Ñ PnZ is in ‘homogeneous coordinate form’, as defined in Chapter
5. The same applies when ? A is a local ring, or any ring with PicpSpec Aq “ 0. On the
other hand,
? PicpSpec Z r ´5sq ‰ 0, and we saw that there were ‘non-obivious’ maps
1
Spec Zr ´5s Ñ PZ . △
Example 16.43. Let X “ P1k “ Proj krs, ts and L “ OP1k p2q. Then L is globally generated
by s2 , st, t2 and the corresponding morphism
ϕ : P1k ÝÝÑ P2k
ps : tq ÞÑ ps2 : st : t2 q
has image V px0 x2 ´ x21 q which is an irreducible conic curve. △
Example 16.44 (Cuspidal cubic). Let X “ A1k and L “ OX . Then, ΓpX, Lq “ krts is
infinite dimensional over k . Choosing the three sections 1, t2 , t3 , we get a map of schemes
ϕ : X ÝÝÑ P2k
t ÞÑ p1 : t2 : t3 q
whose image in P2 is the cuspidal cubic V px0 x22 ´ x31 q minus the point at infinity. △
Given a scheme X with s0 , . . . , sn of an invertible sheaf L, there is aŤmaximal open subset
n
U such that the sections generate L for all points in U , namely U “ i“0 Dpsi q. If we do

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not assume that the si globally generate L, we still get a morphism ϕ : U Ñ PnA . In other
words, the sections define a rational map ϕ : X 99K PnA , which is a morphism when restricted
to U .

Example 16.45 (Pn as a quotient space). Let X “ An`1 k , and L “ OX . Then ΓpX, Lq “
krx0 , . . . , xn s. If we take the sections x0 , . . . , xn , then they generate L outside V px0 , . . . , xn q.
Hence we get a morphism of schemes

An`1
k ´ V px0 , . . . , xn q ÝÝÑ Pnk
px0 , . . . , xn q ÞÑ px0 : ¨ ¨ ¨ : xn q
which is exactly the ‘quotient space’ description of Pn from (5.5). △
Exercise 16.8.1 (The Veronese surface). Let X “ P2k “ Proj krx0 , x1 , x2 s. Show that
L “ OP2 p2q is globally generated by the sections
x20 , x21 , x22 , x0 x1 , x0 x2 , x1 x2
Show that the corresponding morphism ϕ : P2 Ñ P5 is a closed embedding.

Exercise 16.8.2 (The quadric surface). Let X “ P1k ˆ P1k and L “ p˚ OP1k p1q b q ˚ OP1k p1q
where p, q : X Ñ P1 are the two projections. If x0 , x1 is a basis for ΓpX, p˚ OP1k p1qq, and
y0 , y1 is a basis for ΓpX, q ˚ OP1k p1qq, show that p˚ OP1k p1q b q ˚ OP1k p1q is globally generated
by the four sections

s0 “ x0 y0 , s1 “ x0 y1 , s2 “ x1 y0 , s3 “ x1 y1 .
Describe the corresponding morphism X Ñ P3k .

Application: Automorphisms of Pnk

If k is a field, then any invertible pn ` 1q ˆ pn ` 1q matrix A with entires in k acts on
krx0 , . . . , xn s and gives rise to a linear automorphism Pnk Ñ Pnk . Moreover, two matrices A
and A1 determine the same automorphism if and only if m “ λm1 for some non-zero scalar
λ P k ˆ . The projective linear group is defined by as the quotient group
PGLn pkq “ GLn pkq{k ˆ

We will now prove that all automorphisms of Pnk are given by linear transformations.

Theorem 16.46. AutK pPnk q “ PGLn pkq.

Proof The above paragraph shows that there is an injective map from the righthand side to
the left. Conversely, let ϕ : Pnk Ñ Pnk be any automorphism. Then ϕ induces an isomorphism

PicpPn q ÝÝÑ PicpPnk q

Since PicpPnk q is generated by OPnk p1q, we must have either ϕ˚ OPnk p1q “ OPnk p1q or
ϕ˚ OPnk p1q “ OPnk p´1q. The latter case is however impossible, because ϕ˚ OPnk p1q has

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many global sections, whereas OPnk p´1q has none. Hence ϕ˚ OP1k p1q “ OP1k p1q. Evaluating
ϕ˚ over global sections gives a k -linear isomorphism
ΓpPnk , OPnk p1qq ÝÝÑ ΓpPn , OPnk p1qq,
Now, we may choose tx0 , . . . , xn u as a basis for ΓpPnk , OPn p1qq, and so in this basis ϕ˚
gives rise to an invertible pn ` 1q ˆ pn ` 1q-matrix A. By construction A induces the same
linear transformation Pnk Ñ Pnk as ϕ, and so ϕ comes from an element of PGLn pkq.

16.9 Exercises
Exercise 16.9.1. Let R “ Qrx, y, zs with deg x “ 1, deg y “ 2, deg z “ 3. Show that the
map (16.4) is not an isomorphism for M “ Rp1q and N “ Rp2q.
Exercise 16.9.2. Suppose R is a graded ring finitely generated in degree 1 and let M and
N be two finitely generated R-modules. Show that M Ă»N r if and only if Mąd » Nąd for
some d.
Exercise 16.9.3. Let P1k “ Proj R where R “ krx0 , x1 s. Show that the functor F ÞÑ
Γ˚ pFq is not right exact, by considering the exact sequence
0 ÝÝÑ Rp´nq ÝÝÑ R ÝÝÑ R{pxn0 q ÝÝÑ 0.
Exercise 16.9.4. Let k be a field and let R “ krx0 , . . . , xn s. Let π : An`1 ´ 0 Ñ Pnk “
Proj R denote the ‘quotient morphism’ (see page 98). Show that for a graded R-module M ,
we have
à
π ˚ pM
Ă| n`1 q “
Ak M
Ăpdq
´0
nPZ

Exercise 16.9.5. Let R be a graded Noetherian integral domain generated in degree 1 by

x0 , . . . , xn . The aim of this exercise is to show that R1 “ Γ˚ pOX q is an integral extension
of R.
a) Show that α : R Ñ R1 is injective.
b) s P R1 be a homogeneous element of non-negative degree. Show that we can
find an n ą 0, so that αpxni qs P αpRq for every i. Deduce that αpRm qs Ă
αpRq for m large.
c) Let J Ă R be the generated by elements of degree ě kn. Show that J is
finitely generated, and that αpJqs Ă αpJq.
d) Use the Cayley–Hamilton theorem to show that s satisfies an integral equation
over R.
Exercise 16.9.6. Let k be an algebraically closed field and let X Ă Pnk be a projective
k -scheme of finite type. Show that ΓpX, OX q » k .
Exercise 16.9.7. Let X “ ProjpRrx, ys{px2 ` y 2 qq. Show that X is affine, and that
ΓpX, OX q » C.
Exercise 16.9.8. Let P1k “ Proj kru0 , u1 s and t “ u0 {u1 . Consider the closed sub-
scheme Z Ă P1k which is supported at p0, 1q and which is locally given as Spec krts{tn Ă

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D` pu0 q “ Spec krts. Describe the R module Γ˚ OZ and the canonical map R “ Γ˚ OPk1 Ñ
Γ˚ OZ .
Exercise 16.9.9. Check that the saturation I sat is homogeneous if I is.
Exercise 16.9.10. Assume R be generated over R0 by finitely many elements of degree 1.
Let M and N be two finitely generated graded R-modules. In this exercise you will show
that M
Ă»N r if and only if Mąd » Nąd for some d. H INT: Consider the images of M and
N in Γ˚ pM
Ăq “ ΓpN r q.

Exercise 16.9.11. Assume R be generated over R0 by finitely many elements of degree 1.

Let F be a quasi-coherent sheaf on Proj R. Show that F is of finite type if an only if it
is the tilde of a finitely generated R-module. H INT: Write F “ M
Ă and pick surjections
n
À
OUi Ñ M |Ui . Lift these to a map ϕ : i,j Rp´dij q Ñ M and show that the cokernel has
Ă
zero tilde.
Exercise 16.9.12. Let ϕ : R Ñ S be a map of graded rings and let f : U Ñ Proj S be the
morphism given by ϕ, where U “ ProjpSq ´ V pϕpR` qq.
a) Show that f ˚ M
Ă “ MČ bR S|U for each graded R-module M .
b) Show that f˚ pN |U q “ N
r ĂR for each graded S -module N (and where NR is as
usual, N regarded as a graded R-module).

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Divisors

17.1 Weil divisors

Throughout this chapter we will assume that X is a Noetherian integral normal scheme. A
prime divisor is an integral subscheme Z Ă X of codimension 1. A Weil divisor, or simply
divisor, is a finite formal sum
D “ m1 Z1 ` ¨ ¨ ¨ ` mr Zr
where mi P Z and the Zi ’s are prime divisors. The set of divisors form a group, which will
be denoted by DivpXq.
We say that a divisor D is effective if each mi ě 0, and we write D ě 0. The support of
D is defined to be the closed subset SupppDq “ Z1 Y ¨ ¨ ¨ Y Zr .
Example 17.1. If X “ Spec A is affine, then every prime Àdivisor is of the form V ppq where
p is a prime ideal of height 1. Hence DivpSpec Aq “ htp“1 Z ¨ V ppq can be identified
with the free abelian group on all height 1 prime ideals. △
Example 17.2. On X “ P1C , the prime divisors are simply the closed points. Here are some
examples of Weil divisors:
D1 “ 3 ¨ p1 : 0q ´ 5 ¨ p0 : 1q, D2 “ p1 : 1q ` 5 ¨ p0 : 1q
D1 ` D2 “ 3 ¨ p1 : 0q ` p1 : 1q.
△
The reader may wonder why we include the assumption that X is normal. Certainly,
the above definition can be made for any scheme, but the concept of a Weil divisor is not
particularly useful without the normality assumption. The main reason is that there is a
well-behaved notion of ’order of vanishing’, or ‘multiplicity’ for rational functions.
More precisely, as the scheme X is normal, all the local rings OX,x are integrally closed
in their fraction field, KpXq. This means that if Z Ă X is a prime divisor, and ζ P X is
the generic point of Z , then the local ring OX,ζ is a 1-dimensional normal ring, and hence a
discrete valuation ring (see Appendix ?). We write
ordZ : KpXq ÝÝÑ Z Y t8u
for the corresponding valuation. More explicitly, the function ordZ can be defined as follows.
Given a nonzero f P OX,ζ , we can write it as f “ u ¨ tm , where t is the generator for the
maximal ideal of OX,ζ and u is a unit, and define the valuation of f at Z to be ordZ pf q “ m.
Finally, we extend this to elements of KpXq by setting ordZ pf {gq “ ordZ pf q ´ ordZ pgq

333

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and ordZ p0q “ 8. The number ordZ pf q is called the order of vanishing of f at Z . If
ordZ pf q is non-negative we say that f vanishes to that order along Z . If ordZ pf q is negative,
then we say that f has a pole of that order along Z .
If X “ Spec A is an affine scheme, then all prime divisors are of the form V ppq where
p is a prime ideal of height 1. For such a prime p, the local ring Ap is a discrete valuation
ring, consisting of exactly the fractions f “ a{b P KpAq where Ş b R p, or equivalently,
ordV ppq pf q ě 0. In light of Theorem 17.3, which says that A “ ht p“1 Ap , we conclude
that ordV ppq pf q ě 0 for every prime divisor V ppq if and only if f P A. Moreover, f P KpAq
has zero order of vanishing along every prime divisor if and only if f is invertible in Aˆ .
In general, for an integral scheme X , a rational function f P KpXq is regular if and only
if f P OX,x for every x P X , which by the affine case above means that ordZ pf q ě 0 for
every prime divisor Z . Therefore, we have:

Proposition 17.3. For a rational function f P KpXqˆ , we have

(i) ordZ pf q ě 0 for all prime divisors Z Ă X if and only if f P OX pXq.
(ii) ordZ pf q “ 0 for all prime divisors Z if and only if f P OX pXqˆ .

This is illustrates one of the close links between rational functions and codimension 1
subschemes on normal schemes. If a rational function is regular outside a closed subset of
codimension at least 2, then it is regular everywhere.
Using the order of vanishing functions, we can define the divisor of a rational function:

Definition 17.4 (Principal divisors). For a rational function f P KpXqˆ , we define its
corresponding divisor by
ÿ
divpf q “ ordZ pf qZ, (17.1)
Z

where the sum runs over all prime divisors. Divisors of the form divpf q are called
principal divisors, and they form a subgroup of DivpXq.

To ensure that this is well-defined, we need to verify that for a nonzero rational function
f P KpXq, there are only finitely many prime divisors Z Ă X such that ordZ pf q ‰ 0. To
see this, let U “ Spec A be any open affine subset such that f |U P OX pU q. For a prime
divisor Z Ă X , there are two possible cases: (i) Z Ă X ´ U , or (ii) Z X U is a prime
divisor of U . As we assume X is Noetherian and integral, there can be only finitely many
codimension 1 components of the closed subset X ´ U , so there are only finitely many Z ’s in
case (i). For the ones in case (ii), note that ordZ pf q ě 0 automatically, because f is regular
in U , and ordZ pf q ą 0 if and only if f vanishes along Z . Again because X is Noetherian,
the zero set V pf q has only finitely many components, so we conclude.
Example 17.5. On Spec Z, a Weil divisor is an expression of the form
D “ n1 V pp1 q ` ¨ ¨ ¨ ` n1 V ppr q
where the pi are prime numbers. In the function field, Q, the ‘rational function‘ f “
pn1 1 ¨ ¨ ¨ pnr r satisfies divpf q “ D. Hence every divisor is principal on Spec Z. △

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Example 17.6. Let k be an algebraically closed field, and consider the affine line A1k “
Spec krts which has fraction field kptq. Then prime divisors in A1k correspond to k -points
a P A1k associated to maximal ideals pt ´ aq in krts. Consider the rational function
f “ t2 pt ´ 1qpt ` 1q´1 P kptq.
Note that t, pt ´ 1q and pt ` 1q are all invertible in the local ring OA1k ,a “ krtspt´aq
except when a “ 0, ˘1. When a “ 0, then t is a local parameter in OA1k ,a “ krtsptq ,
and we can write f “ t2 punitq. Hence the order of vanishing of f is equal to 2 at the
point a “ 0. Similarly, we find that the non-zero orders of vanishing are ordt“0 pf q “ 2,
ordt“1 pf q “ 1, ordt“´1 pf q “ ´1 and so
divpf q “ 2V ptq ` V pt ´ 1q ´ V pt ` 1q.
More generally, if f ptq “ pt ´ a1 qn1 ¨ ¨ ¨ pt ´ ar qnr , then
divpf q “ n1 V pt ´ a1 q ` ¨ ¨ ¨ ` nr V pt ´ ar q. (17.2)
This means that every divisor D is the divisor of some nonzero rational function. △
Example 17.7. Consider the projective line X “ P1k “ Proj krx0 , x1 s, whose function
field is kptq where t “ x1 {x0 . Consider the rational function f “ t2 pt ´ 1q´1 P kptq. To
compute the divisor of f , we treat the two affine charts D` px0 q and D` px1 q separately:
On U “ D` px0 q “ Spec krts, note that t2 pt ´ 1q´1 can only have nonzero valuation at
t “ 0 or t “ 1. Using the formula (17.2), we have
divpf q|D` px0 q “ 2p1 : 0q ´ p1 : 1q.
In the open chart U “ D` px1 q “ Spec krus, where u “ x0 {x1 “ t´1 , we may write
f “ u´2 pu´1 ´ 1q “ pu ´ u2 q´1 . The only non-zero valuations are: ordu“0 “ ´1 and
ordu“1 “ ´1. Note that the point u “ 1, P D` px1 q is the point p1 : 1q which we found
also in D` px0 q above. It follows that the divisor of f is given by
divpf q “ 2p1 : 0q ´ p1 : 0q ´ p1 : 1q.
△
Example 17.8. One may also consider the function f from Example ?? as a rational function
on P1k . As we have already computed the orders of vanishing for every point in Dpx0 q, we
need only consider the remaining point at infinity, p0 : 1q. In OP1k ,p0:1q , the element s “ t´1
is a local parameter, and expressed in terms s, the function f becomes
f “ s´2 ps´1 ´ 1qps´1 ` 1q´1 “ s´2 p1 ´ sqp1 ` sq´1 .
This has order of vanishing ´2 at s “ 0, and hence
divpf q “ 2p1 : 0q ` p1 : 1q ´ p´1 : 1q ´ 2p0 : 1q.
△
Example 17.9. Let X be the curve V px3 ´y 3 `yq Ă A2k . Then x, y and y{x2 define rational
functions on X . Here x and y are regular, so they have nonnegative orders everywhere. Let
us find the points p P X where ordp pxq ą 0.

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The function x vanishes exactly at the points in V px, x3 ´ y 3 ` yq Ă A2 , i.e., the points
p0, 0q, p0, 1q, p0, ´1q. The local ring at the origin p0, 0q is isomorphic to
` ˘
OX,p0,0q “ krx, ys{px3 ´ y 3 ` yq px,yq

In this ring we have x3 ´ ypy 2 ´ 1q “ 0, and so y “ x3 punitq. Hence x is the uniformizing

parameter. In particular, ordp0,0q x “ 1. Similar computations show that
divpxq “ p0, 0q ` p0, ´1q ` p0, 1q
As for y , this can only have non-zero orders of vanishing at the points in V px3 ´ y 3 , yq “
V px, yq, i.e., at the origin p0, 0q. We just computed that y “ x3 punitq here, so
divpyq “ 3p0, 0q
From this we get that
divpy{x2 q “ 3p0, 0q ´ 2 pp0, 0q ` p0, ´1q ` p0, 1qq “ p0, 0q ´ 2p0, ´1q ´ 2p0, 1q.
△
Example 17.10. For non-Noetherian schemes X one quickly runs into difficulties when
defining the divisor of a rational function. For instance, imitating the construction of the
affine line with two origin, one can construct the affine line X with infinitely many origins:
this scheme is integral, normal, with fraction field kptq, but there are infinitely many closed
points p P X for which ordp ptq “ 1. △

Restrictions of divisors
If U Ă X is an open set and Z Ă X is a prime divisor, the intersection Z X U is either
empty (if Z Ă
řX ´ U ) or a prime divisor on U . This allows us to define the restriction of a
divisor D “ nZ Z to U by the formula
ÿ
D|U “ nZ ¨ Z X U. (17.3)
ZXU ‰H

This defines a map of groups DivpXq Ñ DivpU q. Note that any divisor with support in
X ´ U is sent to 0 via this map.
If f is a rational function on X , the restriction f |U is a rational function on U , and it
holds that ordZXU pf |Z q “ ordZ pf q (the two valuation rings are equal), and consequently
the divisor divpf q restricts to the divisor divpf |U q. Therefore, the restriction map sends
principal divisors to principal divisors.

17.2 Cartier divisors

If f P OX pXq is a nonzero regular function, then divpf q is an effective divisor with support
along the zero set of f , and ‘multiplicity’ ordZ pf q along each component. Generally, we
expect that a codimension 1 subscheme can be defined by a single equation, at least locally.
As we will see this intuition is often correct, at least when X has mild singularities.

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can be described entirely using the collection of rational functions fi , rather than the
underlying prime divisors and their multiplicities.

Definition 17.11. We say that a divisor D is locally principal or a Cartier divisor if there
is an open covering tUi uiPI of X and for each i P I , a rational function fi such that
D|Ui “ divpfi q

If a divisor D is Cartier, then it is completely determined by the collection of rational

functions fi . More precisely, a Cartier divisor D can be specified using its Cartier data,
which consists of:
(i) an open covering tUi uiPI of X ,
(ii) elements fi P K satisfying fi fj´1 P OX ˆ
pUi X Uj q for every i, j P I .
Any set of Cartier data pUi , fi q determines a unique divisor D which is Cartier. Indeed, the
second item reflects the property that fi and fj define the same divisor over Ui X Uj (the
restriction of D). If we restrict fi and fj to Ui X Uj , they have the same orders of vanishing
at every prime divisor which intersects both Ui and Uj . This means that for any prime divisor
Z Ă X , the coefficient nZ of Z in D is determined by nZ “ ordZ pfi q where i is any index
such that Z X Ui ‰ H.
In fact, the second item is required: if ordZ pfi q “ ordZ pfj q for every Z intersecting
Ui X Uj , then by Proposition 17.3, we see that
fj “ cji ¨ fi (17.4)
ˆ
for some units cji P OX pUi X Uj q.
Specifying a Cartier divisor in terms of the fi is often convenient in computations, as one
can directly work with the local equations, locally. We will see examples of this in Section
17.9.
The Cartier data is however not unique, as D may be locally principal both with respect to
several open coverings and different choices of rational functions. The ambiguity is resolved
if we consider two defining Cartier data tpUi , fi quiPI and tpVj , gj qujPJ to be equivalent if
fi gj´1 P ΓpUi X Vj , OX ˆ
q for all i and j .
The set of Cartier divisors forms a subgroup of DivpXq. In terms of Cartier data, the
identity element is represented by pX, 1q, that is, the trivial covering consisting of the open
set X and the rational function ‘1’. If D and E are represented by the data pUi , fi q and
pVj , gj q, then D ` E is represented by pUi X Vj , fi gj q and ´D is represented by pUi , fi´1 q.
Furthermore, pUi , fi q represents a principal divisor if and only if it is equivalent to pX, f q
for some f P KpXqˆ .

Example 17.12. Consider the projective n-space Pnk over a field k . Write Pnk “ Proj R
where R “ krx0 , . . . , xn s. Any homogeneous polynomial of degree d, F px0 , . . . , xn q P Rd
defines a closed subscheme of Pnk of codimension 1. The corresponding Weil divisor D is
Cartier. Concretely, we can write down the Cartier data with respect to the standard covering
Ui “ D` pxi q of Pnk . Note that F px{xi q “ F p xx0i , . . . , xxi´1
i
, 1, xxi`1
i
, . . . , xxni q defines a
non-zero regular function on Ui , and the collection
pUi , F px{xi qq

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forms a Cartier divisor D on X . Indeed, on the overlap Ui X Uj , we have the relation

d
F px{xi q “ pxj {xi q F px{xj q (17.5)

and xj {xi is a regular and invertible function on Ui X Uj . Two homogeneous polynomials

F, G of the same degree d give linearly equivalent divisors, because the quotient F pxq{Gpxq
is a global rational function on Pnk . the coefficient nZ of Z in D is determined by nZ “
ordZ pfi q where i is any index such that Z X Ui ‰ H. △

Cartier divisors have better formal properties than general Weil divisors because of their
close links to invertible sheaves. We will explore this in Section 17.4.
It is not true in general that every divisor is Cartier. We will see some basic counterexamples
in Section 17.9. However, if X is locally factorial, meaning that all the local rings OX,x are
UFD ’s, then the two notions are the same:

Theorem 17.13. Let X be a Noetherian normal scheme such that every local ring OX,x
is a UFD. Then every Weil divisor is Cartier.

Proof ‘Being Cartier’ is a local condition, so we may assume X “ Spec A is affine.

Let Z “ V ppq be a prime divisor on X , where p is a height 1 prime ideal. Note that Z
is automatically principal over the open set U “ X ´ Z , because Z|U “ 0. If x P Z , then
x corresponds to a prime ideal q containing p. As p has height 1 in A, then pAq remains of
height 1 (there is a one-to-one correspondence between primes in A lying in q and primes in
Aq ). As Aq “ OX,x is assumed to be a UFD, pAq is a principal ideal, by Proposition A.58.
Let a, b P A be elements such that a{b P Aq is a generator for pAq . As b R q, we see that in
fact p “ Aq “ paqAq .
If p “ pf1 , . . . , fr q, then fi Aq P paqAq for each i “ 1, . . . , r, so we may write fi {1 “
ai {bi ¨ a. Setting b “ b1 ¨ ¨ ¨ br , we have pAb “ paqAb . Therefore, U “ Dpbq Ă X is an
open neighbourhood of x such that Y X U is defined by a principal ideal in OX pU q. In other
words, Y |U “ V ppq|U “ divpaq is principal over U .

Corollary 17.14. Let X be a Noetherian, integral nonsingular scheme. Then every Weil
divisor is Cartier.

Proof If X is nonsingular, then each local ring OX,x is a regular local ring, hence a UFD
by the Auslander–Buchsbaum theorem (see Theorem 11.5).

17.3 The class group

One of the fundamental invariants of a scheme (or of a ring) is the class group, along with its
close relative, the Picard group. The class group can be interpreted as the group that captures
the obstructions to a divisor being the divisor of a rational function.

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Definition 17.15. The class group of X is defined as the group of Weil divisors modulo
the principal divisors, i.e.,
ClpXq “ DivpXq{xdivpf q | f P KpXqˆ y.
Two Weil divisors D and D1 are said to be linearly equivalent (written D „ D1 ) if they
have the same image in ClpXq, or equivalently, that D ´ D1 is principal.

For an affine scheme X “ Spec A, the divisor of a rational function f P KpXq is equal to
ÿ
divpf q “ ordp pf qV ppq.
htp“1

Hence divpf q “ 0 if and only if ordp pf q “ 0 for all height 1 primes, which by Proposi-
tion 17.3 happens if and only if f P Aˆ . Hence the kernel of the map div : KpXqˆ Ñ
DivpSpec Aq equals Aˆ , and the cokernel is by definition the class group ClpSpec Aq.
Hence we have the exact sequence
div
0 Ñ Aˆ Ñ KpXqˆ ÝÝÑ DivpSpec Aq Ñ ClpSpec Aq Ñ 0. (17.6)
Example 17.16. It follows from Example 17.5 that ClpSpec Zq “ 0. △
Example 17.17. The class group of Spec Zris is also trivial. This follows by the analysis of
Example 2.41, which showed that every prime ideal of Zris is principal. △
Example 17.18. On Ank “ Spec krx1 , . . . , xn s, any prime ideal of height 1 is a principal
ideal, so any prime divisor is of the form D “ divpf q, where f P krx1 , . . . , xn s is an
irreducible polynomial. It follows that the class group of Ank is trivial. △
Example 17.19. Let A be a discrete valuation ring and let X “ Spec A. In A, the only
nonzero prime ideal is the maximal ideal m. Therefore, if x P X denotes the closed point,
we have DivpXq “ Z ¨ x. Any Weil divisor on X is principal: if t is a generator for m, then,
then divptn q “ n ¨ x for each n P Z. Hence ClpXq “ 0. △

Projective space
Write Pnk “ Proj R, with R “ krx0 , . . . , xn s. Prime divisors on Pnk are defined by homoge-
neous height 1 prime ideals in R, that is, ideals p “ pGq where G is a nonzero homogeneous
irreducible polynomial. The generator G is unique up to a scalar, so its degree is well-
defined. We can use this to define the degree of ařdivisor, by taking the sum of degrees of the
corresponding polynomials. Explicitly, if D “ i ni V pGi q, we define
ÿ
deg D “ ni deg Gi .
i

On Pnk ,
any rational function f is the quotient of two homogeneous polynomials of the
ś same
degree. By factoring the numerator and the denominator, we can write f as f “ i Gni i
ř Gi are different irreducible homogeneous polynomials in R and the integers ni
where the
satisfy i ni pdeg Gi q “ 0, because f is homogeneous of degree zero.

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ř
Lemma 17.20. divpf q “ ni V pGi q.

Proof Let Z Ă Pnk be a prime divisor with generic point ζ P Z . Since Z has codimension 1,
it holds that Z “ V pP q for some irreducible polynomial P of some degree d. For any other
polynomial Q of degree d, the quotient P {Q defines a rational function which generates
maximal ideal mξ OX,ζ . We can write f “ pP {Qqr u where u is a unit in OZ,ζ . Then
r “ ni if Gi divides P (and r “ 0 if no Gi divides P ), and u is a rational function which
does not involve
ř P in its numerator or denominator. It follows that ordZ pf q “ ni and so
divpf q “ ni V pGi q.
ř
Now, as deg divpf q “ ni deg Gi “ 0, the degree map descends to a map of groups

deg : ClpPn q ÝÝÑ Z

We claim that it is an isomorphism:

Proposition 17.21. The degree map gives an isomorphism ClpPnk q » Z.

Proof The degree map is clearly surjective becauseřthe degree of any hyperplane, for
instance V px0ř
q, is equal to 1. For injectivity: if Z “ ni V pGś i q lies in the kernel of deg,
we must have ni deg Gi “ 0. Consequently, the product f “ i Gni i is homogeneous of
degree zero and defines a rational function on Pnk . By the lemma above, we have Z “ divpf q,
and hence Z is a principal divisor.

The class group and unique factorization

Both the terms ‘divisor’ and ‘class group’ originate from algebraic number theory and their
origins can be traced back to Kummer’s work on Fermat’s Last Theorem. In this setting,
the class groups are designed to measure how far the base ring is from being a unique
factorization domain.

Proposition 17.22. Let A be a normal Noetherian integral domain. Then the following
are equivalent:
(i) ClpSpec Aq “ 0.
(ii) Every height 1 prime ideal in A is a principal ideal.
(iii) A is a unique factorization domain.

Proof The equivalence of (ii) and (iii) is a fact from commutative algebra (see Proposition
A.58). It remains to show the equivalence (i) ô (ii).
(ii) ñ (i): Write X “ Spec A. If Z is a prime divisor in X , then Z “ V ppq for some
prime ideal p Ă A, and as Z has codimension 1, p is of height 1. By assumption (ii), p “ pf q
for an element f P A, that is, Z “ divpf q. This shows that ClpSpec Aq “ 0.
(i) ñ (ii): Assume that ClpSpec Aq “ 0. Let p be a prime of height 1, and let Z “
V ppq Ă X . By assumption, there is an f P KpXqˆ so that divpf q “ Z . We want to show
that in fact f P A and that p “ pf q. For the first statement: as divpf q “ Z , the valuations of

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f are given by ordq pf q “ 0 for q ‰ p and ordp pf q “ 1, and so f lies in OX pXq “ A by

Proposition 17.3.
Secondly, to prove that f generates p, consider any nonzero element g P p. Then
ordp pgq ě 1 and ordq pgq ě 0 for all q ‰ p. It follows that ordq pg{f q “ ordq pgq ´
ordq pf q ě 0 for every prime ideal q P Spec A. Hence g{f P Aq for every prime q of height
1, and hence g{f P A, by Proposition 17.3. It follows that g P f A, and so p “ pf qA is
principal.

In particular, since A “ krx1 , . . . , xn s is a unique factorization domain, we get a new

proof that the class group of affine space Ank is trivial.

An example from number theory

?
The ring A “ Zr ´5s “ Zrxs{px2 ` 5q has already appeared a few places throughout the
book. As we have seen, A is not a UFD, so ClpSpec Aq ‰ 0. Now we claim the following:

Proposition 17.23. The class group of A is given by ClpSpec Aq “ Z{2, and it is

generated by the class of the prime divisor Y “ V p2, 1 ` xq.

Proof First, note that p2, 1 ` xq is prime, because

A{p2, 1 ` xq » Zrxs{p2, 1 ` xq » Z{2.

Let p be a height 1 prime ideal in A. We will show that V ppq is either principal or equivalent
to Y in ClpSpec Aq.
The idea is to study the morphism π : Spec A Ñ Spec Z induced by the integral extension
Z Ă A. Note that π is surjective (by the Going-Up Theorem), and p lies over a nonzero
prime ppq P Spec Z. There are three cases to consider:
(i) ppq remains prime in A.
(ii) ppq is a square of a prime ideal in A.
(iii) ppq decomposes as

ppq “ pp, x ´ aq X pp, x ` aq (17.7)

where a P Z is an integer such that a2 ` 5 ” 0 pmod pq.

In case (i), pA is prime in A, and so V ppq “ V ppAq is principal. The case (ii) happens if
and only if the minimal polynomial x2 ` 5 “ 0 has a multiple root modulo p. This happens
only for the primes p “ 2 and p “ 5. For p “ 2, we have p2q “ p1 ` xq2 . Therefore,
divp2q “ 2V p2, 1 ` xq “ 2Y , and hence
2Y “ 0 in ClpSpec Aq.
For p “ 5, we have p5q “ pxq2 , and so V ppq “ V p5, xq “ V pxq, which is principal.
We therefore reduce to the case (iii). By adjusting a by a multiple of p, we may assume
0 ď a ă p. From (17.7),
divppq “ V pp, x ` aq ` V pp, x ´ aq, (17.8)

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Which shows that V pp, x ` aq “ ´V pp, x ´ aq in ClpSpec Aq. Hence it suffices to analyze
V pp, x ` aq.
Consider more generally ideals of the form pn, x ` aq where n ě 2 and 0 ď a ă n and
a2 ` 5 ” 0 pmod nq.
If n “ 2, we recover Y “ V p2, 1 ` xq.
If n “ 3, then we must have a “ 1. As p1 ` x, x2 ` 5q “ p1 ` x, 6q, we get
divp1 ` xq “ V p3, 1 ` xq ` V p2, 1 ` xq.
Hence V p3, 1 ` xq “ ´Y in ClpSpec Aq.
If n “ 4, no solutions exist, as a2 ` 1 ” 0 pmod 4q has no integer solutions.
If n “ 5, then a “ 0, and p5, xq “ pxq, which is principal.
If n ě 6, write a2 ` 5 “ bn. Then we get
divpx ` aq “ V px ` a, x2 ` 5q
“ V pa ` x, bnq “ V pa ` x, bq ` V pa ` x, nq
But as a2 ` 5 “ bn and n ě 6, we must have b ă n. Therefore, by induction on n, the
class of V pb, a ` xq can be written as a multiple of Y in the class group. This completes the
proof.
?
More generally, consider the quadratic number field K “ Qp dq, where d is a square-free
integer. The ring of integers OK , that is, the integral closure of Z in K , is a normal integral
domain of dimension 1 (see Exercise 11.6.8 for more details). When d ă 0, it is known that
the class group ClpOK q is trivial if and only if
d P t´1, ´2, ´3, ´7, ´11, ´19, ´43, ´67, ´163u.
It is currently unknown whether there exist infinitely many positive values of d for which
ClpOK q ‰ 0.

17.4 The sheaf associated to a Weil divisor

ř
Let D “ nZ Z be a Weil divisor on a Noetherian, normal scheme X . We will construct a
sheaf, denoted by OX pDq, whose sections are given by the rational functions with ‘poles
at worst along D’. There are several ways to express this. The simplest way is to require
that a rational function f satisfies ordZ pf q ě ´nZ for all Z , meaning that the order of the
pole of f along Z is at most nZ . Another way to express this is to say that divpf q ` D is an
effective Weil divisor; in other words, divpf q ` D ě 0. Informally, this means that the ‘pole
part’ of divpf q is canceled out by D.

Definition 17.24. Let X be a Noetherian integral normal scheme and let D be a Weil
divisor on X . We define the sheaf OX pDq by letting
OX pDqpU q “ t f P KpXq | ordZ pf q ě ´nZ for all Z with Z X U ‰ H u
for each open subset U Ă X .

The condition in the bracket is less restrictive when applied to a smaller subset U 1 Ă U , so

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there is an inclusion OX pDqpU q Ă OX pDqpU 1 q. Hence OX pDq becomes a presheaf with

inclusion maps as the restriction maps. It is not hard to see that the sheaf axioms are satisfied,
so OX pDq is a sheaf. As such, it is a subsheaf of the constant sheaf KpXq on the function
field KpXq. It is moreover an OX -module, because if a P OX pU q is a regular function U ,
we have ordY paf q “ ordY paq ` ordY pf q ě ´nZ for all Z and all f P OX pDqpU q, so
OX pDq is stable under multiplication by elements of OX .

Proposition 17.25. Let X be Noetherian integral normal scheme and let D be a Weil
divisor. Then OX pDq is a coherent sheaf.

Proof We may assume that X “ Spec A is affine. We claim that for any g P A, there is an
equality of subgroups of KpXq

ΓpX, OX pDqqg “ ΓpDpgq, OX pDqq.

Given this, it follows that OX pDq is the tilde of the A-module M “ ΓpX, OX pDqq on U ,
and so it is quasi-coherent.
In any case, we have the containment ‘Ă’, because g is invertible over Dpgq and so
ordZ ph{g m q “ ordZ phq for any h P ΓpX, OX pDqq and any prime divisor Z which
intersects Dpgq.
For ‘Ą’, take any f P ΓpDpgq, OX pDqq such that divpf q ` D ě 0 on Dpgq. This
implies that divpf q ` D ě 0 can only fail over V pgq Ă Spec A. However, as V pgq has
only finitely many components, there exists an m ą 0 such that

divpg m f q ` D ě 0
over Spec A. Then g m f P ΓpX, OX pDqq and f is the image of pg m f q{g m . This shows that
OX pDq is quasi-coherent.
To prove that OX pDq is of finite type, itřsuffices to show that ΓpX, OX pDqq is a finitely
generated A-module. For this, write D “ ni Zi , and pick a nonzero element g P A which
vanishes on all the Zi . This means that we may choose an m so that ordZi pg m q is greater
than all the ni , which implies that ordZi pg m f q ě 0 for any f P ΓpX, OX pDqq. Hence
g m ΓpX, OX pDqq is contained in ΓpX, OX q “ A, and being an A-module, it must be an
ideal of A. As we assume that A is Noetherian, this ideal is finitely generated, and we deduce
that ΓpX, OX pDqq is finitely generated as well.

Example 17.26. If D “ 0, then OX pDq “ OX . Indeed, if U is any open in X , then

ΓpU, OX pDqq consists of the rational functions g P KpXq such that ordZ pgq ě 0 for
every prime divisor Z . Any such must be regular, by Proposition 17.3, so ΓpU, OX pDqq “
ΓpU, OX q. △
Example 17.27. If Z is a prime divisor, then

OX p´Zq » IZ
is the ideal sheaf defining Z . Indeed, over an open set U , ΓpU, OX pDqpU qq consists of the
rational functions g P KpXq so that ordZ pgq ě 1 and ordZ 1 pgq ě 0 for every prime divisor
Z 1 ‰ Z . In other words, g P OX pU q is regular, and vanishes along Z . △

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Example 17.28. Let X be the projective line P1k “ Proj krx0 , x1 s over k and consider the
divisor D “ V px1 q “ p1 : 0q. We have the standard covering of P1k by the distinguished
open sets U0 “ Spec krx1 {x0 s “ Spec krts and U1 “ Spec krx0 {x1 s “ Spec krss (so
s “ t´1 on U0 X U1 ). Let us find the global sections of OX pDq.
Note that the point p1 : 0q does not lie in U1 “ D` px1 q, and this means that a rational
function f P KpXq such that divpf q ` D is effective on U1 , must be regular on U1 ; that is
ΓpU1 , OX pDqq “ krss.
Over the open set U0 , we are looking at elements f P kptq having order of vanishing at least
´1 at t “ 0. This implies that
ΓpU0 , OX pDqq “ t αt´1 ` pptq | α P k, pptq P krts u.
Now, by the usual sheaf sequence, we may think of the elements in ΓpX, OX pDqq as pairs
pf, gq with f and g sections of OX pDq over U0 and U1 respectively, so that f “ g on
U0 X U1 . Here g “ gpsq is a polynomial in s, and
f ptq “ pptq ` αt´1 “ pps´1 q ` αs.
If f “ g in krt, t´1 s, it is clear that p must be a constant. This implies that
ΓpX, OX pDqq “ k ‘ k t´1 .
In fact, we will see in a bit that OX pDq » OP1 p1q. △

Lemma 17.29. If two Weil divisors D and E are linearly equivalent then OX pDq »
OX pEq as OX -modules.

Proof Write D “ E ` divphq for a rational function h P KpXq. Let U Ă X be an

open set. Then f P ΓpU, OX pDqq if and only if divpf q ` D ě 0, which is equivalent to
divpf hq ` E ě 0, i.e., f h P ΓpU, OX pEqq. It follows that multiplication by h induces an
isomorphism of OX pU q-modules
ΓpU, OX pDqq ÝÝÑ ΓpU, OX pEqq.
This is compatible with the restriction maps, so we get that OX pDq » OX pEq.

Lemma 17.30. Let D and E be two Weil divisors on X . Then OX pDq “ OX pEq as
subsheaves of KpXq if and only if D “ E .
ř ř
Proof Write D “ mZ Z and E “ nZ Z . Fix a prime divor Z Ă X . We need to prove
that mZ “ nZ . Let us choose an affine open U “ Spec A Ă X containing the generic point
ζ of Z . Assuming that OX pDq “ OX pEq, we get that OX pDqζ “ OX pEqζ . However,
OX pDqζ “ t f P KpXq | ordV ppq pf q ě ´mZ u (17.9)
OX pEqζ “ t f P KpXq | ordV ppq pf q ě ´nZ u. (17.10)
These cannot be equal unless mZ “ nZ . Indeed, let t P Ap be a generator for the maximal
ideal pAp . If mZ ą nZ , then t´mZ lies in (17.9), but not in (17.10). A similar argument
takes care of the case mZ ă nZ .

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Lemma 17.31. For a nonzero rational function h P KpXq, we have

1
OX pdivphqq “
¨ OX (17.11)
h
Moreover, a Weil divisor D is principal if and only if OX pDq » OX .

Proof Let U Ă X be an open set. We claim that

1
ΓpU, OX pdiv hqq “¨ ΓpU, OX q (17.12)
h
as subgroups of KpXq. Note that a rational function f P KpXq belongs to ΓpU, OX pDqq if
and only if divpf q ` divphq ě 0, or equivalently, divpf hq ě 0. In other words, r “ f h is
a rational function which has non-negative order of vanishing at every prime divisor Z which
intersects U . Therefore r P OV pU q is regular, by Proposition 17.3, and g “ r{f belongs to
the right-hand side. The opposite containment is clear.
For the last statement: If D “ divphq is principal, then (17.12) shows that multiplcation
by h induces an isomorphism of OX pU q-modules
ΓpU, OX pDqq ÝÝÑ ΓpU, OX q
This is compatible with the restriction maps, and so OX pDq » OX .
Conversely, suppose that OX pDq » OX . As OX pDq is an OX -submodule of KpXq
which is free of rank 1, there is a g P KpXq so that OX pDq “ g ¨ OX . Define h “ 1{g .
Then
1
OX pdivphqq “ OX “ g ¨ OX “ OX pDq
h
as subsheaves of KpXq, and hence D “ divphq by Lemma 17.30.
When the divisor D is Cartier, say given by the data pUi , fi q, then the sheaf OX pDq can
be described as the subsheaf of KpXq given by
ΓpV, OX pDqq “ th P KpXq | fi h P ΓpUi X V, OX q @ i P Iu
In this, case the sheaf is invertible:

Proposition 17.32. Let X be a Noetherian integral normal scheme and let D be a Weil
divisor on X .
(i) D is Cartier if and only if OX pDq is an invertible sheaf.
(ii) If D is given by Cartier data tpUi , fi qu, then
1
OX pDq|Ui “ ¨ OUi for each i P I
fi

Proof First of all, if OX pDq is locally free, then it must have rank 1. This is because over
an open set V Ă X ´ SupppDq, the group OX pDqpV q consists of the rational functions
such that ordZ pf q ě 0, for every prime divisor Z that intersects V , and hence f P OX pV q.
This means that OX pDq is isomorphic to OX over an open set, so it has rank 1.
Suppose first that D is Cartier, say given by D|Ui “ divpfi q for rational functions fi

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and an open cover tUi u. Over each Ui , Lemma 17.31 shows that OX pDq|Ui » OX |Ui , so
OX pDq is invertible.
Conversely, if OX pDq is an invertible subsheaf of KpXq, we can define fi P KpXq by
choosing local generators so that OX pDqpUi q “ f1i OX pUi q Ă KpXq. Then it follows that
D|Ui “ divpfi q by Lemma 17.31.
Note that two different Cartier data pUi , fi q and pVj , gj q for the same divisor D give rise
to the same invertible sheaf. This is because over Ui X Vj , we have fi “ uij gj for some
units uij P OX pUi X Vj qˆ . This means that fi´1 OUi XVj “ gi´1 OUi XVj , and so the sheaf is
uniquely determined as a subsheaf of KpXq. Note that the units uij “ fi fj´1 automatically
satisfy the cocylce condition, because fk fj´1 ¨ fj fi´1 “ fk fi´1 . In the notation of Section
15.1, OX pDq is the invertible sheaf determined by the isomorphisms τij : OUij Ñ OUij
given by multiplication by uij .

Proposition 17.33. Let X be a Noetherian integral normal scheme and let D and E be
two Cartier divisors. Then:
(i) OX pD ` Eq » OX pDq bOX OX pEq
(ii) OX pDq » OX pEq if and only if D and E are linearly equivalent.

Proof We may pick a common affine covering Ui so that both D and E are both represented
by data pUi , fi q, pUi , gi q. Then D ` E is determined by the Cartier data pUi , fi gi q. Locally,
over Ui the sheaf OX pD ` Eq is defined as the subsheaf of KpXq given by pfi gi q´1 OUi “
fi´1 gi´1 OUi . As Ui is affine, the tensor product is locally given by fi´1 OUi b gi ´1 OUi ,
which is clearly isomorphic to fi´1 gi ´1 OUi via the map afi´1 b bgi ´1 ÞÑ abfi ´1 gi ´1 .
For the second claim, it suffices (i) to show that OX pDq » OX if and only if D is a
principal Cartier divisor. But this is a consequence of Lemma 17.31.

Example 17.34. Consider again the example of projective space Pnk . In the notation of
Example 17.12, if F is a homogeneous polynomial of degree d on Pnk , the sheaf OPnk pDq
determined by pD` pxi q, F px{xi qq is isomorphic to OPnk pdq. This follows from (17.5), which
shows that OX pDq has exactly the same gluing functions as OPnk pdq. In particular, this shows
that two divisors of different degrees are not linearly equivalent. This, together with the fact
that any Weil divisor is Cartier, gives a new proof of the equality ClpPnk q “ Z. △
Example 17.35. Let A be a discrete valuation ring with local parameter t, and let X “
Spec A. If x “ ptq P X denotes the closed point, we have OX pnxq “ t´n ¨ OX . If
f P KpXq, then divpf q “ ordx f ¨ x. Moreover,
OX pdivpf qq “ t´ ordx f OX .
△

17.5 The divisor associated to a section of an invertible sheaf

A prototype example of a Cartier divisor is the divisor associated to a section of an invertible
sheaf. The construction pararells the definition of a principal divisor.

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Let L be an invertible sheaf on X and let s be a nonzero global section of L. Let Ui be an

open cover of X such that L is trivial when restricted to each Ui . This means that there are
isomorphisms

ϕi : L|Ui ÝÝÑ OX |Ui .

Let fi P OX pUi q be the image of s|Ui via ϕi . As X is integral and fi is a nonzero regular
function over Ui , we may regard it as a nonzero element of the function field KpXq. Then
the collection pUi , fi q defines a Cartier divisor on X , which we denote by divpsq.
The Cartier divisor divpsq is independent of the choice of local isomorphisms ϕi : if
we are given isomorphisms ϕα : L|Uα Ñ OUα and ψβ : L|Uβ Ñ OUβ , then over each
U Ă Uα X Uβ , the rational functions ϕα psq|U and ψβ psq|U are equal up to multiplication
by an invertible element c P OX pU q, and so the orders of vanishing along prime divisors are
the same.
Note that as s is a global section, the rational functions fi are regular functions over each
Ui . This implies that the orders of vanishing ordZ pfi q are non-negative for every prime
divisor Z . In other words, the divisor divpsq is an effective Cartier divisor on X .
Geometrically, the divisor divpsq is supported on the zero scheme Y “ V psq of s. More
explicitly, multiplication by s determines a map of invertible sheaves

s_ : L_ ÝÝÑ OX

the image I is the ideal sheaf defining Y . Over each Ui , where L is trivial, the map s_ |Ui
can be identified with the map OUi Ñ OUi which sends 1 to fi . Hence I is locally generated
by the local equations fi P OX pUi q.
Note also that we have isomorphisms I “ OX p´Dq » L_ . If ι : D Ñ X denotes the
inclusion, the ideal sheaf sequence takes the form

0 ÝÝÑ OX p´Dq ÝÝÑ OX ÝÝÑ ι˚ OD ÝÝÑ 0

One of the benefits of using zero schemes of sections of invertible sheaves is that they can
be defined on any scheme. In particular, we do not really need to assume that X is normal,
integral or Noetherian to be able to work with them. (See Exercise 17.11.14.)

Example 17.36. On X “ P1k , the monomial x30 x21 defines a global section s of the invertible
sheaf L “ OP1 p5q. Over U0 “ D` px0 q, we have trivialization

5
ϕ0 : krxČ
1 {x0 sx0 ÝÝÑ krx
Č 1 {x0 s

2 2 2
given by multiplication by x´5
0 . Hence s|U0 is transported to the rational function t “ x1 {x0
on U0 , which has order of vanishing two at p1 : 0q. Similarly, the order of vanishing of s at
p0 : 1q is equal to 3. Hence the divisor of s is equal to

divpsq “ 2p1 : 0q ` 3p0 : 1q.

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Proposition 17.37. Let L be an invertible sheaf, s P ΓpX, Lq a global section. Then

there is an isomorphism
OX pdivpsqq » L. (17.13)
Two sections s, t give rise to the same divisor if and only if t “ λ ¨ s for some unit
λ P OX pXqˆ .

Proof Let tUi u and ϕi : L|Ui Ñ OUi be as above. Write D “ divpsq. When restricted to
Ui , both L and OX pDq become isomorphic to the structure sheaf OUi . We need to check that
these local isomorphisms glue to the same global sheaf on X . For this, we check what happens
over the intersections. Over Ui X Uj , the two rational functions fi “ ϕi psq and fj “ ϕj psq
are related by a relation of the form fj “ cji fi for some unit cji P OX pUi X Uj qˆ . Hence
we obtain L by gluing OUi to OUj using multiplication by cji “ fj {fi . But these are exactly
the gluing function for the sheaf OX pDq, which equals f1i OUi over Ui . Therefore, the two
sheaves are isomorphic.
For the last statement, suppose that s and t define the same ideal sheaf I of OX , so that
we have isomorphisms
s_ pt_ q´1
L_ ÝÑ I ÝÝÝÝÑ L_ .
Note by Proposition 15.17 on page 303 we have
HomOX pL_ , L_ q “ HomOX pOX , OX q “ OX pXq.
Hence every isomorphism L_ Ñ L_ is given by multiplication by some element in OX pXqˆ .
That is, s and t differ only by a unit.
Example 17.38. Let P1k “ Proj krx0 , x1 s be the projective line over a field k and let P be
the point p1 : 0q. Using the standard covering D` px0 q and D` px1 q, we see that P is the
effective Cartier divisor determined by the data pD` px0 q, x1 {x0 q and pD` px1 q, 1q. Note
that on the intersection D` px0 q X D` px1 q the function x1 {x0 is invertible, so the data yields
an effective Cartier divisor.
On the open set D` px0 q “ Spec krx1 {x0 s “ A1k , the ideal is generated by x1 {x0 which
defines the point P , and on D` px1 q the local equation is 1 which is without zeros, so the
divisor defined is exactly P .
We may also consider the data pD` px0 q, px1 {x0 qn q and pD` px1 q, 1q. In the distinguished
open set D` px0 q “ Spec krx1 {x0 s the ideal ppx1 {x0 qn q which defines a subscheme sup-
ported at P and of length n, and in D` px1 q the ideal will be the unit ideal, whose zero set is
empty. The corresponding divisor is equal to nP . △
The above constuction can in fact be carried out for a section s of L defined over any subset
V Ă X . We call such a section a rational section. Indeed, if s P LpV q, the trivializations of
L still give rational functions fi (working over the open sets Ui XV ) and we have well-defined
orders of vanishing ordZ psq for any prime divisor Z Ă X .
Unlike the previous construction, when s was a global section, the divisor divpsq of a
rational section may no longer be effective. Here is a typical example:
x30
Example 17.39. Continuing the example of X “ P1k , consider the quotient s “ x1
which

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defines a section of L “ OP1k p2q over D` px1 q, hence a rational section on X . Let us compute
the divisor associated to s: Let t “ xx01 be the coordinate on U “ D` px1 q “ Spec krts.

OX p2qpU q “ krx0 {x1 sx21 “ krtsx21 .

x3
So the rational function f “ ϕpsq is given by x03 “ t3 which has non-zero order of vanishing
1
only at the point t “ 0 P U , where we have ordt pf q “ 3. To compute divpsq, we must also
consider the point outside D` px1 q. On U “ D` px0 q, we use the coordinate u “ xx10 , and
we have
OX p2qpU q “ krusx20 .
So the rational function ϕpsq is given by f “ xx01 “ u´1 . This has order of vanishing
ordt pf q “ ´1 at t “ 0 (and ordZ pf q “ 0 at all other points). Hence we obtain
divpsq “ 3p0 : 1q ´ p1 : 0q.
△

Proposition 17.40. Let X be a Noetherian integral normal scheme.

(i) The divisor associated to a non-zero rational section s of L is a Cartier
divisor.
(ii) Any Cartier divisor is the divisor associated to a rational section s of an
invertible sheaf L.
ř
Proof The divisor divpsq is Cartier, by definition. For the second statement, if D “ nZ Z
is a Cartier divisor, then the sheaf L “ OX pDq is invertible. The element ‘1 P KpXq1 gives
a distinguished rational section s P ΓpV, Lq where V is the open set V “ X ´ SupppDq. It
is clear that divpsq “ D: over the open set Ui , the section ‘1’ is transported to the rational
function 1 ¨ fi , and the associated Cartier divisor is exactly D.

Corollary 17.41. Let X be a Noetherian integral normal scheme. Then the map D ÞÑ
OX pDq induces an isomorphism
ρ : CaDivpXq ÝÝÑ PicpXq. (17.14)

Proof By the item (i) and (ii) in Proposition 17.33, the map D ÞÑ OX pDq is additive,
and has the subgroup of principal divisors as its kernel. This means that the induced map ρ
is injective. By Proposition 17.40, any invertible sheaf L is isomorphic to one of the form
OX pdivpsqq, so ρ is also surjective.

Corollary 17.42. On a nonsingular variety X , then every Weil divisor is Cartier, and
there are natural bijections between
(i) Weil divisors (up to linear equivalence)
(ii) Cartier divisors (up to linear equivalence)
(iii) Invertible sheaves (up to isomorphism)

From our previous computation of ClpAnk q, we get the following theorem:

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Theorem 17.43. For a field k , we have

PicpAnk q “ ClpAnk q “ 0.

We previously computed that ClpPnk q “ Z, so Corollary 17.42 gives the following:

Corollary 17.44. On Pnk any invertible sheaf is isomorphic to some OPnk pmq.

17.6 Linear systems

When D is an effective Cartier divisor, the set of effective divisors D1 linearly equivalent to
D is denoted by |D|. This is called the complete linear system of D.
The name ‘linear system’ comes from the special case when X is a projective variety X
over a field k . In this case, we have OX pXqˆ “ k ˆ , and Proposition 17.37 shows that the
linear system |D| is given by
|D| “ t D1 | D1 ě 0 and D1 „ D u
“ pΓpX, OX pDqq ´ 0q {k ˆ
“ PΓpX, OX pDqq
When X is projective over k , the groups ΓpX, OX pDqq are finite dimensional as k -vector
spaces (Theorem 18.31), so the set of effective divisors D1 linearly equivalent to D is
parameterized by a projective space Pn pkq.
We define a linear system of divisors as a linear subspace of a complete linear system |D|.
Example 17.45. Let X “ P2k “ Proj krx0 , x1 , x2 s and D “ 2H , where H “ V px0 q.
Then the complete linear system |2H| is given by the set of homogeneous polynomials of
degree 2 modulo scalars, i.e.,
␣ (
|D| “ a200 x20 ` a110 x0 x1 ` ¨ ¨ ¨ ` a002 x22 {k ˆ » P5 pkq
is a projective 5-space with homogeneous coordinates a200 , a110 , a101 , a020 , a011 , a002 . The
points of this projective space parameterize the degree 2 curves in P2 .
Inside |2H|, there are smaller linear systems, e.g., the projective line
t λx20 ` µx21 | pλ : µq P P1 pkq u.
More generally, the complete linear system dH on Pnk is given by the projective space
# +
ÿ
|D| “ ai0 ,...,in xi00 ¨ ¨ ¨ xinn {k ˆ » PN pkq
i0 `¨¨¨`in “d
`n`d˘
of dimension N “ d
´ 1. △

17.7 Pullbacks of divisors

Given a morphism ϕ : X Ñ Y and a divisor D on Y , we can ask whether we can define a
Weil divisor on X supported on ϕ´1 D. In general, this is not possible. Consider for instance,

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the case where ϕ is the inclusion of a closed subscheme, say Y “ P2k and f : X Ñ Y is the
inclusion of a line X “ V px0 q. Then of course D “ V px0 q defines a Weil divisor on Y , but
there is no reasonable definition of ϕ´1 D that defines a codimension 1 subscheme of X .
There is a situation where we can always define the pullback of a divisor D. This is when
ϕ : X Ñ Y is a dominant morphism and D is a Cartier divisor. In that case, there is a
covering Ui such that D|Ui is given by divpfi q over Ui . The fact that f is dominant, means
that there is an induced map on function fields ϕ7 : KpY q Ñ KpXq. We can therefore define
a divisor ϕ˚ D by
ÿ
ϕ˚ D “ ordZ pϕ7 fi qZ
ZĂX

where Z runs over all the prime divisors in X .

As before, this is well-defined, as the rational functions ϕ7 pfi q are related by invertible
elements in OX pϕ´1 Ui X ϕ´1 Uj q.
On the level of invertible sheaves, we have
ϕ˚ OY pDq “ OX pϕ˚ Dq.
Note that there is always a pullback map for invertible sheaves, even if f is not dominant.
Furthermore, if s P ΓpY, Lq is a global section, there is also a well-defined pullback f ˚ s P
ΓpX, f ˚ Lq. However, as f ˚ s may be zero section, it may not define a divisor divpsq.
Example 17.46. Consider the curve X as in Figure 17.1, given by
X “ V py 2 z ´ x3 ´ z 3 q Ă P2 .
For a line L “ V pyq on P2k , let L|X denote the restriction of L to X (i.e., the Weil divisor
L X X on X which is of codimension 1 as X is integral). Moreover, for another line
L1 “ V pzq, the two restrictions L|X and L1 |X are linearly equivalent divisors on X , since
L|X ´ L1 |X “ divp xz |X q. This argument applies for any two lines L, L1 in P2 , so we
get many relations between divisors on X . The figure below shows one example where
L|X “ P ` Q ` R and L1 “ 2S ` T . △

17.8 The class group of an open set

Given a Noetherian, normal scheme X and an open subset U , the restriction of a prime
divisor on X is a prime divisor on U , so it is natural to ask how the two class groups are
related. The answer is given by the theorem below.

Theorem 17.47. Let X be a normal integral Noetherian scheme. Let W Ă X be a closed

subscheme and let U “ X ´ W . If Z1 , . . . , Zr are the prime divisors corresponding to
the codimension 1 components of W , there is an exact sequence
r
à
ZZi ÝÝÑ ClpXq ÝÝÑ ClpU q ÝÝÑ 0, (17.15)
i“1

where the map ClpXq Ñ ClpU q is defined by rZs ÞÑ rZ X U s.

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Figure 17.1 Two linearly equivalent divisors on a plane cubic

Proof If Z is a prime divisor on U , the closure in X is a prime divisor in X , so the

right-most map in (17.15) is surjective, and we just need to check exactness in the middle.
Suppose Z is a prime divisor which is principal on U . Then Z|U “ divpf q for some
f P KpU q “ KpXq. Now D “ divpf q is a divisor on X such that D|U “ divpf q|U .
Hence D ´Z is a Weil divisor supported in X ´U , and hence it must be a linear combination
of the Zi ’s. Therefore D ´ Z is in the image of the left-most map, and we are done.

Corollary 17.48. If Z Ă X is a closed subset of codimension at least 2, then

ClpX ´ Zq “ ClpXq

Example 17.49. If P denotes the origin in Ank , then ClpAnk ´ P q “ ClpAnk q “ 0 for n ě 2.
If n “ 1, we have ClpA1 ´ P q “ ClpSpec krx, x´1 sq “ 0 because krx, x´1 s is a UFD.
△

Example 17.50. Consider the projective line P1k over a field k , and let P be a k -point. We
have the exact sequence

Z ¨ rP s ÝÝÑ ClpP1k q ÝÝÑ ClpA1k q ÝÝÑ 0.

We saw that ClpA1k q “ 0, so the map Z Ñ ClpP1k q is surjective. It is also injective: If

rnP s “ 0 in ClpP1k q for some n, then nP “ divpf q for some f P KpP1 q. We have
nP |A1k “ 0, so we must have divpf q|A1k “ 0. Therefore, f must have neither zeros nor poles
on A1k , implying that f is a constant, and hence n “ 0.
The same proof works in higher dimensions, and gives a new proof of the formula
ClpPnk q “ Z. △

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17.9 Quadrics
The circle
Let X “ Spec A, where A “ Rru, vs{pu2 ` v 2 ´ 1q. Let us show that
ClpXq “ Z{2.
Let P be the point p0, 1q corresponding to the maximal ideal m “ pu, v ´ 1q. Then
X ´ P “ Dpv ´ 1q is given by the spectrum of the ring
` ˘
Av´1 “ Rru, vs{pu2 ` v 2 ´ 1q v´1 .
There is an isomorphism of rings
Av´1 » Rrts1`t2
given by the usual parameterization of the circle
ˆ ˙
2t 1 ´ t2
pu, vq ÞÑ ,
1 ` t2 1 ` t2
with inverse sending t to u{pv ´ 1q. As Rrts1`t2 is the localization of a UFD, we have
ClpDpv ´ 1qq “ 0.
This means that P generates ClpXq, by the exact sequence (17.15) applied to the open set
U “ Dpv ´ 1q.
It is clear that divpv ´ 1q “ 2P , because u is a local parameter at P and v ´ 1 “
pv ` 1q´1 u2 in OX,p . Hence 2P “ 0 in ClpXq.
On the other hand, the divisor P is not a principal divisor. The easiest way to see this is
perhaps to show that OX pP q ‰ OX , by showing that there are non-constant global sections.
Consider the covering X “ U0 Y U1 where U0 “ Dpv ´ 1q and U1 “ Dpv ` 1q. Then
ΓpU0 , OX pP qq “ Av´1 , ΓpU1 , OX pP qq “ u´1 ¨ Av`1
u
and the two local sections s0 “ 1´v and s1 “ 1`v
u
glue to a global section of OX pP q.
2
Note that the ring A is not a UFD, as u “ pv ´ 1qpv ` 1q and u, v ´ 1 and v ` 1 are
not units. In contrast, for XC “ Spec Cru, vs{pu2 ` v 2 ´ 1q » Spec Crx, ys{pxy ´ 1q has
trivial class group, as Crx, ys{pxy ´ 1q is a UFD.

The quadric cone

A singular quadric surface

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Consider the quadratic cone X “ Spec A where A “ krx, y, zs{pxy ´ z 2 q, and the field k
has characteristic ‰ 2. Let Z “ V py, zq be the closed subscheme corresponding to the line
ty “ z “ 0u. Note that
Z » Spec krx, y, zs{pxy ´ z 2 , y, zq “ Spec krxs,
so it is integral of codimension 1.
The complement X ´ Z “ X ´ V pyq “ Dpyq is isomorphic to
Spec krx, y, y ´1 , zs{pxy ´ z 2 q “ Spec kry, y ´1 srt, us{pt ´ u2 q “ Spec kry, y ´1 , us.
As this is the spectrum of a UFD, we have ClpX ´ Zq “ 0. By the exact sequence
Z ¨ Z ÝÝÑ ClpXq ÝÝÑ ClpX ´ Zq ÝÝÑ 0,
we see that ClpXq is generated by rZs.
Let us first show that 2Z “ 0 in ClpXq. In fact, divpyq “ 2Z . Note that, because of the
equation xy “ z 2 , we have V pyq “ V py, zq “ Z , as sets. Therefore, the only prime divisor
where y has non-zero order of vanishing is Z . Letting p “ py, zq, the local ring at the generic
point of Z is given by
Ap “ pkrx, y, zs{pxy ´ z 2 qqpy,zq
Since x is invertible in this ring, we see that y P pz 2 q and that z is a local parameter. This
also gives ordZ pyq “ 2, so 2Z “ divpyq, as we wanted.
Next, let us show that Z is itself not a principal divisor. It suffices to prove that this is not
principal in Spec OX,p where p P X is the origin p0, 0, 0q, corresponding to the maximal
ideal m “ px, y, zq. The local ring at p equals
OX,p “ pkrx, y, zs{pxy ´ z 2 qqpx,y,zq
In this ring p “ py, zq is a height 1 prime ideal, but it is not principal. Indeed, the vector
space m{m2 (that is, the Zariski cotangent space at x) is 3-dimensional, spanned by tx, y, zu
and y, z define a 2-dimensional subspace of m{m2 . If p were principal, one could write
y “ ah and z “ bh for some a, b, h P OX,p . But then the span of ȳ and z̄ are contained in
the submodule generated by h in m{m2 , a contradiction.
This means that rZs ‰ 0 in ClpXq and hence
ClpXq “ Z{2.
Note that the open subscheme U “ X ´ p0, 0, 0q is nonsingular, and the divisor Z is
Cartier. This example shows that removing a codimension 2 subset has no effect on Weil
divisors, as ClpU q “ ClpXq “ Z{2, but the subgroup of Cartier divisors might change.

Here is a direct way to see that PicpXq “ 0. The key point is that there is a natural action
of the multiplicative group Gm on X , given by
Gm ˆ X Ñ X, pt, px, y, zqq ÞÑ ptx, ty, tzq.
This extends to a morphism
φ : A1 ˆ X Ñ X

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This morphism is induced by the ring map A ÞÑ Arts defined by x ÞÑ tx, y ÞÑ ty , and
z ÞÑ tz .
Now let ι0 and ι1 be the closed embeddings X Ñ A1k ˆ X defined by the fibers over
t “ 0 and t “ 1 respectively. Note that φ ˝ ι1 “ idX , so the composition
φ˚ ι˚
PicpXq ÝÑ PicpA1 ˆ Xq ÝÑ
1
PicpXq
is equal to the identity. As ι˚1 is an isomorphism, we conclude that φ˚ is an isomorphism.
On the other hand, φ ˝ ι0 equals the zero map px, y, zq ÞÑ p0, 0, 0q and so φ˚ ˝ ι˚0 is
the zero map PicpXq Ñ PicpXq. As both ι˚0 and φ˚ are isomorphisms, this shows that
PicpXq “ 0 must be 0.

Nonsingular quadric surfaces

Let k be an field, and let X “ P1k ˆ P1k (where the fiber product is taken over k ). Recall that
X embeds as a quadric surface in P3k via the Segre embedding. So we can view X both as a
fiber product P1k ˆ P1k and the quadric V pu3 u0 ´ u1 u2 q Ă P3k .
Since X is a product of two P1 ’s there are natural ways of constructing divisors on it from
those on P1 . For instance, we can let
L1 “ p0 : 1q ˆ P1 Ă X,
which is a prime divisor on X corresponding to the ‘vertical fiber’ of X . Similarly, L2 “
P1 ˆ p0 : 1q is a Weil divisor on X . From these we obtain an exact sequence

ZL1 ‘ ZL2 ÝÝÑ ClpXq ÝÝÑ ClpX ´ L1 ´ L2 q ÝÝÑ 0

Note that X ´ L1 ´ L2 “ U11 “ Spec krx´1 , y ´1 s. The latter is isomorphic to A2k , so
ClpX ´ L1 ´ L2 q “ 0. This shows that ClpXq is generated by the classes of L1 and L2 .
We claim that the first map is also injective, so that in fact that
ClpXq “ ZL1 ‘ ZL2 .
If L1 and L2 are related by aL1 ´ bL2 „ 0, then
OX paL1 q » OX pbL2 q (17.16)
for some integers a, b P Z. We will show that this is not the case, by showing
(i) OX pL1 q|L1 » OP1k .
(ii) OX pL2 q|L1 » OP1k p1q
Then restricting both sides of (17.16) to L1 , the two statements imply that b “ 0. We must
also have a “ 0, by switching the roles of L1 and L2 .
Let x0 , x1 be homogeneous coordinates on the first P1k factor, and y0 , y1 homogeneous
coordinates on the second.
To prove i): Note that L1 „ L11 where L1 “ p1 : 0q ˆ P1 . This follows because x0 {x1
defines a rational function with
ˆ ˙
x0
div “ p0 : 1q ˆ P1k ´ p1 : 0q ˆ P1k “ L1 ´ L11 (17.17)
x1

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Note that L1 is contained in the open set U “ X ´ L11 . Clearly, OX pL11 q|U » OU , so (i)
holds by (17.17).
To prove (ii), note that L2 “ pr˚2 p0 : 1q, where pr : X Ñ P1k is the second projection.
Moreover, the divisor p0 : 1q corresponds to OP1 p1q on P1k . Therefore,
OX pL2 q|L1 » ι˚ pr˚2 OP1k p1q “ ppr2 ˝ιq˚ OP1k p1q “ OP1k p1q
because pr2 ˝ι is the identity map P1k Ñ P1k .
Alternatively, one can note that pD` py0 q, 1q and pD` py1 q, y0 {y1 q defines the Cartier data
of L2 , and that this defines the same sheaf as OP1k p1q when restricted to L1 .
This completes the proof that
ClpXq » ZL1 ‘ ZL2 .
If D is a divisor on X , D „ aL1 `bL2 and we call pa, bq the ‘type’ of D. A divisor of type
p1, 0q or p0, 1q is a line on the quadric surface X Ă P3 . We have i˚ OP3 p1q » OX pL1 ` L2 q,
so a p1, 1q-divisor is represented by a hyperplane section of X (a conic). A prime divisor of
type p1, 2q or p2, 1q is a twisted cubic curve.

17.10 The 3-dimensional quadratic cone

Let k be a field and consider X “ Spec A, where
A “ krx, y, z, ws{pxw ´ yzq.
Then the ring A is a normal integral domain, but it is not a UFD (as xw “ yz ). Let us show
that ClpXq “ Z.
Consider the principal divisor H “ div y “ V pyq, which equals D ` D1 where D “
V px, yq and D1 “ V py, wq. We have X ´H “ SpecpAy q “ Specpkrx, y, y ´1 , wsq which
has trivial class group, as the ring Ay is a UFD. Applying the exact sequence (17.15) we see
that ClpXq is generated by D and D1 . As H is principal, we have D ` D1 „ 0 in ClpXq,
so ClpXq is in fact generated by D.
To conclude, we will show that nD ȷ 0 for every n ‰ 0. In fact, we will show that D is a
Weil divisor which has no multiple which is Cartier!
We showed in Exercise 9.9.29 that the open subset U “ X ´ D is not affine. Note that
the complement X ´ D does not see the scheme structure on D; in particular, if nD were
Cartier, say given by some section s P ΓpX, OX pnDqq, then X ´ D “ X ´ V psq. To
conclude, we use the following criterion:

Proposition 17.51. Let X be an affine, integral, Noetherian, normal scheme, and let
V psq be the zero scheme of a section s of an invertible sheaf L. Then the open set
U “ X ´ V psq is also affine.

Proof Let ϕi : L|Vi Ñ OVi be trivializing isomorphisms for L and let fi P OX pVi q be
the images of the section s|Vi . Then if X “ Spec A, then V psq X Vi “ SpecpA{pf qi q and
Vi ´ V psq “ SpecpAfi q.
Now consider the inclusion ι : U Ñ X . As ι´1 pVi q “ Vi ´ V psq is affine, and the open

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sets ι´1 pVi q form a cover of X ´ V psq, we see that ι is an affine morphism. But then,
U “ ι´1 pX ´ V psqq is affine as well.

Quadric hypersurfaces in higher dimension

Let A “ krx1 , . . . , xn , y, zs{px21 ` ¨ ¨ ¨ ` x2n ´ yzq. We will prove that A is a UFD for
n ě 3. A is a domain, because the defining quadric polynomial is irreducible. Let us show
that A is normal. Note that
Ay “ krx1 , . . . , xn , y, z, z ´1 , zs{pz ´1 px21 ` ¨ ¨ ¨ ` x2m q ´ yq » krx1 , . . . , xn , z, z ´1 s
This is a UFD, hence normal. Therefore, if w P KpAq, satisfies an integral relation with
coefficients in A, then at least we have w P Ay . We may write w “ a{z l . Then the relation
takes the form
pa{z l qn ` b1 pa{y l qn´1 ` ¨ ¨ ¨ ` bn “ 0
Multiplying by y ln , we get that an P pyq, hence y|a, as the ideal pyq is prime for n ě 3.
This shows that A is normal.
This shows that X “ Spec A is a Noetherian normal scheme. Now the divisor D “ div y
is an effective divisor on X . Then the exact sequence (17.15) takes the form
ZD ÝÝÑ ClpSpec Aq ÝÝÑ ClpSpec Ay q “ 0 ÝÝÑ 0
The image of the left-most map is 0, so ClpAq “ 0. Therefore, A is a UFD by Proposition
17.22.
Note that for n “ 1, 2, the above argument does not work, as the ideal pzq may not be
prime. In the case n “ 1, X is the quadric cone, which has class group ClpXq “ Z{2. If
k contains a square root of ´1, then X is isomorphic to the affine quadric xy ´ zw “ 0,
which has Class group equal to Z. If n “ 2, and k does not have a square root of ´1, the
same argument works to show that ClpXq “ Z{2.
Applying a change of variables, we find the following description of the class groups of
diagonal quadrics in any dimension:

Proposition 17.52. Let k be a field containing a square root of ´1 and let X “ V px20 `
¨ ¨ ¨ ` x2n q Ă An`1
k “ Spec krx0 , . . . , xn s.
(i) n “ 2, ClpXq “ Z{2
(ii) n “ 3, ClpXq “ Z
(iii) n ě 4, ClpXq “ 0

The projective quadric cone

Let X “ Proj R where R “ krx, y, z, ws{pxy ´ z 2 q. Let H “ V pwq be the hyperplane
determined by w. We have the exact sequence
0 ÝÝÑ ZH ÝÝÑ ClpXq ÝÝÑ ClpX ´ Hq ÝÝÑ 0
Here H is a divisor corresponding to the restriction of OP3 p1q, hence it is non-torsion in
ClpXq, so the first map is injective. Note that X ´ H is isomorphic to the affine quadric cone

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from earlier, hence ClpX ´ Hq “ Z{2. Hence ClpXq is either Z, generated by a divisor D
such that 2D “ H or Z ‘ Z{2. It is the first case which is the case: D is the divisor V px, zq,
which is supported on a line on X , and H “ divpxq “ 2D.
The Weil divisor D is not Cartier: being Cartier is a local condition, so this follows from the
example of the affine quadric cone above. Here is an alternative way to see it: if D “ V px, zq
were Cartier, the sheaf L “ OX pDq would be invertible, and the same would be true for
for the restriction to the line ℓ “ V px, zq » P1k . The Picard group of P1k is Z, generated by
OP1 p1q, so we would have L|ℓ » OP1 paq for some a P Z. On the other hand, we know that
the divisor H “ 2D is Cartier, and in fact OX pHq » OP3 p1q|X (the local generator is given
by x). Restricting further to ℓ, we obtain OP3 p1q|ℓ » OP1 p1q (as the divisor of w is just one
point on ℓ). But these two observations imply that 2a “ 1, which is not possible. Hence D is
not Cartier.
There is also the following statement for projective quadrics of higher dimension:

Proposition 17.53. Let X “ V px20 ` ¨ ¨ ¨ ` x2m q Ă Pn “ Proj krx0 , . . . , xn s.

(i) n “ 2, ClpXq “ Z.
(ii) n “ 3, ClpXq “ Z2 .
(iii) n ě 4, ClpXq “ Z.

17.11 Exercises
Exercise 17.11.1. Verify that the set of principal divisors form a subgroup of DivpXq.

Exercise 17.11.2. Consider the curve y 2 “ x3 ´ 1 in A2k where k is algebraically closed of

characteristic different from 2 and 3. If pa, bq P X we let σpa, pq “ pa, ´bq, which also lies
in X .
a) Show that for any P P X , it holds that P ` σpP q „ 0.
b) Show that if P , Q and R are three collinear points on X , then P ` Q ` R „ 0.
c) Show that any Weil divisor on X is linearly equivalent to a prime divisor.

Exercise 17.11.3. Let X be an integral normal scheme and let KX denote the constant sheaf
ˆ
on K “ KpXq. Note that the sheaf OX of invertible sections of OX embeds as a subsheaf of
KX . Show that a Cartier divisor is the same thing as a global section of the sheaf KXˆ {OXˆ
.

Exercise 17.11.4. Check that the inverse of a Cartier divisors and the sum of two are well-
defined; that is, that all cocycle conditions are fulfilled and that the inverse, respectively the
sum, is independent of choices of representatives.

Exercise 17.11.5. Check that the ideal sheaf InP of the divisor nP in Example 17.38 is
isomorphic to OP1 p´nq.

Exercise 17.11.6. Describe Cartier data that defines the hyperplane V pxi q in Pnk .

Exercise 17.11.7. Show that all the local rings OX,p of the curve X given by y 2 “ x3 ´ 1
in A2k are discrete valuation rings, and hence X is a normal variety. We assume that k is
algebraically closed and of characteristic different from three and two. More precisely, if

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pa, bq is a point on X show that x ´ a is a parameter if b ‰ 0 and that y is one when b “ 0.

H INT: Note that y 2 ´ b2 “ x3 ´ a3 .
Exercise 17.11.8. Let X “ Spec Crx, ys{py 2 ´ x3 ´ xq. Compute the divisors of the
rational functions x, y and x2 {y .
Exercise 17.11.9. Consider the curve Z “ V pF q Ă P2k defined by an irreducible homoge-
neous polynomial F of degree d ě 1. Consider the open set U “ P2k ´ Z . Show that the
exact sequence (17.15) takes the form
¨d
0 ÝÝÑ Z ÝÝÑ Z ÝÝÑ ClpU q ÝÝÑ 0.
Deduce that ClpU q » Z{d.
Exercise 17.11.10. Show that for the weighted projective space P “ Pp1, 1, dq we have
ClpPq “ ZD and CaClpPq “ ZH where H “ dD.
Exercise 17.11.11. The same reasoning as for P1k can be applied to the affine line X with
two origins. Compute PicpXq for this example.
Exercise 17.11.12. The aim of this exercise is to prove the following statement, known as
“Nagata’s lemma”: Let A be a Noetherian integral domain, and let x P A ´ 0. Suppose that
pxq is prime, and that Ax is a UFD. Then A is a UFD.
a) Show that Ax is normal.
b) Show that A is normal. H INT: If t P KpAq is integral over A, then t P Ax .
c) Show that there is an exact sequence
ZD Ñ ClpSpec Aq Ñ ClpSpec Ax q “ 0.
d) Use the above sequence to show that ClpSpec Aq “ 0, and conclude that A is
a UFD.
Exercise
? 17.11.13. Let d be a square free integer and assume
? that d ı 1 mod 4 so that
Zr ds is a Dedekind ring. Show that the
?class group of Zr ds is finite. H INT: Imitate the
technique used for the example with Zr ´5s.
Exercise 17.11.14. Let X be a scheme and let D Ă X be a closed subscheme with ideal
sheaf I . Show that the following statements are equivalent:
(i) I is an invertible sheaf.
(ii) For every x P X , the ideal Ix Ă OX,x is principal and generated by a nonzero-
divisor.
(iii) There is an open covering Ui of X and nonzerodivisors fi P OX pUi q such that
fi generates IpUi q.
(iv) For every x P X , there is an open affine neighbourhood U “ Spec A of x such
that U X D “ Spec A{pf q where f P A is a nonzerodivisor.
(v) D is the zero scheme of a global section s of an invertible sheaf L.
When X is normal integral Noetherian, these are also equivalent to
(vi) D is an effective Cartier divisor.
Exercise 17.11.15. Show that the Picard group of the general linear group GLn is trivial.
H INT: Use an exact sequence.

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Exercise 17.11.16. Let X be a normal integral scheme and let π : X ˆ A1 Ñ X . We will

consider the map
π ˚ : PicpXq ÝÝÑ PicpX ˆ A1 q
We will also need a section ι : X Ñ X ˆ A1 defined by x ÞÑ px, 0q.
a) Show that the composition ι˚ ˝ π ˚ is the identity. Deduce that π ˚ is injective.
b) Let L be an invertible sheaf on X ˆ A1 and assume that there is an affine
covering Ui of X such that L is trivial when restricted to each π ´1 pUi q. Show
that L » π ˚ A for some A invertible sheaf on X .
c) Using b), deduce that π ˚ is surjective.
Exercise 17.11.17. Let k be a field. Use the fact that A2k has trivial Picard group to show that
the morphism A2k ´ tp0, 0qu Ñ P1k does not extend to a morphism A2k Ñ P1k .
Exercise 17.11.18 (Norms). Let f : X Ñ Y be a finite morphism of integral Noetherian
normal schemes. Then f induces an inclusion of function fields KpY q Ñ KpXq, making
KpXq into a finite extension of KpY q.
a) Show that the multiplication g P KpY q induces a KpXq-linear map of KpXq-
vector spaces mg : KpY q Ñ KpY q.
b) We define the norm to be the map N : KpY q Ñ KpXq to be the determinant
of mg . Show that N is multiplicative, i.e., N pghq “ N pgqN phq for each
g, h P KpY q.
c) Show that if h P KpXq, then f˚ f ˚ divphq “ pdeg f q ¨ divphq.
Exercise 17.11.19. Let X “ A2k ´ p0, 0q and let f : X Ñ P1k be the quotient morphism.
Show that f ˚ OP1k p1q » OX . Deduce that the induced map f ˚ : PicpP1k q Ñ PicpXq may
fail to be injective.
Find a dominant morphism f : X Ñ Y so that PicpY q Ñ P icpXq is not surjective.

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First steps in sheaf cohomology

One of the main challenges when working with sheaves is that surjective maps of sheaves do
not always induce surjections on global sections. Given a short exact sequence of sheaves

0 F1 F F2 0,

one has a sequence

0 ΓpX, F 1 q ΓpX, Fq ΓpX, F 2 q (18.1)

which is exact at each stage except on the right, but the right-most map may fail to be
surjective. In many situations in algebraic geometry, knowing that ΓpX, Fq Ñ ΓpX, F 2 q is
surjective is of fundamental importance. For instance, if U Ă X is an open subscheme, it is
useful to know when a regular function defined on U extends to a regular function on all of
X.
Cohomology groups can be seen as a partial response to this behavior of Γ, and in good
situations, they allow us to say something about the missing cokernel. More precisely, the
sequence (18.1), induces a long exact sequence of cohomology groups

0 ΓpX, F 1 q ΓpX, Fq ΓpX, F 2 q

H 1 pX, F 1 q H 1 pX, Fq H 1 pX, F 2 q

H 2 pX, F 1 q H 2 pX, Fq H 2 pX, F 2 q ÝÑ ¨ ¨ ¨

This means that the failure of surjectivity of the above is controlled by the group H 1 pX, F 1 q
and the other groups in the sequence.
In addition to problems such as lifting, cohomology groups allow us to define many
geometric invariants of F and X . These in turn allow us to distinguish schemes, that is, if
two schemes have different cohomology groups they can not be isomorphic.
Cohomology groups can be defined in a completely general setting, for any topological
space and a (pre)sheaf on it. There are several ways to define them. The modern approach uses
the theory of derived functors. This is in most respects the ‘right way’ to define the groups in
general, but going through the whole machinery of derived functors and homological algebra

361

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would take us too far astray. We therefore begin with taking a more down-to-earth approach
using Cech cohomology which is better suited for computations.

18.1 Some homological algebra

Recall that a complex of abelian groups A‚ is a sequence of groups Ai together with maps
between them
di´2 di´1 di di`1
¨¨¨ Ai´1 Ai Ai`1 ¨¨¨
f
such that di ˝ di´1 “ 0 for each i. A map of complexes A‚ Ý
Ñ B ‚ is a collection of maps of
i i
groups fi : A Ñ B making the following diagram commutative:

di´1 diA
¨¨¨ Ai´1 A
Ai Ai`1 ¨¨¨
fi´1 fi fi`1
di´1 diB
¨¨¨ B i´1 B
Bi B i`1 ¨¨¨
In this way, we can talk about kernels, images, cokernels, exact sequences of complexes, etc.
We say that an element σ P Ap is a cocycle if it lies in the kernel of the map dp i.e.,
dp σ “ 0. A coboundary is an element in the image of dp´1 , i.e. σ “ dp´1 τ for some
τ P Ap´1 . Since dp pdp´1 aq “ 0 for all a, we have
Im dp´1 Ă Ker dp ,
and so all coboundaries are cocycles. The cohomology groups of the complex A‚ are set up
to measure the difference between these two notions. We define the p-th cohomology group
as the quotient group
H p A‚ “ Ker dp {Im dp´1 .
One thinks of H p A‚ as a group that measures the failure of the complex A‚ of being exact at
stage p: A‚ is exact if and only if H p A‚ “ 0 for every p.
The following result is fundamental in the theory of cohomology groups:

f g
Proposition 18.1. Suppose that 0 Ñ A‚ ÝÑ B ‚ ÝÑ C ‚ Ñ 0 is an exact sequence of
complexes. Then there is a long exact sequence of cohomology groups

¨¨¨ H p A‚ H pB‚ H pC ‚

H p`1 A‚ H p`1 B ‚ H p`1 C ‚ ÝÑ ¨ ¨ ¨

Proof For each p P Z, consider the commutative diagram

fp gp
0 Ap Bp Cp 0
dp
F dp
G dp
C

fp`1 gp`1
0 Ap`1 B p`1 C p`1 0

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where the rows are exact by assumption. By the Snake Lemma, we obtain a sequence
fp gp
0 Ker dpA Ker dpB Ker dpC

fp`1 gp`1
Ap`1 {Im dpA B p`1 {Im dpB p`1
H p {Im dC 0.

Consider now the diagram

fp gp
Ap {Im dp´1
A
p´1
B p {Im dB H p {Im dpC 0
dp
A dp
C dp
C

fp`1 gp`1
0 Ker dp`1
A Ker dp`1
G Ker dp`1
C

where the rows are exact by the above. For the maps in this diagram, H p A‚ “ Ker dpA and
H p`1 A‚ “ Coker dpA etc. Hence applying the Snake Lemma one more time, we get the
desired exact sequence.
A map of complexes f : C ‚ Ñ D‚ is a chain map if f ˝ dC “ dD ˝ f . Such a map induces
a well-defined map between cohomology groups
f : H i pC ‚ q Ñ H i pD‚ q
A chain homotopy between two chain maps f, g : C ‚ Ñ C ‚ is a collection of maps
h : C ‚ Ñ D‚´1 such that
f ´ g “ dD ˝ h ` h ˝ dC
If f and g are related by a chain homotopy, they induce the same map H i pC ‚ q Ñ H i pD‚ q.
Indeed, if c P KerpC p Ñ C p`1 , then
rf pcq ´ gpcqs “ rdD phpcqqs “ 0.
Example 18.2. To show that H i pC ‚ q “ 0 (e.g., that C ‚ is exact), it is enough find a
chain homotopy between the identity map and the zero map. Concretely, the chain map
h : C ‚`1 Ñ C ‚ should satisfy
pdp ˝ h ` h ˝ dp`1 qpcq “ 0 (18.2)
for every c P C p`1 . △

18.2 Cech cohomology

Let X be a topological space. For simplicity, we will assume that X admits an open cover U
consisting of finitely many open sets U1 , . . . , Ur . We will index the intersections
UI “ Ui0 X ¨ ¨ ¨ X Uip
using strictly increasing sequences of positive integers I “ pi0 ă i1 ă ¨ ¨ ¨ ă ip q.

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For a sheaf F on X , we have the sheaf exact sequence (3.1)

ź ź
0 ÝÝÑ FpXq ÝÝÑ FpUi q ÝÝÑ FpUi X Uj q. (18.3)
i i,j

The Cech complex is essentially the continuation of this sequence; it is a complex obtained by
adjoining all the groups FpUi1 X ¨ ¨ ¨ X Uir q over all possible intersections Ui1 X ¨ ¨ ¨ X Uir .

Definition 18.3. For a sheaf F on X , we define the Cech complex C ‚ pU, Fq of F (with
respect to the open covering U ) as
d0 d1 d2
C 0 pU, Fq C 1 pU, Fq C 2 pU, Fq ...
where
ź
C p pU, Fq “ FpUi0 X ¨ ¨ ¨ X Uip q,
i0 ăi1 ă¨¨¨ăip

and the coboundary maps dp : C p pU, Fq ÝÝÑ C p`1 pU, Fq by

p`1
ÿ
pdp σqi0 ,...,ip`1 “ p´1qj σi0 ,...iˆj ,...,ip`1 |Ui0 X¨¨¨XUip`1
j“0

where i0 , . . . iˆj , . . . , ip`1 means i0 , . . . , ip`1 with the index ij omitted.

Note that since we assume that the open cover is finite, say having r elements, C p pU, Fq “
0 for every p ě r.

Example 18.4. The two first groups in the Cech complex are given by
ź ź
C 0 pU, Fq “ FpUi0 q and C 1 pU, Fq “ FpUi0 X Ui1 q.
i0 i0 ăi1

An element σ P C 0 pU, Fq is an r-tuple of sections σ “ pσ1 , . . . , σr q, where σi P FpUi q

for each i. Likewise, an element σ “ pσij q P C 1 pU, Fq is a collection of sections σij P
FpUi X Uj q, one for each pair i ă j .
The coboundary map d0 : C 0 pU, Fq Ñ C 1 pU, Fq sends an element σ “ pσi q, to the
element d0 σ P C 1 pU, F q whose ij -th component is equal to
ˇ
pd0 σqij “ σj ´ σi ˇU ij
(18.4)

The coboundary map d1 : C 1 pU, Fq Ñ C 2 pU, Fq sends σ “ pσij q, to the element with
ijk -th component equal to
ˇ
pd1 σqijk “ σjk ´ σik ` σij ˇU (18.5)
ijk

Substituting (18.4) into (18.5), there are many cancellations, and we see that d1 ˝ d0 “ 0.
The same happens also in higher degrees:

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Lemma 18.5. For every p, we have

dp`1 ˝ dp “ 0.

Proof
p
ÿ
dpdpσqqi0 ,...,ip “ dσi0 ,...,iˆk ,...,ip |Ui0 ,...,ip
k“0
p
˜p´1 ¸
ÿ ÿ
“ p´1qk p´1ql σi0 ,...,îl ,...,îk ,...,ip |Ui0 ,...,ip
k“0 lăk
p
˜p´1 ¸
ÿ ÿ
` p´1qk p´1ql σi0 ,...,îk ,...,îl`1 ,...,ip |Ui0 ,...,ip .
k“0 lěk

For integers m ă n, the term σi0 ,...,m̂,...,n̂,...,ip appears twice in the sum (for pl, kq “ pm, nq
and pl, kq “ pn ´ 1, mq). But as the signs are different, the terms cancel, and so dpdσq “
0.
Therefore, the Cech complex is indeed a complex of abelian groups. The Cech cohomology
groups of F with respect to U is defined to be the cohomology of this complex:

Definition 18.6. The p-th Cech cohomology of F with respect to U is defined as

H p pU, Fq “ Ker dp {Im dp´1 .

The Cech cohomology groups depend on the open cover U , but not on the choice of
the ordering of the open sets Ui . Given two orderings, there is an isomorphism of the two
associated Cech complexes given by multiplication by ˘1 on each C p , so in particular, the
cohomology groups are the same.
A sheaf homomorphism F Ñ G induces maps C p pU, Fq Ñ C p pU, Gq (it does so
component-wise), and a straightforward computation shows that the induced maps commute
with the coboundary maps, and hence they pass to the cohomology. So we obtain functors
H p pU, ´q from sheaves to abelian groups.
Example 18.7. The group H 0 pU, Fq is the kernel of the map d0 : C 0 pU, Fq Ñ C 1 pU, Fq,
which is simply the usual map
ź ź
FpUi q Ñ FpUi X Uj q.
i iăj

This kernel is equal to FpXq by the sheaf axioms, so H 0 pU, Fq “ FpXq. △

Example 18.8 (H 1 and lifting of sections). The most interesting cohomology group is
arguably H 1 pU, Fq. It is the group of elements pσij q such that σik “ σij ` σjk modulo the
elements of the form σij “ τj ´ τi (restricted to Ui X Uj ). As mentioned in the introduction,
this group is closely related to the lifting of sections, as we now explain.
Suppose that we have a short exact sequence of sheaves
0 ÝÝÑ A ÝÝÑ B ÝÝÑ C ÝÝÑ 0

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Suppose we want to try to lift a section c P CpXq to a section of BpXq. Since the sequence
is exact, we can at least find lifts locally, i.e. there is an open covering U “ tUi u and sections
bi P BpUi q that map to c|Ui over each Ui . Now we ask if we can assemble the bi to a section
b P BpXq. For this to be the case, we must have bj |Uij ´ bi |Uij “ 0. In any case,
` ˘
σ “ bj |Uij ´ bi |Uij
defines an element of C 1 pU, Aq (because bi and bj map to the same element in CpUij qq.
Furthermore, dσ “ 0, because
pdσqijk “ pbk ´ bj q ´ pbk ´ bi q ` pbj ´ bi q “ 0
(all terms restricted to Uijk ). When is σ zero in H 1 pU, Aq? This occurs if and only if there is
an element a “ pai q P C 0 pU, Aq such that
bj |Uij ´ bi |Uij “ aj |Uij ´ ai |Uij ,
which is equivalent to saying that the elements bi ´ ai P BpUi q agree over the overlaps Uij ,
or in other words, that they glue together to a section b P BpXq. Note that since ai P ApUi q,
the image of bi ´ ai is the same as that of bi , i.e. b maps to c.
In summary, the section c P CpXq can be lifted if and only if the associated element in
H 1 pU, Aq equals 0. If the latter group is zero, any section of CpXq lifts.
In Example 18.11 we will see a concrete example of a section which does not lift. △

18.3 Examples
Example 18.9. If the cover U consists of two open sets U0 and U1 , then the Cech complex
takes the form
d0
0 ÝÝÑ FpU0 q ˆ FpU1 q ÝÝÑ FpU0 X U1 q ÝÝÑ 0 (18.6)
Therefore, the group H 1 pU, Fq can be identified with the cokernel of d0 , and there is an
exact sequence
d0
0 Ñ H 0 pU, Fq Ñ FpU0 q ˆ FpU1 q ÝÝÑ FpU0 X U1 q Ñ H 1 pU, Fq Ñ 0. (18.7)
Concretely, H 1 pU, Fq is the group of sections FpU0 X U1 q modulo the sections of the form
s1 ´ s0 where s0 and s1 are restrictions of sections in FpU0 q and FpU1 q respectively.
If 0 Ñ F 1 Ñ F Ñ F 2 Ñ 0 is an exact sequence, we can understand the connecting map
δ in the long exact sequence by applying the Snake Lemma to the diagram

0 C 0 pU, F 1 q C 0 pU, Fq C 0 pU, F 2 q 0

(18.8)

0 C 1 pU, F 1 q C 1 pU, Fq C 1 pU, F 2 q 0

△
Example 18.10 (The projective line). Consider the projective line P1 “ P1k over a field
k . It is covered by the two standard affines U0 “ Spec krts and U1 “ Spec krt´1 s with

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intersection U0 X U1 “ Spec krt, t´1 s. For the structure sheaf OP1 , the Cech-complex takes
the form
d0
0 OP1 pU0 q ˆ OP1 pU1 q OP1 pU0 X U1 q 0
» »

krts ˆ krt´1 s d
krt, t´1 s,

where d sends a pair ppptq, qpt´1 qq to qpt´1 q ´ pptq. We saw in Chapter 5 (during the proof
of Proposition ??) that Ker d “ k . On the other hand, it is clear that each element of krt, t´1 s
is a sum of a polynomial in t and one in t´1 . Hence d is surjective, and we have
H 1 pU, OP1 q “ Coker d “ 0.
△
Example 18.11 (The sheaves OP1 pmq). Continuing the above example, let us compute the
Cech cohomology groups of OP1 pmq. We use the same affine cover, and the Cech complex
still takes the form
d
0 krts ˆ krt´1 s krt, t´1 s 0,

but the coboundary map d is different: there is a multiplication by tm in one of the restrictions,
so the coboundary map is now given by
dppptq, qpt´1 qq “ tm qpt´1 q ´ pptq.
(see Section 5.1). As we saw in Proposition ??, the kernel of d is generated by the m ` 1
elements p1, t´m q, pt, t´m`1 q, . . . , ptm , 1q if m ě 0, and Ker d “ 0 otherwise. Hence
dimk H 0 pU, Opmqq “ m ` 1
if m ě 0 and H 0 pU, Opmqq “ 0 if m ă 0.
We can also compute H 1 pU, OP1 pmqq, which is given by the cokernel of d. Consider
first the case when m ě 0. As before, it is easy to see that any polynomial in krt, t´1 s
can be written in the form tm qpt´1 q ´ pptq. In fact, this also works for m “ ´1, because
t´k “ t´1 ¨ t´k`1 ´ 0 and tk “ t´1 ¨ 0 ´ tk . Hence H 1 pU, OP1 pmqq “ 0 for m ě ´1.
For m ď ´2 however, no linear combination of the monomials
t´1 , t´2 , . . . , tm`1
lies in the image of d, but combinations of all the others do. It follows that H 1 pU, OP1 pmqq
is a k -vector space of dimension ´m ´ 1 in this case. △
Example 18.12. Let Z Ă P1k be the subscheme associated to two closed points p, q in P1 .
We saw in Example 16.33 that the ideal sheaf sequence takes the form
0 ÝÝÑ OP1k p´2q ÝÝÑ OP1k ÝÝÑ i˚ OZ ÝÝÑ 0
Consider the element p0, 1q P k ‘ k , which defines a section of i˚ OZ pP1k q “ k ‘ k . One
can ask whether this section lifts to a global section s of OP1k . In fact, this is not possible,

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because OP1k pP1k q “ k : any regular function on P1k is constant so it can not take the value 0 at
one point and 1 at another.
This failure of ability to lift is of course explained by the cohomology group H 1 pU, OP1k p´2qq
which is 1-dimensional. Since H 1 pU, OP1k q “ 0, one can think of the elements of this group
as the group of elements of k ‘ k modulo those that lift to OP1k . Here it is clear that an
element pa, bq P k ‘ k lifts if and only if a “ b. In fact, in this example, the connecting map
δ : H 0 pU, i˚ OZ q Ñ H 1 pU, Op´2qq
can be identified with the map k ‘ k Ñ k sending pa, bq to a ´ b. △
Example 18.13 (The cuspidal cubic). The curve X “ Proj krx0 , x1 , x2 s{px32 ´ x0 x21 q
admits an open cover U with two open sets, U0 “ D` px0 q and U1 “ D` px1 q. We have
OX pU0 q “ krx1 {x0 , x2 {x0 s{ppx2 {x0 q3 ´ px1 {x0 q2 q
OX pU1 q “ krx0 {x1 , x2 {x1 s{ppx2 {x1 q3 ´ px0 {x1 qq “ krx2 {x1 s
OX pU01 q “ krx2 {x1 , x1 {x2 s.
where we have used the defining equation to identify x0 {x1 “ px2 {x1 q3 and x0 {x2 “
px1 {x2 q2 . The coboundary d1 sends ppx1 {x0 , x2 {x0 q and qpx2 {x1 q to
qpx2 {x1 q ´ pppx1 {x2 q3 , px1 {x2 q2 q.
From these expressions we can obtain any monomial xa1 {xa2 except x1 {x2 . Therefore,
H 1 pU, OX q “ Coker d1 “ k ¨ x1 {x2
△
Example 18.14. Let U be a finite open cover such that one of the members is the whole
space X . In this case, the higher cohomology groups of any sheaf are all zero; that is
H p pU, Fq “ 0 for all p ě 1
To see this, suppose for simplicity that U0 “ X , where 0 P I denotes the smallest element
(otherwise, rename the indexes), and define the map h : C p`1 pU, Fq ÝÝÑ C p pU, Fq by
#
σ0,j0 ,...,jp if j0 ‰ 0
hpσqj0 ,...,jp “
0 if j0 “ 0.
Then for i0 ‰ 0, we have
ÿp
pdh ` hdqpσqi0 ,...,ip “ p´1qj hpσqi0 ,...,îj ,...,ip ` dpσq0,i0 ,...,ip
j“0
ÿp ÿp
“ p´1qj σ0,i0 ,...,îj ,...,ip ` σi0 ,...,ip ` p´1qj`1 σ0,i0 ,...,îj ,...,ip
j“0 j“0

“ σi0 ,...,ip .
Likewise, if i0 “ 0, we have
ÿp
pdh ` hdqpσq0,i1 ,...,ip “ p´1qj hpσq0,i1 ,...,îj ,...,ip ` 0
j“0

“ σ0,i1 ,...,ip .

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Hence h is a homotopy between the identity map on C p`1 pU, Fq and the zero map, and the
cohomology group H p`1 pU, Fq is zero by Example 18.2 △
Example 18.15 (The unit circle). Here is an example from topology. Consider the unit circle
X “ S 1 (with the Euclidean topology), and equip it with a standard covering U “ tU, V u
consisting of two intervals intersecting in two intervals as shown in the figure. Let F “ ZX
be the constant sheaf on Z.

Here we have
C 0 pU, Fq “ ZX pU q ˆ ZX pV q » Z ˆ Z C 1 pU, Zq “ ZX pU X V q » Z ˆ Z.
The map d0 : C 0 pU, ZX q Ñ C 1 pU, ZX q is the map Z2 Ñ Z2 given by
d0 pa, bq “ pb ´ a, b ´ aq.
Hence
H 0 pU, ZX q “ Ker d0 “ Zp1, 1q » Z,
and
H 1 pU, ZX q “ Coker d0 “ Z2 {Zp1, 1q » Z.
Readers familiar with algebraic topology may recognize that this gives the same answer as
singular cohomology. In fact, it is a general fact that the cohomology groups H p pU, Zq agree
p
with the usual singular cohomology groups Hsing pX, Zq for any topological space homotopy
equivalent to a CW complex, provided that the open sets in the covering U are contractible
(see e.g., (Griffiths and Harris, 1979, p. 42)). △
Example 18.16 (Constant sheaves on irreducible spaces). Suppose that X is an irreducible
topological space and let A be an abelian group. We claim that for any finite covering U of
X,
H p pU, AX q “ 0 for all p ě 1.
The Cech complex takes the form
ź ź ź
AÑ AÑ A Ñ ¨¨¨ (18.9)
i iăj iăjăk

Note that this complex does not depend on X nor on the covering U ; only the index set I
plays a role. We can therefore use a cover consisting of pn ` 1q opens, all equal to X , and
the higher cohomology groups vanish by Example 18.14.
For this reason, constant sheaves do not reveal much about the geometry of a scheme
equipped with the Zariski topology. For a scheme, it is the quasi-coherent sheaves which give
us richer and more interesting geometric invariants.
△

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18.4 Cech cohomology on schemes

As the previous examples illustrate, the cohomology groups H p pU, Fq can be computed if
we have adequate information on the sections of F over the open sets in the finite cover U . In
such cases, the maps in the Cech complex are entirely explicit, and computing their kernels
and images can often be done quite algorithmically.
On the other hand, the current definition of the cohomology groups is unsatisfactory for a
few reasons. First of all, the groups H p pU, Fq depend on the open cover U , whereas we want
something canonical that only depends on F . This dependency is problematic: for instance,
U could consist of the single open set X , and so H i pU, Fq “ 0 for all i ě 1.
The solution to these issues is typically to use more refined coverings, where the open sets
are ‘simple’ in some sense. In the context of schemes, the most natural approach is to consider
an open covering U consisting of affine open sets. We will show that in good situations, that
is, X is Noetherian and separated and the sheaf F is quasi-coherent, the group H i pU, Fq
will, in fact, turn out to be independent of the covering U .

Theorem 18.17 (Main properties of Cech cohomology). Let X be a Noetherian sepa-

rated scheme, and let U “ tUi u be a finite affine cover of X . Then
(i) The Cech cohomology groups are functors H i pU, ´q : AbpXq Ñ Ab.
(ii) H 0 pU, Fq “ FpXq.
(iii) Short exact sequences of quasi-coherent sheaves induce long exact sequences
of cohomology
¨ ¨ ¨ Ñ H p pU, F 1 q Ñ H p pU, Fq Ñ H p pU, F 2 q Ñ H p`1 pU, F 1 q Ñ ¨ ¨ ¨
(iv) If V “ tVi u is another affine cover, then there is a natural isomorphism
H p pU, Fq “ H p pV, Fq
for every p and every quasi-coherent sheaf F .
(v) If X has dimension n, then H p pU, Fq “ 0 for all p ą n and all quasi-
coherent F .

In light of item (iv), we make the following definition.

Definition 18.18. For a Noetherian, separated scheme X , we write H p pX, Fq for the
group H p pU, Fq, where U is an affine cover of X .

We have already proved the first two of these properties. For these statements, we do not
need to assume that X is separated (in fact, not even that the cover is finite). The other items
will require a little more work.

18.5 Cohomology of sheaves on affine schemes

The following result is fundamental in the study of sheaf cohomology groups. It is the first
example of a ‘vanishing theorem’ for cohomology. Recalling that cohomology groups were
defined to measure the ‘failure’ of certain desirable statements (e.g. restriction maps being
surjective), we are in general happy if cohomology groups are zero.

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Theorem 18.19. Let X “ Spec A and let F be a quasi-coherent sheaf on X . Then for
any affine cover U of X ,
H p pU, Fq “ 0 for all p ą 0.

Proof Let Ui “ Spec Ai be the affines in U . As X is affine, it is separated, so all intersec-

tions UI are also affine. We want to show that the complex of A-modules

0 ÝÝÑ FpXq ÝÝÑ C 0 pU, Fq ÝÝÑ C 1 pU, Fq ÝÝÑ ¨ ¨ ¨ (18.10)

is exact. We know that the theorem holds in the ‘trivial case’ when one of the Ui , say, U0 is
equal to X (see Example 18.14). In general, we reduce to the trivial case as follows.
As (18.10) is a complex of A-modules, exactness can be checked by localizing at each
prime ideal p P Spec A. We may assume without loss of generality that p P U0 .
We will compare the complex (18.10) to that of F|U0 with respect to the covering U X U0
of U0 .

0 ÝÝÑ FpU0 q ÝÝÑ C 0 pU X U0 , F|U0 q ÝÝÑ C 1 pU X U0 , F|U0 q ÝÝÑ ¨ ¨ ¨ (18.11)

This complex is exact by the ‘trivial case’ by Example 18.14, and hence the localization

0 ÝÝÑ FpU0 qp ÝÝÑ C 0 pU X U0 , F|U0 qp ÝÝÑ C 1 pU X U0 , F|U0 qp ÝÝÑ ¨ ¨ ¨ (18.12)

is also exact. However, as F is quasi-coherent, the localization of (18.10) at p coincides the

complex (18.12). As the latter is exact, so is (18.10), and we are done.

The fact that the higher cohomology groups vanish for every quasi-coherent sheaf is
quite special for affine schemes. In fact, affine schemes are characteristed by this property
(see (Stacks Project Authors, 2018, Tag 01XE)). Example 18.11 showed that even the most
basic non-affine scheme, P1k , admits sheaves with non-vanishing higher cohomology. Here is
another example:

Example 18.20 (The affine line with two origins). Consider the ‘affine line with two origins’
X from Example ?? on page ??. It is covered by two affine subsets X1 “ Spec krus and
X2 “ Spec krus and these are glued together along their common open set X12 “ Dpuq “
Spec kru, u´1 s with the identity as gluing map. The Cech complex for this covering looks
like
d1 d2
0 krus ˆ krus kru, u´1 s 0

where d1 pppuq, qpuqq “ qpuq ´ ppuq, and is nothing but the standard sequence that appeared
in the example, and as we checked in there, it holds that OX pXq “ Ker d1 “ krus.
More strikingly, H 1 pX, OX q, i.e. theÀ
cokernel of the map krus ‘ krus Ñ kru, u´1 s is
rather big. It equals kru, u s{krus “ ią0 k u´i , so that H 1 pX, OX q is not even finite-
´1

dimensional as a k -vector space. This gives another proof that X is not isomorphic to an
affine scheme. △

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The long exact sequence

Proving item (ii) in Theorem 18.17 is not so difficult. Consider a short exact sequence of
quasi-coherent sheaves

0 F1 F F2 0.
In Corollary 14.12 we proved that whenever the U “ Spec A is an open affine in X , the
sequence
0 F 1 pU q FpU q F 2 pU q 0 (18.13)

is exact. This means that if an affine cover U “ tUi uiPI has the property that each intersection
Ui0 X¨ ¨ ¨XUip is affine, as taking products do not disturb exactness, there is an exact sequence

0 C p pU, F 1 q C p pU, Fq C p pU, F 2 q 0,

and consequently the sequence of Cech complexes

0 C ‚ pU, F 1 q C ‚ pU, Fq C ‚ pU, F 2 q 0

is also exact. We are therefore in position to apply Lemma 18.1 to obtain a long exact
sequence of Cech cohomology groups

¨¨¨ H i pU, F 1 q H i pU, Fq H i pU, F 2 q ¨¨¨ .

18.6 Independence of the cover

Let us embark on the proof of item (iv). Let U “ tU1 , . . . , Ur u and V “ tV1 , . . . , Vs u
be two finite affine covers, and form the following group of sections over all the mixed
intersections:
ź
C m,n “ ΓpUI X VJ , Fq
|I|“m,|J|“n

Note that for m fixed

ź
C m,‚ » C ‚ pUI X VJ , F|UI q
|I|“m

is the Cech complex of F|UI with respect to the cover Vj X UI . Likewise,

ź
C ‚,n » C ‚ pVj X Ui , F|UI q
|J|“n

is the Cech complex of F|VJ with respect to the cover UI X VJ

One says that C m,n forms a double complex. It has two differentials, one written d in
the ‘rightwards’ direction, and one in the ‘upwards’ direction, δ . These are induced by the
differentials of the two Cech complexes above. The figure below illustrates this.
The key point is that the intersections UI X VJ are affine, as X is separated. That means
that all the higher cohomology groups in each direction are zero, i.e., the complexes C n,‚
and C ‚,m are exact in degrees ě 1.

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In degree 0, the cohomology group of the complex C m,‚ is equal to

ź
H 0 pC m,‚ q “ ΓpUI , Fq “ C m pU, Fq
|I|“m

and likewise,
ź
H 0 pC ‚,n q “ ΓpVJ , Fq “ C n pV, Fq
|J|“n

To show the independence of the cover, i.e., item (iv), we want to show that the cohomology
of these two complexes are equal. This is a formal consequence of the following fact from
homological algebra:

Lemma 18.21. Let C n,m be a double complex with H i pC m,‚ q “ H i pC ‚,n q “ 0 for all
m, n ě 1. Then Am “ H 0 pC m,‚ q and B n “ H 0 pC ‚,n q are complexes and there is a
canonical isomorphism between their cohomology:
H i pA‚ q “ H i pB ‚ q.

Proof We augment the double complex by adding Ai “ KerpC i,0 Ñ C i,1 q and B i “
KerpC 0,i Ñ C 1,i q to get the diagram below.
d d d
B2 C 0,2 C 1,2 C 2,2 ¨¨¨
δ δ δ δ

d d d
B1 C 0,1 C 1,1 C 2,1 ¨¨¨
δ δ δ δ

d d d
B0 C 0,0 C 1,0 C 2,0 ¨¨¨

d d d
A0 A1 A2 ¨¨¨
‚ ‚
Now all rows and columns are exact except along A and B . One now applies a diagram
chase to construct a map
H i pA‚ q ÝÝÑ H i pB ‚ q.
To see how this works, consider the case i “ 1, and the diagram

B1 C 0,1 C 1,1 Dc3 P B 1 c2 0

B0 C 0,0 C 1,0 C 2,0 Dc1 c 0

A0 A1 A2 c P A1 0
Starting with an element c P A1 so that dpcq “ 0, then send it to c P C 1,0 , which by
commutativity of the diagram must map to 0 in C 2,0 . By exactness, we may therefore lift
c to an element c1 P C 0,0 , which in turn maps to an element c2 P C 0,1 . By commutativity

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of the diagram, c2 maps to zero in C 1,1 . Hence by exactness, there is an c3 P B 1 such that
dpc3 q “ c2 . The element c3 satisfies δpc3 q “ 0 because δpc2 q “ 0, and hence it defines an
element in H 1 pB ‚ q. One checks that this assignment is independent of the choices of lifts,
and that the map H 1 pA‚ q Ñ H 1 pB ‚ q is a map of groups. By a symmetric argument, one
constructs a map H 1 pB ‚ q Ñ H 1 pA‚ q which defines the inverse.
For more details of this argument, see Exercise 18.19.19. For a full proof, see (?, p. 64).

In particular, we get independence of H p pU, Fq for any affine covering on a Noetherian

separated scheme.

18.7 Pushforwards and cohomology

The following lemma is very useful, and will be applied several times in the book. Given an
affine morphism f : X Ñ Y , it will allow us to transport computations of cohomology of
sheaves on X to computations on Y , at the cost of replacing F with f˚ F .

Lemma 18.22. Let f : X Ñ Y be an affine morphism of Noetherian, separated schemes.

Then for each quasi-coherent sheaf F on X , and i ě 0, we have a canonical isomorphism
H i pX, Fq “ H i pY, f˚ Fq. (18.14)

Proof Let V “ tVi u be a finite affine covering of Y such that H i pX, f˚ Fq is computed by
the Čech complex C ‚ pVi , f˚ Fq. Note that the latter complex equals C ‚ pf ´1 Vi , Fq. As f is
affine, the affine subsets f ´1 pUi q forms an affine covering U of X , and the lemma follows
simply because the Cech complexes computing the two sides of (18.14) are the same.

Example 18.23. If ι : Y Ñ X is a closed embedding, we have

H i pY, Fq “ H i pX, ι˚ Fq.

In particular, H i pY, OY q “ H i pX, ι˚ OX q. △

18.8 Cohomology and dimension

The next result is another ‘vanishing theorem’ for cohomology groups. It is a general result,
due to Grothendieck, that the cohomology groups vanish above the dimension of X , at least
for spaces X that are Noetherian and the dimension is interpreted as the Krull dimension.

Theorem 18.24 (Grothendieck). Let X be a Noetherian topological space and let F be

a sheaf. Then
H i pX, Fq “ 0 for all i ą dim X

We will contend ourselves to proving this in the case when X is a quasi-projective scheme
over a ring A. For the general case, see (Stacks Project Authors, 2018, XXX) or (Godement,
1960, Theorem 4.5.12).

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Theorem 18.25. Let X be a quasi-projective scheme of finite type over a Noetherian

ring A of dimension n. Then X admits an open cover U consisting of at most n ` 1
affine open subsets. In particular, for any quasi-coherent sheaf on X ,
H p pX, Fq “ 0 for p ą n.

Proof We may write X “ X ´ W where X, W Ă PrA are closed subschemes, and we

may assume that no irreducible component of X is contained in W (simply by discarding
such components).
We proceed by induction on n to prove that X can be covered by n ` 1 open affines
induced from open affines in PrA .
Let IX and IW denote the homogeneous ideals of X and W in Arx0 , . . . , xn s respectively.
Let p1 , . . . , ps denote the minimal primes of IX , which correspond to the irreducible compo-
nents of X . By prime avoidance there is an f P IW such that f R pi for any i “ 1, . . . , s.
Let H “ V pf q Ă PnA be the corresponding hypersurface. Note that Pr ´ H “ D` pf q is an
affine scheme and hence so is X ´ H “ X X D` pf q.
By construction X ´ H Ă X ´ W “ X and H Č V ppi q for any i by the choice of
f . Therefore dimpYi X Hq ă dim Yi , so we may use induction on the dimension to cover
Y X H by at most n open affines, all induced from the ambient projective space, which
together with Dpf q gives a covering of X with n ` 1 open affine subsets. This proves the
first claim.
For the second, note that in a Cech complex built on a covering consisting of at most
n ` 1 affines open subsets, terms C p pX, Fq with p ą n will vanish, from which follows
that 0 “ H p pU, Fq “ H p pX, Fq for each F and each p ą n.

18.9 Cohomology of sheaves on projective space

In Examples 18.10 and 18.11 we computed the cohomology groups of the sheaves OP1k pmq.
For m ě 0, we found that H 0 pP1k , OP1k pmqq could be identified with the space of homoge-
neous polynomials of degree m, and H 1 pP1k , OP1k pmqq “ 0. On the other hand, for m ď ´2,
H 0 pP1k , OP1k pmqq “ 0, while H 1 pP1k , OP1k pmqq was non-zero.
We will now carry out a more general computation of the cohomology groups of OPnA pmq
for any projective space PnA over a ring A. The strategy is however the same: we have a
distinguished cover U of open sets D` pxi q, and we use the Cech complex associated to this
cover to compute the cohomology.
Write R “ Arx0 , . . . , xn s. Then the groups in the Cech complex are
ź
C 0 pU, OPnA pmqq “ pRxi qm
0ďiďn
ź ` ˘
1
C pU, OPnA pmqq “ Rxi xj m
0ďiăjďn

..
.
n
C pU, OPnA pmqq “ pRx0 x1 ¨¨¨xn qm

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

and the Cech complex takes the form

d0 d1 ˘ d2
ź ź ` ˘ ź `
pRxi qm ÝÑ Rxi xj m
ÝÑ Rxi xj xk m ÝÑ ¨ ¨ ¨ (18.15)
0ďiďn 0ďiăjďn 0ďiăjăkďn

where the maps are as usual composed of alternating sums of localization maps.
In degree 0, we recover the following isomorphism:

H 0 pPnA , OPnA pmqq “ Ker d0

“ Arx0 , . . . , xn sm .

Example 18.26. The Cech complex of OP1A pmq on P1A takes the form
„ ȷ „ ȷ „ ȷ „ ȷ
x1 m x0 m x1 x0 m x1 x0 m
0ÑA x ˆA x ÝÝÑ A , x “A , x Ñ0
x0 0 x1 1 x0 x1 0 x0 x1 1

The cohomology group H 1 pU, OP1A pmqq can be identified with the cokernel of the map d0 .
For m ě 0, this group is zero, whereas for m ă 0, it is generated as an A-module by the
m`1 m`2 m`1
monomials x´1
0 x1 , x´2
0 x1 , . . . , x´1
0 x1 . In other words,

H 1 pU, Opmqq “ px´1 ´1 ´1 ´1

0 x1 Arx0 , x1 sqm .

The next fundamental theorem gives similar formulas for higher-dimensional projective
spaces.

Theorem 18.27 (Cohomology of sheaves on Pn ). Let PnA “ Proj Arx0 , . . . , xn s, where

A is a ring.
(i) For each m P Z,
H 0 pPnA , OPnA pmqq “ Arx0 , . . . , xn sm .
(ii) For each 0 ă p ă n and m P Z,
H p pPnA , OPnA pmqq “ 0.
(iii) For each m P Z, we have
` ˘
H n pPnA , OPnA pmqq “ x´1 ´1 ´1 ´1
0 ¨ ¨ ¨ xn Arx0 , . . . , xn s m (18.16)
In particular, there is a canonical isomorphism
H n pPnA , OPnA p´n ´ 1qq » A.

Proof We have already shown (i). To prove (iii), observe that

C n pU, OPnA pmqq “ pArx0 , . . . , xn sx0 ¨¨¨xn qm

is a free graded A-module spanned by monomials of the form xe00 ¨ ¨ ¨ xenn of degree ei “ m.
ř
The image of dn´1 is spanned by such monomials where at least one ei is non-negative.

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Therefore,
H n pX, OPnA pmqq “ Coker dn´1
! ˇ ÿ )
e0 en ˇ
“ A x0 ¨ ¨ ¨ xn ei ă 0 for every i and ei “ m
` ´1 ´1 ´1 ´1
˘
“ x0 ¨ ¨ ¨ xn Arx0 , . . . , xn s m
In degree m “ ´n ´ 1, there is only one such monomial, namely x´1 ´1
0 ¨ ¨ ¨ xn , so

H n pPnA , OPnA p´n ´ 1qq “ A ¨ x´1 ´1

0 ¨ ¨ ¨ xn .

To prove (ii), we use induction on n. For n “ 0 and n “ 1, there is nothing to prove. Let
n ě 2, and write R “ Arx0 , . . . , xn s. Rewriting to single out the variable xn , we express
the Cech complex as the degree m part of the following complex:
n´1
ź ź ź
C‚ : 0Ñ Rxi ˆ Rxn Ñ ¨ ¨ ¨ Ñ RxI ˆ RxI xn Ñ ¨ ¨ ¨ (18.17)
i“0 |I|“p |I|“p´1
ISn ISn

To finish the proof, it is enough to prove that H p pC ‚ q “ 0 for p “ 1, . . . , n ´ 1.

If we localize the complex with respect to xn , we see that C ‚ embeds as a subcomplex of
the following complex
n´1
ź ź ź
D‚ : 0Ñ Rxi xn ˆ Rxn Ñ ¨ ¨ ¨ Ñ RxI xn ˆ RxI xn Ñ ¨ ¨ ¨ (18.18)
i“0 |I|“p |I|“p´1
ISn ISn

We recognize this as the same complex as the one given by the Cech complex of ODpxn q on
Dpxn q “ SpecpRxn q associated to the covering with n ` 1 open affines
Dpx0 xn q, . . . , Dpxn´1 xn q, Dpxn q.
In particular, since Dpxn q is affine, the complex (18.18) is exact in all positive degrees, i.e.,
H p pD‚ q “ 0 for all p ą 0.
Let S “ Arx0 , . . . , xn´1 s. Decomposing the elements in C ‚ and D‚ according
À to their xn -
degree, we see that the quotient complex E ‚ “ D‚ {C ‚ decomposes as E ‚ “ lą0 El‚ ¨ x´l n ,
‚
and El is the complex
n´1
ź ź ź
0Ñ Sxi ˆ 0 Ñ ¨ ¨ ¨ Ñ SxI ˆ 0 Ñ ¨¨¨ , (18.19)
i“0 |I|“p |I|“p´1
ISn ISn

Taking the long exact sequence of cohomology associated to the sequence

0 Ñ C ‚ Ñ D‚ Ñ E ‚ Ñ 0, (18.20)
we obtain the exact sequence
0 Ñ H 0 pC ‚ q Ñ H 0 pD‚ q Ñ H 0 pE ‚ q Ñ H 1 pC ‚ q Ñ 0
and isomorphisms
H p pC ‚ q » H p´1 pE ‚ q for p ą 1. (18.21)

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

The first sequence takes the form

à 1
0 Ñ R Ñ Rxn Ñ S ¨ x´l ‚
n Ñ H pC q Ñ 0
lą0
À
Here the map Rxn Ñ lą0 S ¨ is clearly surjective, so we deduce that H 1 pC ‚ q “ 0.
x´l
n
On the other hand, disregaring the zeroes in (18.19), we see that the degree-m part of the
complex El‚ is exactly the Cech complex of OPAn´1 pmq on Pn´1 A . Therefore, by induction, the
complex E ‚ is exact in degrees p “ 1, . . . , n ´ 2 and so (18.21) implies that H p pC ‚ q “ 0
for p “ 2, . . . , n ´ 1, as well. This completes the proof of (ii).
The proof of Theorem 18.27 also an important duality between H 0 and H n on Pn . To
state the result, we recall the following definition:

Definition 18.28. Let A be a ring and let M and N be A-modules. A bilinear map
M ˆ N Ñ A is a called a perfect pairing if the induced map M ÞÑ HomA pN, Aq is an
isomorphism.

Consider the pairing of A-modules defined by

` ˘
p , q : Arx0 , . . . , xn s ˆ x´1 ´1 ´1 ´1
0 ¨ ¨ ¨ xn Arx0 , . . . , xn s Ñ A, (18.22)
sending a pair of Laurent polynomials pp, qq to the coefficient of . . . x´1 x´1
n in the product
0
pq . Note that this mapping is A-linear in each factor. In terms of the standard monomial basis,
we have #
1 if di ` ei “ ´1 for all i
pxd00 . . . xdnn , xe00 . . . xenn q “
0 otherwise
This pairing is perfect and allows us to canonically identify
` ´1 ˘
x0 ¨ ¨ ¨ x´1 ´1 ´1
n Arx0 , . . . , xn s d “ HomA pArx0 , . . . , xn s´n´1´m , Aq.

This allows us to regard the n-th cohomology group H n pPnA , Opmqq as the dual of a corre-
sponding H 0 :

Corollary 18.29 (Serre duality for Pn ). For each m P Z, there is a canonical isomor-
phism
H n pPnA , Opmqq “ HomA pH 0 pPn , Op´m ´ n ´ 1qq, Aq. (18.23)

When A “ k is a field, the dimensions of the cohomology groups are easily computed:

Corollary 18.30. Let k be a field. Then for m ě 0

˙ ˆ
0 m`n
dimk H pPnk , OPnk pmqq
“
n
ˆ ˙
n n m´1
dimk H pPk , OPnk p´mqq “ .
n
All other cohomology groups are 0.

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18.10 Cohomology of sheaves on projective schemes

By the results of the previous section, the cohomology groups of Opmq on Pnk over a field k
are always finite-dimensional k -vector spaces. This is part of a more general result, saying
that on projective schemes of finite type over a ring, the cohomology groups of quasi-coherent
sheaves of finite type are always k -vector spaces of finite dimension. Note that this is
definitely not the case for affine schemes: even the H 0 of the structure sheaf on A1k is infinite
dimensional, as it equals krts.

Theorem 18.31 (Serre). Let X Ă PnA be a projective scheme of finite type over a
Noetherian ring A and let F be an OX -module of finite type. Then:
(i) For each i, the cohomology group H i pX, Fq is a finite A-module.
(ii) There exists an m0 ą 0 such that
H i pX, Fpmqq “ 0.
for all m ě m0 and i ą 0.

Proof Let ι : X Ñ PnA denote the closed embedding and consider the sheaf ι˚ F . Since ι is
finite, the sheaf ι˚ F is again of finite type (Exercise 14.12.35) and
H i pX, Fpdqq “ H i pX, F bOX ι˚ OPn pdqq
“ H i pPn , ι˚ pF bOX ι˚ OPn pdqqq
“ H i pPn , ι˚ F bOPn OPn pdqq
“ H i pPn , pι˚ Fqpdqq. (18.24)
Therefore we reduce to the case X “ PnA .
Note that both parts of the theorem are trivially satisfied if i ą n because the cohomology
groups are zero in this case (PnA can be covered by n ` 1 affines). The proof will take this as
the base case and proceed by downwards induction on i.
By Theorem 16.25, any finite type OPnA -module is of the form M Ă for some finitely graded
module M over R “ Arx0 , . . . , xn s. À
(i): As M is finitely genenerated, we may pick a graded surjection j Rp´aj q Ñ M
for M . Letting K be the kernel, we have an exact sequence of finitely generated graded
R-modules
à
0 ÝÝÑ K ÝÝÑ Rp´aj q ÝÝÑ M ÝÝÑ 0
i

Applying tilde, we have an exact sequence

à
0 ÝÝÑ K ÝÝÑ OPnA p´aj q ÝÝÑ F ÝÝÑ 0.
j

If we take the long exact sequence of cohomology, we get

à i n
¨ ¨ ¨ Ñ H i pPnA , Kq Ñ H pPA , OPnA p´aj qq Ñ H i pPnA , Fq Ñ H i`1 pPnA , Kq Ñ . . .
j

By induction on i, the cohomology group H i`1 pPnA , Kq is a finitely generated A-module,

and the same holds for each H i pPnA , OPnA p´aj qq by Theorem 18.27. H i pPnA , Fq is therefore

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

squeezed between two finitely generated A-modules, so by exactness, it must itself be finitely
generated.
(ii): Twist the above sequence by OPnA pmq and take the long exact sequence in cohomology
to get
à
H i pPnA , OPnA pm ´ aj qq ÝÝÑ H i pPnA , Fpmqq ÝÝÑ H i`1 pPnA , Kpmqq
j

Again, by downward induction on i, and the fact that H i pPnA , OPnA pm ´ aj qq “ 0 for all
i ą 0 and m ą aj , we find that H i pPnA , Fpmqq “ 0.

Corollary 18.32. Let f : X Ñ Y be a projective morphism of Noetherian schemes and

let F be an OX -module of finite type. Then f˚ F is of finite type on Y .

Proof It suffices to consder the case when Y “ Spec A is affine. The sheaf f˚ F is in any
case quasi-coherent by Theorem 14.33. In fact, f˚ F “ ΓpX, Č Fq, so it suffices to show that
0
ΓpX, Fq “ H pX, Fq is finitely generated as an A-module. This is clear if f is a closed
embedding X Ñ PnA , because f is finite and f˚ F is of finite type on PnA (Exericse 14.12.35).
It is also clear if f is the projection PnA Ñ Spec A, by item (i) in the above theorem. Now the
general case follows by factoring f as a closed embedding followed by the projection.

The Euler characteristic

If X is a projective scheme of finite type over a field k and F is an OX -module of finite type,
Serre’s theorem tells us that the cohomology groups H i pX, Fq are finite-dimensional k -
vector spaces. In particular, we can ask about their dimensions. It turns out that the alternating
sum of these dimensions has very good formal properties, so we make the following definition:

Definition 18.33. Let X be a projective scheme of finite type over a field k . We define
the Euler characteristic of F as
ÿ
χpFq “ p´1qk dimk H k pX, Fq.
kě0

Note that the sum is well-defined, as there are only finitely many non-zero cohomology
groups appearing on the right-hand side.

Proposition 18.34. The Euler characteristic χ is additive on exact sequences, i.e., if

0 Ñ F 1 Ñ F Ñ F 2 Ñ 0 is an exact sequence of OX -modules of finite type, then
χpFq “ χpF 1 q ` χpF 2 q.

Proof This follows because

ř if 0 Ñ V0 Ñ V1 Ñ ¨ ¨ ¨ Ñ Vn Ñ 0 is an exact sequence of
k -vector spaces, then i p´1qi dimk V “ 0. Applying this to the long exact sequence in
cohomology gives the claim.
` ˘
Example 18.35. Let X “ Pnk and F “ Opdq for d ě 0. Then dimk H 0 pPnk , Fq “ n`d n
and all of the higher cohomology groups are zero. In the case when d ă 0, only H n pX, Fq

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Corollary 18.36. Let X Ă Pnk be a projective scheme of finite type over k and let Op1q
be the Serre twisting sheaf. Then the function
PF pmq “ χpFpmqq
is a polynomial in m, and for large m, PF pmq “ H 0 pX, Fpmqq.

This polynomial is called the Hilbert polynomial of F . While χpFq is an intrinsic invariant
of F , the Hilbert polynomial is not, as it depends on the choice of embedding X Ă Pnk .
When F “ M Ă for a graded module M , PF pmq coincides with the usual Hilbert polyno-
mial of M as defined in commutative algebra.

18.11 Example: Plane curves

Let X “ V pf q Ă P2k be a plane curve, defined by an homogeneous polynomial f px0 , x1 , x2 q
of degree d. Let us compute the groups of the structure sheaf H i pX, OX q. We have the ideal
sheaf sequence
0 ÝÝÑ IX ÝÝÑ OP2 ÝÝÑ i˚ OX ÝÝÑ 0
1 Recall that tensoring by a locally free sheaf preserves exactness.

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

where the ideal sheaf IX is the kernel of the restriction OP2 Ñ i˚ OX . By Example 16.33,
IX » OP2 p´dq, and the sequence can be rewritten as

0 ÝÝÑ OX p´dq ÝÝÑ OP2 ÝÝÑ i˚ OX ÝÝÑ 0 (18.25)

From the short exact sequence, we get the long exact sequence as follows:

0 H 0 pP2 , Op´dqq H 0 p P 2 , OP2 q H 0 pX, OX q

H 1 pP2 , Op´dqq H 1 p P 2 , OP2 q H 1 pX, OX q

H 2 pP2 , Op´dqq H 2 p P 2 , OP2 q 0.

Using the results on cohomology of line bundles on P2 , we deduce the equality H 0 pX, OX q »
k and hence
H 1 pX, OX q » k pd´1qpd´2q{2 .

The dimension of the cohomology group on the left is the genus of the curve X (it will be
introduced properly in Chapter 21). So the above can be rephrased as saying the genus of a
plane curve of degree d is pd ´ 1qpd ´ 2q{2.
Tensoring the sequence (18.25) by OP2 pmq, we obtain

0 ÝÝÑ OP2 pm ´ dq ÝÝÑ OP2 pmq ÝÝÑ i˚ OX pmq ÝÝÑ 0

and the long exact sequence gives that the Hilbert polynomial of OX equals
ˆ ˙ ˆ ˙
m`2 m´d`2 d2 ´ 3d
P pmq “ ´ “ dm ´ .
2 2 2

Example 18.37. A plane curve of degree 1, i.e., a projective line, is isomorphic to P1k , and
the genus is 0 in accordance with the above result. △

Example 18.38. Suppose k is a field of characacteristic ‰ 2 and assume there is an i P k

with i2 “ ´1. Then the curve

X “ V px20 ` x21 ` x22 q Ă P2k

is isomorphic to P1k . This follows because the variable change u1 “ x1 , u0 “ x0 ` ix2 ,

u2 “ x0 ´ ix2 , brings the equation into the form u21 ´ u0 u2 “ 0, and so the image is
isomorphic to P1k embedded into P2k via the standard embedding ps : tq Ñ ps2 : st : t2 q.
A plane curve of degree 3, i.e., an elliptic curve, has genus 1. It follows for instance that
the curve
X “ V px30 ` x31 ` x32 q Ă P2k

is not isomorphic to P1k , over any field. △

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18.12 Example: The twisted cubic

Let k be a field and consider P3 “ Proj R where R “ krx0 , x1 , x2 , x3 s. We will continue
Example 16.35 and consider the twisted cubic curve X “ V pIq where I Ă R is the ideal
generated by the 2 ˆ 2-minors of the matrix
ˆ ˙
x0 x1 x2
M“ .
x1 x2 x3
Let us compute the group H 1 pX, OX q. Of course we know what the answer should be,
because X » P1 , and H 1 pP1 , OP1 q “ 0.
Now, to compute H 1 pX, OX q on X , it is convenient to relate it to a cohomology group on
P . We have H 1 pX, OX q “ H 1 pP3 , i˚ OX q where i : X Ñ P3 is the inclusion. The sheaf
3

i˚ OX fits into the ideal sheaf sequence

0 ÝÝÑ I ÝÝÑ OP3 ÝÝÑ i˚ OX Ñ 0
where I is the ideal sheaf of X in P3 . Applying the long exact sequence in cohomology, we
get

¨¨¨ H 1 pP3 , Iq H 1 pP3 , OP3 q H 1 pP3 , i˚ OX q

H 2 pP3 , Iq H 2 pP3 , OP3 q ¨¨¨

By our description of sheaf cohomology on P3 , H 1 pP3 , OP3 q “ H 2 pP3 , OP3 q “ 0, which

implies that H 1 pX, OX q “ H 2 pP3 , Iq. We can compute the latter cohomology group using
the exact sequence of Example 16.35:

0 Ñ OP3 p´3q2 Ñ OP3 p´2q3 Ñ I Ñ 0.

Now, taking the long exact sequence we get

¨¨¨ H 2 pP3 , Op´3q2 q H 2 pP3 , OP3 p´2q3 q H 2 pP3 , Iq

H 3 pP3 , Op´3q2 q H 3 pP3 , OP3 p´2q3 q H 3 pP3 , Iq.

Here H 2 pP3 , Op´2qq “ 0 and H 3 pP3 , Op´3qq “ 0 by our previsous computations. Hence
by exactness, we find H 2 pP3 , Iq “ 0. It follows that H 1 pX, OX q “ 0 also, as expected.

18.13 Example: Hyperelliptic curves

Let us recall the hyperelliptic curves defined in Chapter 5. For simplicity, we work over an
algebraically closed field k and consider hyperelliptic curve associated to the polynomial
f pxq in krxs. More precisely, if the degree is n “ 2g ` 1 or n “ 2g ` 2, we consider the
scheme X glued together by the affine schemes U “ Spec A and V “ Spec B , where
krx, ys kru, vs
A“ and B “ 2 .
py 2 ´ f pxqq pv ´ u2g`2 f puqq

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and before, we glue Dpxq Ă U to Dpuq Ă V using the identifications u “ x´1 and
v “ x´g´1 y .
Let us compute the Cech cohomology groups of OX with respect to the affine covering
U “ tU, V u. As U has only two elements, the Cech complex has only two terms, OX pU q ˆ
OX pV q “ A ˆ B and OX pU X V q “ Ax . To simplify the computation, we use the relation
y 2 “ f pxq to decompose A as a krxs-module as
krx, ys
» krxs ‘ krxsy
py 2´ f pxqq
and similarly B » krus ‘ krusv as a krus-module. With these identifications, the Cech
complex takes the form
` ˘ d
pkrxs ‘ krxsyq ‘ krx´1 s ‘ krx´1 sx´g´1 y ÝÝÑ krx˘1 s ‘ krx˘1 sy,
where the differential d is given by

d0 pppxq ` qpxqy, rpx´1 q ` spx´1 qx´g´1 yq

“ ppxq ´ rpx´1 q ` pqpxq ´ spx´1 qx´g´1 qy.
Comparing monomials on each side, we deduce that
H 0 pX, OX q “ Ker d0 “ k
and
H 1 pX, OX q “ Coker d0 “ k x´1 y ‘ ¨ ¨ ¨ ‘ k x´g y » k g .
For g “ 2, we get a particularly interesting curve: it is an irreducible projective curve which
cannot be embedded in P2 . Indeed, we showed that for any irreducible curve in P2 of degree d,
the corresponding cohomology group H 1 pX, OX q has dimension equal to 12 pd ´ 1qpd ´ 2q,
but this does not take the value 2 if d is an integer.

Proposition 18.39. There exist nonsingular projective curves which cannot be embedded
in the projective plane P2 .

We still haven’t yet proved that X is projective. This follows because X can be embedded
into the weighted projective space Pp1, 1, g ` 1q “ Proj krx0 , x1 , ws given by the equation
w2 “ a2g`2 x2g`2
0 ` a2g`1 x2g`1
0 x1 ` ¨ ¨ ¨ ` a0 x2g`2
1 . (18.26)
Note that this makes sense because w has degree g ` 1; (18.26) does not define a subscheme
of P2k . Then the two open sets U and V are isomorphic to the distinguished opens D` px0 q
and D` px1 q respectively. The weighted projective plane Pp1, 1, g ` 1q is a projective variety,
and hence so is X .

18.14 Example: Complete intersections

Let k be a field and let F and G be two homogeneous polynomials in krx0 , . . . , xn s of
degrees d and e respectively. Let X be the closed subscheme of Pnk defined by I “ pF, Gq.
Let us use the exact sequence (16.16) to compute the cohomology groups of X .

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0 ÝÝÑ I ÝÝÑ OPnk ÝÝÑ ι˚ OX ÝÝÑ 0.

Note that H 0 pPnk , Iq “ ΓpPnk , Iq “ 0, as there are no nonconstant global sections which
vanish along X . Taking the long exact sequence, we get that
‚ dimk H 0 pX, OX q “ dimk H 1 pPnk , Iq ` 1,
‚ dimk H p pX, OX q “ H p`1 pPnk , Iq for p “ 1, . . . , n ´ 1
‚ H n pPnk , Iq “ H n´1 pPnk , ι˚ OX q “ H n pX, OX q “ 0, because X has dimension n ´ 2.
We proceed to study the cohomology groups H p pPnk , IZ q. Recall the exact sequence from
Section 16.34,
0 Ñ OPnk p´d ´ eq Ñ OPnk p´dq ‘ OPnk péq Ñ IZ Ñ 0.
Taking the long exact sequence of cohomology, we obtain that
H p pPnk , Iq “ 0
for p “ 0, . . . , n ´ 2, and
0 Ñ H n´1 pIq Ñ H n pOPnk p´d ´ eqq Ñ H n pOPnk p´dqq ‘ H n pOPnk péqq Ñ 0.
Therefore, we get
ˆ ˙ ˆ ˙ ˆ ˙
n´1 dè´1 d´1 e´1
dimk H pX, OX q “ ´ ´ . (18.27)
n n n
Example 18.40. When n “ 3, X Ă P3k is a curve, with H 1 pX, OX q of dimension
ˆ ˙ ˆ ˙ ˆ ˙
dè´1 d´1 e´1 1
´ ´ “ depd ` e ´ 4q ` 1
3 3 3 2
For instance, when d “ 1, e “ 2, X is a conic and we recover h1 pX, OX q “ 0. △

18.15 Example: Bezout’s theorem

Let k be an algebraically closed field. Let C and D be two curves in P2k defined by homo-
geneous polynomials F and G of degrees d and e respectively, and assume that C and D
have no common component, so that closed subscheme Z “ V pF, Gq is a 0-dimensional
subscheme. By a linear coordinate change, we may assume Z “ tx1 , . . . , xr u is contained
in Dpx0 q » krx, ys. Then the classical form of Bezout’s theorem says that
r
ÿ
multxi pC, Dq “ de, (18.28)
i“1

where ˆ ˙
krx, ys
multxi pC, Dq “ dimk OZ,xi “ dimk
pf, gq mxi

is the intersection multiplicity of C and D at xi , and f and g are the dehomogenizations of

F and G.

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There is a quick proof of the formula (18.28) by computing the cohomology group
0
H pZ, OZ q in two different ways. First of all, this group is the space of global sections
OZ pZq, so it decomposes as
r r
ˆ ˙
à à krx, ys
OZ pZq “ OZ,xi “ (18.29)
i“1 i“1 pf, gq mx
i

where f, g are the dehomogenized equations for C and D. Therefore,

r ˆ ˙
0
ÿ krx, ys
dimk H pZ, OZ q “ dimk .
i“1
pf, gq mx
i

On the other hand, from (18.27), we find dimk H 0 pZ, OZ q “ dimk H 1 pP2k , IZ q ` 1 and
hence
ˆ ˙ ˆ ˙ ˆ ˙
0 d`e´1 d´1 e´1
dimk H pZ, OZ q “ ´ ´ ` 1 “ de.
2 2 2

18.16 Example: Interpolation problems

Let k be a field and consider projective space Pnk . If Z is a subscheme of Pnk , it is natural to
ask: How many hypersurfaces of degree d contain Z ? The precise answer to this involves
studying the cohomology group
H 0 pPnk , IZ pdqq (18.30)
where IZ is the ideal sheaf defining Z . Indeed, this group is exactly the degree d part of the
homogeneous ideal IZ defining Z , that is, the space of homogeneous polynomials of degree
d vanishing on Z .
While we are mostly interested in the global sections of the sheaf IZ pdq, we can get
information about this space using the higher cohomology groups as well, using the ideal
sheaf sequence
0 Ñ IZ pdq Ñ OPnk pdq Ñ i˚ OZ pdq Ñ 0. (18.31)
In good cases, sufficiently many of these cohomology groups are zero, allowing us to compute
the rank of (18.30).
In the case when Z be a finite set of points p1 , . . . , pr in Pnk , (with the reduced scheme
structure), the cohomology groups of OPnk pdq and i˚ OZ pdq are all zero. Therefore, by
the long exact sequence to (18.31), IZ pdq have at most two non-zero cohomolgy groups,
H 0 pIZ pdqq and H 1 pIZ pdqq. If the latter group is zero, we are happy, because then we have
a short exact sequence
0 Ñ H 0 pIZ pdqq Ñ H 0 pOPnk pdqq Ñ H 0 pOZ pdqq » k r Ñ 0. (18.32)
so that ˆ ˙
0 n`d
dimk H pIZ pdqq “ ´r
d
We say that the points impose independent conditions on the sections of OPnk pdq; adding a
new point drops the dimension by exactly 1.

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Example 18.41. Consider the case of two points Z “ tp1 , p2 u in P2k and d “ 1. In this
case, the equation (18.32) tells us that there is a degree 1 polynomial vanishing on both of
the points, that is, the points determine a line L. Of course, we know that this line has to be
unique, but one can also prove it cohomologically as follows. We have an exact sequence
relating the ideal sheaves IZ|P2k , IZ|L and IZ|P2k :
0 ÝÝÑ IL|P2k ÝÝÑ IZ|P2k ÝÝÑ j˚ IZ|L ÝÝÑ 0
where j : L Ñ P2k is the inclusion of the line. If we identify L » P1k , we have IZ|L »
OP1k p´2q (the ideal sheaf of two points on P1 ). Moreover, IL|P2k » OP2 p´1q, so twisting the
above sequence by OP2k p1q, we get the sequence
0 ÝÝÑ OP2 ÝÝÑ IZ|P2k p1q ÝÝÑ j˚ OP1k p1q ÝÝÑ 0,
from which we deduce that H 1 pIZ|P2k p1qq “ 0, and hence H 0 pIZ p1qq has dimension 1. △
Example 18.42. Consider the case of three points Z “ tp1 , p2 , p3 u in P2k and d “ 1. Then
by an analysis similar to the previous example, we see that H 1 pIZ p1qq ‰ 0 if and only if
p1 , p2 , p3 lie on a line. △
In general, it can certainly happen that H 1 pIZ pdqq ‰ 0. In this case, the cohomology
group H 0 pPnk , IZ pdqq will be bigger than expected.

18.17 Example: Non-split locally free sheaves

A locally free sheaf is said to be split if it is isomorphic to a direct sum of invertible sheaves.
We have seen several examples of locally free sheaves that are not free, even on affine
schemes, but a priori it is not so clear whether these are direct sums of invertible sheaves. In
this section we will study the sheaf E from Section ?? and show that it is indeed non-split.
The sheaf E is the locally free sheaf of rank n on Pnk sitting in the exact sequence
0 ÝÝÑ OPnk p´1q ÝÝÑ OPn`1
n
k
ÝÝÑ E ÝÝÑ 0.
Suppose that E is split, i.e., E is isomorphic to a direct sum of invertible sheaves. Since
PicpPnk q “ Z is generated by the class of Op1q, this would mean that
E » OPnk pa1 q ‘ ¨ ¨ ¨ ‘ OPnk pan q
for some integers a1 , . . . an P Z.
Recall that for n ě 2, we have H n´1 pPnk , Opmqq “ 0 for any m P Z. So if we could
show that H n´1 pPnk , Eq ‰ 0, we would have a contradiction. Actually, it is the case that
H n´1 pPnk , Eq “ 0, but we can instead consider F “ Ep´nq, which fits into the sequence

0 ÝÝÑ OPnk p´n ´ 1q ÝÝÑ OPnk p´nqn`1 ÝÝÑ F ÝÝÑ 0.

Taking the long exact sequence in cohomology, we get
δ
¨ ¨ ¨ Ñ H n´1 pOPn`1
n pńqq Ñ H
k
n´1
Ý H n pOPnk pń´1qq Ñ H n pOPn`1
pFq Ñ n pńqq Ñ ¨ ¨ ¨
k

Here the two outer cohomology groups are zero, by Theorem 18.27. Hence, by exactness,

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we find that H n´1 pPnk , Fp´1qq » H 0 pPnk OPnk q “ k . Therefore, we find that F “ Ep´nq
is not split, and hence E is not split either.
This gives an example of a non-split locally free sheaf of rank n on Pnk . However, coming
up with examples of non-split sheaves of low rank on projective space is a famously difficult
problem. In fact, a famous conjecture of Hartshorne says that any rank 2 vector bundle on Pnk
for n ě 5 is split.

18.18 Example: The rational quartic curve

Let us consider the rational quartic curve X of Example 16.22. In other words, X Ă P3k “
Proj krx0 , x1 , x2 , x3 s is the closed subscheme defined by the ideal
I “ px1 x2 ´ x0 x3 , x32 ´ x1 x23 , x0 x22 ´ x21 x3 , x31 ´ x20 x2 q (18.33)
We showed in the previous example that the natural map
krx0 , x1 , x2 , x3 s ÝÝÑ Γ˚ pOX q (18.34)
is not surjective and that ΓpX, OX p1qq is 5-dimensional as a k -vector space. We may explain
this fact by looking at the cohomology of the ideal sheaf sequence
0 ÝÝÑ I ÝÝÑ OP3k ÝÝÑ ι˚ OX ÝÝÑ 0 (18.35)
More precisely, we claim that the map OP3k p1q Ñ pι˚ OX qp1q is not surjective on global
sections, and that H 1 pP3k , Ip1qq is 1-dimensional.
Using a computer, we find that the ideal has a graded resolution given by
U¨ V¨ W¨
0 Ñ Rp´5q ÝÑ Rp´4q4 ÝÑ Rp´2q ‘ Rp´3q3 ÝÝÑ I Ñ 0 (18.36)
where the matrices are given by
¨ ˛ ¨ 2 ˛
x3 ´x1 ´x0 x2 ´x1 x3 ´x22
˚´x2 ‹ ˚ x2 x3 0 0 ‹
U “˚ ˝´x1 ‚, V “ ˝ x0
‹ ˚ ‹
x1 ´x2 ´x3 ‚
x0 0 0 x0 x1
and W is the 4 ˆ 1-matrix with entries equal to the generators of I in (18.33). Note that
W ¨ V “ 0 and V U “ 0.
If we twist the sequence by 1 and apply tilde, we get the following exact sequence of
sheaves on P3k .
v
0 Ñ Op´4q Ñ Op´3q4 Ñ
Ý Op´1q ‘ Op´2q3 Ñ Ip1q Ñ 0 (18.37)
To compute the cohomology of Ip1q, we split this sequence into
0 Ñ Op´4q Ñ Op´3q4 Ñ Im v Ñ 0
0 Ñ Im v Ñ Op´1q ‘ Op´2q3 Ñ Ip1q Ñ 0
Taking the long exact sequence of the first sequence, we deduce that H 2 pP3k , Im vq “
H 3 pP3k , Op´4qq “ k . Then doing the same for the second, we see that H 1 pP3k , Ip1qq “ k .
The cohomology group H 1 pP3k , Ip1qq explains the failure of surjectivity of the map

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(18.34). The fact that it is 1-dimensional explains the missing monomial x2 y 2 in the definition
of X .

18.19 Exercises
Exercise 18.19.1. Generalize Example 18.13 to show that the curve V pxd2 ´ x0 xd´1
1 q Ă P2k
1 1
has an H pX, OX q of dimension 2 pd ´ 1qpd ´ 2q as a k -vector space.
Exercise 18.19.2. Let X “ S 1 and let U be the covering of X with three pairwise intersecting
open intervals with empty intersection. Show that the Cech complex is of the form
d0
Z3 ÝÑ Z3 Ñ 0.
Compute the map d0 and use it to verify again that H i pU, ZX q “ Z for i “ 0, 1 as above.
Exercise 18.19.3. Let X “ Ank ´ t0u be the complement of the origin.
a) Compute H i pX, OX q for all i.
b) Give a new proof that X is not an affine scheme for n ě 2.
Exercise 18.19.4. Complete the details of the proof of the ‘Zig-zag Lemma’.
Exercise 18.19.5. Fill in the details in the proof of Lemma 18.22.
Exercise 18.19.6. With reference to the twisted cubic example in Section 18.12, show that
‚ H 0 pP3 , Ip2qq “ k 3 (find a basis!)
‚ H 1 pP3 , Ipmqq “ 0 for all m P Z.
‚ H 2 pP3 , Ip´1qq “ k .
Exercise 18.19.7 (Künneth formula). Let X and Y be Noetherian schemes of finite type over
a field. Let F two two finite type quasi-coherent sheaves on X and Y respectively. Show that
there is a canonical isomorphism
à
H n pX ˆk Y, p˚1 F b p˚2 Gq » H p pX, Fq bk H q pX, Gq.
p`q“n

H INT: Use a Chech complex using products of affine opens.

Exercise 18.19.8. Let X Ă P5 denote a quadric hypersurface (i.e., X “ V pqq for a
homogeneous degree 2 polynomial). Recall the exact sequence 15.8
0 Ñ OP5 p´1q4 Ñ OP45 Ñ i˚ E Ñ 0
where E is a locally free sheaf of rank 2.
Use this exact sequence to show that E is not split.
Exercise 18.19.9. Let n ą 0 be an integer and consider the integral projective scheme
X “ ProjpRq, where R is the ring
R “ krx, y, z, ws{px2 , xy, y 2 , un x ´ v n yq.
a) Show that X is irreducible, non-reduced, and of dimension 1.
b) Compute H 0 pX, OX q and H 1 pX, OX q.

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Exercise 18.19.10. Let Yr Ă Pnk be a complete intersection of r hypersurfaces H1 , . . . , Hr

of degrees d1 , . . . , dr .
a) Show that there is an exact sequence of the form
0 ÝÝÑ OYr´1 p´Hr q ÝÝÑ OYr´1 ÝÝÑ OYr ÝÝÑ 0
b) The natural maps
H i pPnk , OPnk pdqq ÝÝÑ H i pYr , OYr pdqq
are isomorphisms for all i ă n ´ r and all d P Z.
c) Show that Yr is connected if dim Yr ě 1.
Exercise 18.19.11. Let E be a locally free sheaf of rank r on P1k .
a) Show that the set of integers n such that there is an injection Opnq Ă E is
non-empty and bounded from above.
b) Show that if n is maximal as in a) then E{Opnq is locally free of rank r ´ 1
c) Show that if r “ 2, and the maximal n is ´1, then E{Op´1q » Opmq for
some m ă 0.
d) Show that, in general, there exists a a filtration
e)
0 “ E0 Ă E1 Ă ¨ ¨ ¨ Ă Er “ E
such that Ei is locally free of rank i and Ei {Fi´1 » Opai q for some a1 ě a2 ě
¨ ¨ ¨ ě ar .
f) Show that for every E with a filtration as above, we have E “ Opa1 q ‘ ¨ ¨ ¨ ‘
Opar q and that the ai are uniquely determined by E .
g) Show that any locally free sheaf on P1k splits as a direct sum of invertible
sheaves.
Exercise 18.19.12. Let X be a projective scheme and let Y, Z be closed subschemes of X
with Z Ă Y . Write PY and PZ for the Hilbert polynomials of OY and OZ respectively.
Show that PZ pmq ď PY pmq for m " 0.
Exercise 18.19.13 (Length 2 subschemes). If Z is a 0-dimensional subscheme, then we
define the length of Z as the length of OZ pZq, which is an Artinian ring.
Use a cohomological argument to show that any 0-dimensional subscheme of length 2 in
P2 is a complete intersection of two curves of degree 1 and 2.
ˆ
Exercise 18.19.14. Let X be a scheme and let OX denote the sheaf of units (Exercise 9.9.23).
Show that there is a natural isomorphism
PicpXq “ H 1 pX, OX
ˆ
q
H INT: Invertible sheaves can be specified using units in OX pUi X Uj q.
Exercise 18.19.15. Continuing Exercise 17.11.3, show that the map from Cartier divi-
sors to PicpXq (induced by D ÞÑ OX pDq) can be identified with the connection map
H 0 pX, KpXqˆ {OX ˆ
q Ñ H 1 pX, OX
ˆ
q from the exact sequence
ˆ ˆ
0 ÝÝÑ OX ÝÝÑ KpXqˆ ÝÝÑÝÝÑ KpXq{OX ÝÝÑ 0.

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Exercise 18.19.16 (Double structures). Let X be a scheme such that Y “ Xred Ă X is

defined by an ideal sheaf I . Let us assume that I 2 “ 0.
ˆ
a) Show that there is an exact sequence 0 Ñ I Ñ OX Ñ OYˆ Ñ 0, where the
first map is f ÞÑ 1 ` f .
b) Assume that X is projective. Show that there is an exact sequence
0 Ñ H 1 pX, Iq Ñ PicpXq Ñ PicpY q Ñ H 2 pX, Iq
c) Compute the Picard group of the scheme
X “ Projpkrx, y, z, ws{px2 , xy, y 2 qq.
Exercise 18.19.17 (P2 minus a point). Consider X “ P2k ´ tp0 : 0 : 1qu, with the
affine covering given by U0 “ Spec krx, ys and U1 “ Spec krx´1 , x´1 ys.. Show that
H 0 pX, OX q “ k and
à
H 1 pX, OX q “ k ¨ xa y b .
bą´a, aă0

Deduce that P2k ´ tp0 : 0 : 1qu is not an affine scheme.

Exercise 18.19.18. Continuing the previous exercise, let C be the curve defined by y 3 `x2 `x
in U0 and px´1 yq3 ` x´2 ` x´1 in U1 . Show that H 0 pC, OC q “ k and
H 1 pX, OC q “ k ¨ x´1 y 2 .
More generally, if C is the curve defined by y d `xd´1 `x in U0 , and px´1 yqd `x´d`1 `x´1
in U1 , then
d´1
H 1 pC, O q » k p 2 q .
C

Exercise 18.19.19. Consider a double complex C m,n as in the text, with horizontal and
vertical coboundary maps d and δ . Assume that all rows C m,n are exact except when n “ 0,
À m “ 0.
and that all columns are exact except when
For each p, consider the group Z p Ă m`n“p C m,n of ‘zig-zags’ pcp,0 , cp´1,1 , . . . , c0,p q
such that
δpci,j q “ dpci´1,j`1 q @i ě 1, and δpc0,p q “ dpcp,0 q “ 0.
Consider the quotient Hi pCq “ Z i pCq{B i pCq where B i pCq Ă Z i pCq is the subgroup
generated by the ‘trivial zig-zags’
pdpcp´1,0 q, dpcp´2,1 q ` δpcp´1,0 q, . . . , dpc0,p´1 q ` δpc1,p´2 q, δpc0,p´1 qq,
a) Define maps αp : Hp pCq Ñ H p pA‚ q and βp : Hp pCq Ñ H p pB ‚ q.
b) Use a diagram chase similar to the one above to show that αp is surjective, and
then injective.
c) Prove Lemma 18.21.

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Proper and projective morphisms

19.1 Proper morphisms

Properness is an important property of morphisms serving as a replacement for ‘proper maps’
in topology. In topology, a map f is called proper if f ´1 pKq is compact for every compact
subset K Ă Y . In the setting of Hausdorff spaces X , one can show that X is compact if and
only if for any other topological space Z the projection map X ˆ Z Ñ Z is closed. (Here
we are using the topological fiber product.) More generally, a continuous map f : X Ñ Y
is proper if and only if for any map g : Z Ñ Y , the induced map X ˆY Z Ñ Z is closed.
In scheme theory, the latter property, along with a finiteness condition, is adopted as the
definition of proper maps.

Definition 19.1. A morphism f : X Ñ S is universally closed if it is closed and stays

closed when pulled back; that is, given any morphism T Ñ S , the pulled back map fT
in the Cartesian diagram below is closed.

X ˆS T X
fT f

T S
The morphism f is said to be proper if it is separated, of finite type and universally closed.
When S “ Spec A, we say that X is proper over A.

In Section 19.2, we will show that any morphisms f : X Ñ Y be between projective

schemes over a field k are proper. In particular, any projective scheme over k is proper
over Spec k . Admittedly, the main class of proper morphisms one meets in practice are the
projective ones1 . However, properness is a condition which is a bit more flexible and easier
than projectivity to check.

Example 19.2. A simple example of a morphism that is not proper, is the structure map
π : A1k Ñ Spec k . If we pull back π along itself, we get the projection map onto the first
factor A1k ˆk A1k “ A2k . In A2k “ Spec krx, ys there are lots of closed sets which project to
non-closed sets, for instance the ‘hyperbola’ V pxy ´ 1q, which maps to the non-closed set
A1k ´ t0u. △
1 There are non-projective varieties which are proper. Over C, the first examples were constructed by Hironaka in
1960.

392

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Example 19.3. The standard example of a proper morphism is π : PnA Ñ Spec A. While
this aligns with our intuition that ‘projective varieties compact, whereas affine varieties are
not’, this statement is a non-trivial fact, which we will prove in Theorem 19.8. △

Example 19.4. Closed embeddings are proper. Indeed they are universally closed by Propo-
sition 8.17 on page 167, separated by (i) of Proposition ?? on page ?? and of finite type by
definition. △

Proposition 19.5 (Properties of properness).

(i) Pullbacks of proper maps are proper.
(ii) Compositions of composable proper maps are proper.
(iii) The product f ˆ g of two proper maps is proper.
(iv) If f : X Ñ Y is a morphism, and tUi uiPI is a cover of Y so that each
f ´1 pUi q Ñ Ui is proper, then f is proper.
(v) If X is proper over a field k , and Y is separated over k , then any morphism
X Ñ Y over k is proper.

The proposition also holds if ‘proper’ is replaced by ‘universally closed’. In fact, we know
the corresponding statements hold for ‘separated’ and ‘finite type’, so we only need to check
it for ‘universally closed’.

Proof (i): Suppose that f : X Ñ Y is proper. Let g : Z Ñ Y be a morphism and considere

the base change morphism fZ : X ˆY Z Ñ Z . If h : T Ñ Z is any morphism, we consider
the following commutative diagram:

X ˆY Z ˆZ T X ˆY Z X
p fZ f

h g
T Z Y
Each square is Cartesian, and hence the outer rectangle is, as well. As f is universally it
closed, follows that p is closed. This shows that fZ is universally closed as well.
(ii): Let f : X Ñ Y and g : Y Ñ Z be universally closed. Given a morphism h : T Ñ Z ,
consider the following commutative diagram:
p q
X ˆY Y ˆZ T Y ˆZ T T

f g
X Y Z
The squares are Cartesian, and hence the outer rectangle is Cartesian as well. As f and g are
universally closed, p and q are closed, and hence q ˝ p is closed as well.
(iii): If f : X Ñ Y and g : X 1 Ñ Y 1 are both proper, then it follows directly from (i) and
(ii) that f ˆ g “ pf ˆ idY 1 q ˝ pidX ˆ gq is proper.
(iv): Let g : T Ñ Y be a morphism and consider the induced morphism p : X ˆY T Ñ T .
Note that T is covered by the open sets g ´1 pUi q and the fiber product is covered by the open
sets p´1 pg ´1 pUi qq “ f ´1 pUi q ˆUi g ´1 pUi q. As f ´1 pUi q Ñ Ui is universally closed, we

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see that
f ´1 pUi q ˆUi g ´1 pUi q ÝÝÑ g ´1 pUi q

is closed. This implies that p is closed as well.

(v): We can factor any morphism f : X Ñ Y as πY ˝ Γf where Γf : X Ñ X ˆk Y is
the graph morphism X Ñ X ˆk Y , and πY : X ˆk Y Ñ Y is the second projection. Note
that Γf is the pullback of ∆Y via the morphism pf, idY q : X ˆk Y Ñ Y ˆk Y . As ∆Y is a
closed embedding, and closed embeddings are stable under base change, Γf is also a closed
embedding, hence proper. Also πY is proper because it is a base change of X Ñ Spec k .
Therefore f is proper by (ii).

Proposition 19.6. Any finite morphism is proper.

Proof Finite morphisms are by definition affine, hence separated, and of finite type. More-
over, as finite morphisms are closed and being finite is stable under base change, finite
morphisms are universally closed.

19.2 Projective morphisms

A morphism f : X Ñ Spec A is said to be projective if it factors as

X ÝÝÑ PnA ÝÝÑ Spec A,

where the first map is a closed embedding of A-schemes and the second is the structure
morphism.
Like separatedness and properness, projectivity is a relative notion: it is the morphism
X Ñ Spec A which is projective, not X itself. Intuitively, it is the fibers of X Ñ Spec A
which are projective. For instance, P1krts is projective over Spec krts, but it is not over Spec k .
In the example, the (scheme-theoretic) fiber over s P S “ Spec krts equals the projective
line P1κpsq over κpsq. Still, if we are working in the category of schemes over, say, a field k or
Z, we still refer to a scheme X being ‘projective’ when it is projective over the base scheme.2

Example 19.7. For A “ Crts, the scheme X “ V` pzy 2 ´ x3 ´ txz 2 q in P2A is projective
over A1C “ Spec A. The fiber of X Ñ Spec Crts over any closed point a P Spec Crts is an
integral projective subscheme of dimension 1, namely V` pzy 2 ´ x3 ´ axz 2 q Ă P2C . X is
however not projective over C (e.g., as it admits many non-constant maps to A1C ). △

Projective morphisms are proper

We are ready to prove the following important theorem:
2 The notion of projectivity can be extended to more general morphisms f : X Ñ S . In fact, there exist two
slightly different definitions in the literature, see (Stacks Project Authors, 2018, Tag 01W7). These notions
agree in most cases, for instance when S is affine or when S satisfies some reasonable finiteness conditions.

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Theorem 19.8. Let S be a scheme and let X Ă PnS be a closed subscheme. Then the
structure morphism X Ñ S is proper.

Proof Projective morphisms are separated and of finite type, so to prove that they are proper
it remains to show that they are universally closed.
By the base change property (item (i) of Proposition 19.5) it suffices to show that PnZ Ñ
Spec Z is universally closed. So let T be any scheme, and consider the base change π : PnT Ñ
T . Since ‘being closed’ is a local property on T , it suffices to assume that T “ Spec A is
affine.
Let Z Ă PnA be a closed subscheme. We need to show that πpZq is closed in Spec A. By
Proposition 16.26 in Chapter ??, Z is given by a homogeneous ideal I Ă Arx0 , . . . , xn s.
Let τ : Z Ñ Spec A denote the composition Z Ñ PnA Ñ Spec A.
Let p P Spec A be a point which is not in the image of Z . We want to show that there is
some open neighbourhood U of p so that π ´1 pyq X Z is empty for all y P U . Equivalently,
we want to show that τ ´1 pyq “ H for all y P U .
Note that the fiber of p in PnA is given by the Proj of the ring
Arx0 , . . . , xn s{I bA κppq (19.1)
To say that this Proj is empty means that the graded ring (19.1) is zero in high degrees. In
other words, we have
IN bA κppq “ Arx0 , . . . , xn sN bA κppq (19.2)
for some large N ą 0. Note that IN and Arx0 , . . . , xn sN are both finitely generated modules
over A. Tensoring these by Ap and using Nakayama’s lemma, (19.2) implies that
pIN qp “ pArx0 , . . . , xn sN qp
In particular, there exists an f R p so that
pIN qf “ pArx0 , . . . , xn sN qf “ Af rx0 , . . . , xn sN
But then
τ ´1 pDpf qq “ Proj pAf rx0 , . . . , xm s{Iq “ H
because pIN qf contains all monomials of degree N . This completes the proof.
A consequence of this is that the ‘image of a projective variety is projective’:

Corollary 19.9. Let k be a field, and let f : X Ñ Y be a morphism of projective

schemes over k . Then the image f pXq Ă Y is closed in Y .

This statement is a priori not at all obvious, especially given how badly it fails with
‘projective’ replaced by ‘affine’ (see Example 19.2). It is yet another reason why one prefers
projective varieties to non-projective ones.
Proof of Corollary 19.9 By assumption X and Y are closed subvarieties of projective
spaces over k , and so they are proper over k . But then the morphism f : X Ñ Y is also
proper, and hence closed.

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Example 19.10. Let A “ Cra0 , a1 , . . . , am , b0 , b1 , . . . , bn s be the polynomial ring, and

consider the closed subscheme of P1A “ Proj Arx, ys given by
` ˘
Z “ V a0 xm ` a1 xm´1 y ` ¨ ¨ ¨ ` am y m , b0 xn ` b1 xy n´1 ` ¨ ¨ ¨ ` an y n .
If π : P1A Ñ Spec A denotes the structure morphism, the image πpZq Ă Spec A contains the
set of points pa0 , a1 , . . . , am , b0 , b1 , . . . , bn q P Am`n`2 pCq such that the two polynomials
of degrees m and n have a common root. This locus is indeed closed; it is given by V pdet Rq,
where det R is the resultant of the two polynomials, i.e., the determinant of the Sylvester
matrix. This matrix is shown below for m “ 3 and n “ 2:
¨ ˛
a0 a1 a2 a3 0
˚ 0 a0 a1 a2 a3 ‹
˚ ‹
R“˚ ˚ b 0 b1 b2 0 0 ‹.
‹
˝ 0 b0 b1 b2 0 ‚
0 0 b0 b1 b2
△
Example 19.11. Let X be a proper scheme over an algebraically closed field k . Let us show
that OX pXq “ k . We showed in Example ?? that elements of OX pXq can be understood
as the set of morphisms f : X Ñ A1k , so it suffices to show that all such morphisms are
constant. Let f : X Ñ A1k be a morphism and consider the morphism f¯: X Ñ P1k given by
composing f with the embedding A1k Ñ P1k . Then since P1k is proper, the image f¯pXq is
closed. But the closed subsets of P1k are either closed points or all of P1k . The latter case is
impossible because f¯ factors via A1k Ñ P1k , and so f maps to a point. △

Proposition 19.12. Let f : X Ñ Y be a morphism of projective schemes over an

algebraically closed field k . Suppose that f ´1 pyq is finite for every k -point y P Y . Then
f is a finite morphism.

Proof Since the property of ‘being finite’ is local on the target, and the k -points are dense
ι π
in Y , we may assume that Y “ Spec A is affine. As f factors as X Ñ Ý PnA Ý Ñ Y “ Spec A,
we may regard X as a closed subscheme of PnA “ Proj Arx0 , . . . , xn s and f is the restriction
of π : PnA Ñ Spec A to X .
We first show that f is affine. Let y P Y be a k -point. By assumption, the topolocal fiber
f pyq is a finite set in π ´1 pyq » Pnk . Pick an element h P Arx0 , . . . , xn s, so that V phq
´1

is disjoint from f ´1 pyq. By Corollary 19.9, the image f pV phq X Xq is a closed set in Y ,
and this set does not contain y . Therefore, there exists a distinguished open set Dpgq Ă Y
containing y , so that Dpgq Ă Y ´f pV phqq. This means that f ´1 pDpgqq Ă X ´V phq. Note
that U “ X ´ V phq “ X X D` phq is an affine scheme, being a closed subset of D` phq. If
ϕ : A Ñ U is the ring map which induces f |U : U Ñ Y , we have f ´1 pDpgqq “ Dpϕpgqq
in U , and so f is affine.
Now, write X “ Spec B for some A-algebra B . To conclude, we need to show that B is
a finite A-module. This follows from Serre’s theorem 18.31, because X is projective over A
and H 0 pX, OX q “ B .
Example 19.13. The projectivity assumption here is essential. For instance, if f : U Ñ X

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is the inclusion of an open subset, then certainly f ´1 pxq is finite for every x P X , but f is
typically not finite (see Example 9.24). △
Example 19.14. The morphism f : SpecpQq Ñ Spec Q is not finite, but of course the
fibers are finite. △

19.3 Blow-ups
Let A be a k -algebra. We are interested in morphisms
SpecpAq Ñ Pnk “ Projpkrx0 , . . . , xn sq.
Interpreting A as P0A “ Proj Arts, the map of graded k -algebras
ϕ : krx0 , . . . , xn s Ñ Arts; xi ÞÑ fi t
defines an A-valued point pf0 : ¨ ¨ ¨ : fn q of Pnk .
The base locus Bspϕq is empty if and only if pf0 , . . . , fn q “ p1q, which happens if and
only if pf0 t, . . . , fn tq “ ptq in Arts, that is, f0 t, . . . , fn t generate the irrelevant ideal.
If pf0 , . . . , fn q “ pgq is a principal ideal, then we have fi “ gci for some elements
ci P A. This means that we can extend the morphism over Bspϕq by defining the A-valued
point pc0 , . . . , cn q.
In general, let a “ pf0 , . . . , fn q Ă A, and consider the graded ring
à i i
Rpaq “ at,
iě0
ř
where t is a variable, i.e., R is the subring Arts of polynomials iě0 ai ti with ai P ai . In
the ring R, t has degree 1, while the elements of A have degree 0. Since R0 “ A, Proj R is
a scheme over Spec A with structure morphism

π : Proj R Spec A,
(this was introduced just after Definition ?? on page ??). We claim that π is an isomorphism
outside the closed subset π ´1 V paq, and so π merits being called the *blow-up* of V paq.
Indeed, if f P a, it holds that aAf “ Af and consequently ai Af “ Af for all i. Therefore,
we have the equality Rf “ Af rts. By Exercise 5.4.13 and Example ??, we then find that π
induces an isomorphism
π ´1 Dpf q “ Proj Rf “ Proj Af rts » Spec Af “ Dpf q.
Note that the elements f0 t, . . . , fn t by definition generate the irrelevant ideal of R. This
means that they induce a morphism
Fr : Proj R Ñ Pnk ,
which fits into the diagram

Proj R
π Fr

F
Spec A Pnk

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The scheme Proj R is called the blow-up of Spec A along a.

The blow-ups can be computed as follows. Let a Ă A be an ideal, generated by f0 , . . . , fr .
Let J be the kernel of the map of graded rings
à n n
Aru0 , . . . , ur s ÝÝÑ Rpaq “ a t , (19.3)
ně0

sending ui to fi t. Then the blow-up of a is the closed subscheme

Bla pXq “ V` pJq Ă PrA ,
where X “ Spec A.

Proposition 19.15. Let a Ă A be an ideal and let X “ Spec A. The blow-up Bla pXq
and the morphism π : Bla pXq Ñ X have the following properties:
(i) E “ π ´1 pV paqq is a Cartier divisor on Bla pXq.
(ii) π is surjective, and if U “ X ´ V paq, the restriction π|U : π ´1 pU q Ñ U
is an isomorphism.

Let a “ pf0 , . . . , fn q Ă A. We define the morphism

Fr : Bla pXq Ñ Pnk

by the composition
Bla pXq Ñ PnA Ñ Pnk .
Then π is proper, because it is a composition of a closed embedding and PnA Ñ Pnk , which is
proper. Ť
Let U “ X ´ V paq. Then U “ f Pa Dpf q. For any f P a, we have

π ´1 pDpf qq “ ProjpRpaq bA Af q “ ProjpAf rtsq » Spec Af .

These isomorphisms glue to an isomorphism
π : π ´1 pU q » U.
Therefore, π is surjective, being closed and ”the image containing
ı a dense open U .
f0 t
Now consider Di “ D` pfi tq “ Spec A fi t
, . . . , ffni tt . Then π ´1 pV paqqXDi “ V pfi q,
f t
as fj “ fi ¨ fji t .
Proposition ?? gives a description of the scheme-theoretic inverse image of V paq, as
à à i i`1
p ai ti q bA A{a “ a {a ,
iě0 iě0

´1
À i i`1
and hence π V paq “ Proj iě0 a {a .
Example 19.16 (The blow-up of the plane). Consider the polynomial ring A “ krx, ys and
the ideal a “ px, yq. Let R be the graded ring
à i i
R“ at,
iě0

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where the graded piece of degree i equals ai ti . The irrelevant ideal R` is generated by xt
and yt, and consequently Proj R is the union of the two open affine subschemes SpecpRxt q0
and SpecpRyt q0 .
There is a map of graded rings ϕ : Aru, vs Ñ R, where u and v are of degree 1, given by
u ÞÑ xt and v ÞÑ yt. This map is surjective because a is generated by x and y . The kernel of
ϕ contains the element xv ´ yu, and by Exercise 5.4.27, we have

R » Aru, vs{pxv ´ yuq. (19.4)

From this description, we see that Proj R is covered by the two distinguished open sets

D` puq “ SpecpRu q0 and D` pvq “ SpecpRv q0 ,

where
pRu q0 » krx, vu´1 s and pRv q0 » kry, uv ´1 s.

These affine schemes are glued together along their intersection, which is

SpecpRuv q0 » pAru, vsuv {pxv ´ yuqq0 .

One finds that

pAru, vsuv {pxv ´ yuqq0 » krx, y, uv ´1 , vu´1 s{pxvu´1 ´ yq » krx, uv ´1 , vu´1 s.

Therefore, Proj R coincides with the blow-up construction described in Section 6.2 on
page 126. △

Example 19.17. Let A “ Zrts and R “ Aru0 , u1 s{ptu1 ´ 7u0 q. The projective scheme
Proj R with its structure morphism

π : Proj R ÝÝÑ A1Z “ Spec A,

can be viewed as the blow-up of A1Z in the closed point m “ p7, tq P A1Z . Then
` ˘
π ´1 pDptqq “ Proj R bA Ar 1t s “ ProjpAru0 , 1t sq » Dptq
` ˘
π ´1 pDp7qq “ Proj R bA Ar 17 s “ ProjpAru1 , 17 sq » Dp7q

This shows that π restricts to an isomorphism over the open subset Proj R ´ π ´1 V pmq. On
the other hand, the schreme-theoretic fiber of m is given by

ProjpAru0 , u1 s bA A{mq “ ProjpAru0 , u1 s{p7, tqq “ P1F7 .

Proposition 19.18. Let X “ Spec A and Y “ V paq, and let X r “ Bla pXq.
(i) If X is reduced/irreducible/integral, then so is X
r.
n
(ii) We have X Ă Z “ V` pfi uj ´ fj ui q Ă PA . If f0 , . . . , fn are a minimal
r
set of generators for a and Z is integral, then X
r “ Z.

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Proof (i): Let X r “ Bla pXq “ Proj Rpaq, where Rpaq “ À d d

dě0 a t is the Rees algebra
of a. If X “ Spec A is reduced, then A has no nilpotent elements. Since Rpaq is a graded
subring of Arts, it inherits the reduced property from A. Thus, X r “ Proj Rpaq is reduced.
The same argument shows that X is also irreducible, integral.
r
r Ă Z “ V` pfi uj ´ fj ui q Ă Pn , and if f0 , . . . , fn are a minimal set of generators
(ii): X A
for a and Z is integral, then X r “ Z.
The blow-up X r is defined by the kernel J of the graded surjection (19.3). This kernel
contains the relations fi uj ´fj ui for all i, j , since fi pfj tq “ fj pfi tq. Thus, X
r “ Proj Rpaq
n
is a closed subscheme of Z “ V` pfi uj ´ fj ui q Ă Pk .
Suppose f0 , . . . , fn are a minimal set of generators for a, and Z is integral. Since X r ĂZ
and both are integral schemes, it suffices to show that the map X Ñ Z is an isomorphism on
r
an open dense subset.
Let U “ X ´ V paq. On U , the blow-up X r restricts to an isomorphism π ´1 pU q » U , as
shown earlier. Similarly, Z restricts to an open subset isomorphic to U because the relations
fi uj “ fj ui define the same structure as the blow-up on U . Since Z is integral and X r Ă Z is
a closed subscheme that agrees with Z on a dense open subset, we conclude that X “ Z . r

Example 19.19. Let A “ krx, ys{py 2 ´x3 q and a “ px, yq. The blow-up X r “ Bla pSpec Aq
embeds into P1A “ Proj ArU, V s. We compute X r on the affine charts D` pU q and D` pV q.
On D` pU q: Let u “ V {U . The blow-up is defined by the relation y “ ux. Substituting
into y 2 “ x3 , we get:
puxq2 “ x3 ùñ u2 x2 “ x3 ùñ u2 “ x.
r X D` pU q “ Spec Arus{pu2 ´ xq, with y “ u3 . The morphism Spec krus Ñ
Thus, X
Spec A is given by:
x ÞÑ u2 , y ÞÑ u3 .
On D` pV q: Let v “ U {V . The blow-up is defined by the relation x “ vy . Substituting
into y 2 “ x3 , we get:
y 2 “ pvyq3 ùñ y 2 “ v 3 y 3 ùñ 1 “ v 3 y.
r X D` pV q “ Spec Arvs{p1 ´ v 3 yq, with x “ vy . The morphism Spec krvs Ñ
Thus, X
Spec A is given by:
x ÞÑ v 3 , y ÞÑ v 2 .
The blow-up morphism π : X r Ñ Spec A is obtained by gluing these two charts. On
D` pU q, it is induced by x ÞÑ u2 , y ÞÑ u3 , and on D` pV q, by x ÞÑ v 3 , y ÞÑ v 2 . △

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Differentials

20.1 Derivations
In this section, we discuss derivations of a k -algebra A. Here we allow ‘k ’ to be any base
ring, but of course the main example we have in mind is when k is a field.

Definition 20.1. Let k be a ring and let A be a k -algebra. An k -derivation from A with
values in M is a k -linear map D : A Ñ M satisfying the Leibniz rule
Dpf gq “ g ¨ Dpf q ` f ¨ Dpgq
for every f, g P A.

Note that even though M is an A-module, a derivation D is usually not a map of A-modules;
we only require it to be linear over k .
If D is a derivation, then Dpaq “ 0 for every element a P k . This follows because
Dpa ¨ aq “ a ¨ Dpaq and Dpa ¨ aq “ a ¨ Dpaq ` a ¨ Dpaq “ 2Dpaq. We think of the
elements in A coming from k as the ‘constants’. However, a derivation can be zero on other
elements in A as well (a silly example is the zero map, which is a derivation).

Example 20.2. If Ař“ krts , then we can define a derivation D : A Ñ A by sending a

i
polynomial f ptq “ i ai t to f 1 ptq. Here f 1 ptq is the formal derivative, defined by
ÿ
f 1 ptq “ iai ti´1 . (20.1)
i

More generally, the partial differential operators BxB 1 , . . . , BxBn , as well as their k -linear
combinations, are k -linear derivations on the polynomial ring krx1 , . . . , xn s. △
Many of the familiar formulas for derivatives from calculus hold also for general derivations.
For instance, an easy induction shows that Dpxn q “ nxn´1 Dpxq for every x P A, and in
case g is invertible in A, Dpf {gq “ pDpgqf ´Dpf qq{g 2 . Moreover, if f pxq is a polynomial,
one has the chain rule Dpf pgqq “ f 1 pxqDpgq.
The set of k -derivations D : A Ñ M is denoted by Derk pA, M q. While derivations need
not be A-linear, the set Derk pA, M q has an A-module structure, where an element a P A
acts by D ÞÑ a ¨ D.
If ρ : M Ñ N is a map of A-modules, and D : A Ñ M is a k -derivation, then the
composition ρ ˝ D : A Ñ N is again a k -derivation with values in N . This means that
Derk pA, ´q is a functor from ModA to itself.
401

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The set of derivations Derk pA, M q is also functorial in the base ring k and the k -algebra
A. If k Ñ k 1 is a ring map, any k 1 -derivation A Ñ M can be regarded as a k -derivation,
so we obtain an inclusion Derk1 pA, M q Ă Derk pA, M q. Likewise, if A Ñ A1 is a map of
k -algebras, then a derivation A1 Ñ M induces a derivation A Ñ M by composition.

20.2 Kähler differentials

For a k -algebra A, there exists an A-module ΩA{k and a derivation dA : A Ñ ΩA{k which is
universal among derivations from A into modules M . This means that for any k -derivation
D : A Ñ M there exists a unique A-linear map α : ΩA{k Ñ M such that D “ α ˝ dA :
dA
A ΩA{k

α
D

M.
In other words, the map sending α ÞÑ α ˝ dA is a bijection
HomA pΩA{k , M q ÝÝÑ Derk pA, M q. (20.2)
We can construct the A-module ΩA{k explicitly as follows.
À For each element a P A introduce
a symbol da and consider the free A-module G “ aPA A da. Inside G, we have the
submodule H generated by the expressions
dpa ` bq ´ da ´ db, dpabq ´ a ¨ db ´ b ¨ da, dc
for a, b P A and c P k . We then define ΩA{k “ G{H , and the map dA : A Ñ ΩA{k by
dA paq “ da. Note that dA is additive, because any Z-linear relation among the da’s maps to
zero in G{H , and it is a derivation because the Leibniz rule dpabq “ a db ` b da is imposed
to hold in G{H . Finally, it will be k -linear because dpc ¨ aq “ c ¨ da ` 0 “ c ¨ da in G{H .
The module G{H also satisfies the universal property above: given any k -derivation
D : A Ñ M , we define the A-linear map α : ΩA{k Ñ M by αpdaq “ Dpaq, which is
well-defined precisely because D is a derivation.

Definition 20.3. The A-module ΩA{k is called the module of differentials of A over k .

Here is an important example, which will serve as a basis for essentially all explicit
computations involving ΩA{k .

Proposition 20.4 (Polynomial rings). Let k be a ring and let A “ krx1 , . . . , xn s. Then
ΩA{k is the free A-module with basis dx1 , . . . , dxn :
n
à
ΩA{k “ A dxi .
i“1
řn Bf
The universal derivation is defined by dA pf q “ i“1 Bxi dxi .

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Proof The universal property follows from the general chain rule: for any k -derivation
D : A Ñ M into a A-module M , we have
n
ÿ Bf
Dpf q “ Dpxi q. (20.3)
i“1
Bxi
Indeed, an easy induction using the Leibniz rule, shows that (20.3) holds when f is a
monomial, and then we conclude using k -linearity. Àn
Given a derivation D, we can define the A-linear map α : i“1 A dxi Ñ M by sending
dxi to Dpxi q. Then (20.3) implies that we have the equality D “ α ˝ dA , so the required
universal property holds.

Differentials and the diagonal

While the construction in the previous section is useful for efficiently defining ΩA{k , there is
an alternative description of ΩA{k , which is a bit more geometric. This description will be
useful when we construct the sheaf of differentials, as the previous construction does not
behave well with respect to gluing.
Let A be a k -algebra, let µ : A bk A Ñ A be the multiplication map µpx b yq “ xy
and let I “ Kerpµq Ă A bk A. Then as we saw in Chapter ??, I is the ideal describing the
diagonal embedding
∆ : Spec A ÝÝÑ SpecpA bk Aq.
The next result shows that the module of differentials ΩA{k can be identified with the quotient
I{I 2 . Just to clarify, there are a priori two A-module structures on I Ă A bk A, one given
by A Ñ A bk A via a ÞÑ a b 1 and the other defined by a ÞÑ 1 b a. But these two coincide
when passing to the quotient I{I 2 , because for any x P I ,
pa b 1qx ´ p1 b aqx “ pa b 1 ´ 1 b aq ¨ x P I 2 .
Another useful fact is that I is generated (as an ‘A bk 1-module’) ř by the elements of the
ř a b 1 ´ 1 b a where
form ř a P A. Indeed, for a given element i ai b bi P I , we have
i ai bi “ 0 in A , hence i 1 b ai bi “ 0 in A bk A, and this implies that
ÿ ÿ
ai b bi “ pai b 1qpbi b 1 ´ 1 b bi q.
i i

Proposition 20.5. There is an isomorphism of A-modules

ΩA{k » I{I 2 . (20.4)
The universal derivation δ : A Ñ I{I 2 is defined by δpaq “ a b 1 ´ 1 b a.

Proof Let us first check that δ is a derivation. Note that δ is k -linear, because for a, b P k
and x, y P A, we have
δpax ` byq “ pax ` byq b 1 ´ 1 b pax ` byq
“ ax b 1 ´ 1 b ax ` by b 1 ´ 1 b by
“ a ¨ δpxq ` b ¨ δpyq.

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Moreover, the Leibniz rule holds:

δpabq “ ab b 1 ´ 1 b ab
“ pa b 1qpb b 1 ´ 1 b bq ` p1 b bqpa b 1 ´ 1 b aq
“ a ¨ δpbq ` b ¨ δpaq.
Next, we check that I{I 2 and δ satisfy the universal property. Let D : A Ñ M be a k -linear
derivation. This defines an ‘A bk 1’-linear map

α : A bk A ÝÝÑ M, a b x ÞÑ a ¨ Dpxq
We have αpI 2 q “ 0, because

αppx b 1 ´ 1 b xqpy b 1 ´ 1 b yqq “ αpxy b 1q ´ αpx b yq ´ αpy b xq ` αp1 b xyq

“ xy ¨ Dp1q ´ x ¨ Dpyq ´ y ¨ Dpxq ` Dpxyq “ 0
The last expression is zero because D is a derivation.
Therefore we get an induced map ᾱ : I{I 2 Ñ M so that d “ ᾱ ˝ δ :
δ
A I{I 2
ᾱ
D

M
The map δ is surjective, because I is generated by the expressions x b 1 ´ 1 b x. This implies
that the induced map ᾱ is unique. Finally, ᾱ is A-linear because α is A b 1-linear.

20.3 Properties of differentials

Let k be a ring and let ρ : A Ñ B be a map of k -algebras. There are two important
exact sequences that relate the three modules ΩA{k , ΩB{k and ΩB{A . The maps involved are
defined as follows. Note first that there is a natural A-linear map ΩA{k Ñ ΩB{k defined by
da ÞÑ dpρpaqq. Taking the tensor product, we obtain a natural map of B -modules
ρ˚ : ΩA{k bA B Ñ ΩB{k ; dA a b b ÞÑ b ¨ dB ρpaq. (20.5)

We also have the ‘change of constants’ map

β : ΩB{k ÝÝÑ ΩB{A ; dB b ÞÑ dB b.

Note that the elements of the form dB ρpxq all map to zero in ΩB{A . The next proposition
shows that these elements in fact generate the kernel of β :

Proposition 20.6 (Cotangent sequence). There is an exact sequence of B -modules

ρ˚ β
ΩA{k bA B ΩB{k ΩB{A 0 (20.6)

Proof The surjectivity of β is easy, because both B -modules are generated by symbols of

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the form dB b, but there are more relations imposed in ΩB{A . In fact, from the description of
ΩB{A in terms of generators and relations, we see that
ΩB{A “ ΩB{k { B dB a : a P A
Moreover, the submodule B dB a : a P A is exactly the image of the map ρ˚ , so we get
also exactness in the middle.
In general, the map ρ˚ needs not be injective nor surjective. There is one important
case where it is surjective, and where one can describe the kernel explicitly, namely when
B “ A{I and ρ is the quotient map. In this case ΩB{A “ 0, because for each a P A, we
have dB{A paq “ a ¨ dB{A p1q “ 0 in ΩB{A , by A-linearity. Therefore, the left-most map
appearing in the cotangent sequence (20.6), ρ˚ , is surjective. By the description of ΩB{k in
terms of generators and relations, we see that the kernel of ρ˚ is generated by elements of the
form da b 1 where a P A. Note that this is also the image of the map
δ : I{I 2 ÝÝÑ ΩA{k bk B; f ÞÑ df b 1.
This map is well-defined, because if f, g P I , then dpf gq b 1 “ pg df ` f dgq b 1 “ 0 by
the Leibniz rule. Moreover, it is B -linear, because
δpa ¨ f q “ dpa ¨ f q b 1 “ pf da ` a df q b 1 “ a df b 1 “ a ¨ δpf q.
We have proved the following proposition.

Proposition 20.7 (Conormal sequence). Let A be a k -algebra and let B “ A{I for
some ideal I Ă A. Then there is an exact sequence of B -modules
δ ρ˚
I{I 2 ΩA{k bA B ΩB{k 0. (20.7)

The conormal sequence allows us to find explicit presentations of modules of differentials. If

A is a finitely generated k -algebra, say A “ krx1 , . . . , xn s{I where I “ pf1 , . . . , fr q, then
we have
Àn
Ωkrx1 ,...,xr s{k bk A » i“1 A dxi .
and the sequence (20.7) takes the form
Àn
I{I 2 δ i“1 A dxi ΩA{k 0.
As I{I 2 is generated as a A-module by the classes of the f1 , . . . , fr modulo I 2 , there is a
surjection Ar Ñ I{I 2 which gives the exact sequence
Àn
Ar i“1 A dxi ΩA{k 0. (20.8)
řn Bfi
Here the left-most map sends the i-th basis vector ei to dfi “ j“1 Bxj dxj . This means that
it can be identified with the A-linear map J t : Ar Ñ An defined by multiplication by the
transpose of the Jacobian matrix
¨ Bf1 Bf1 ˛
Bx1
¨ ¨ ¨ Bx n

J “ ˝ ... .. ‹
˚
. ‚
Bfr Bfr
Bx1
¨¨¨ Bxn
.

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We summarize the discussion in the following theorem:

Theorem 20.8. Let k be a ring and let A “ krx1 , . . . , xn s{pf1 , . . . , fr q. Then

Àn
i“1 A dxi
ΩA{k “ “ Coker J t (20.9)
A df1 ` ¨ ¨ ¨ ` A dfr
řn Bf
and the universal k -derivation is given by dA pf q “ i“1 Bxi dxi .

The description in (20.9) allows for easy computations of differentials for a large class of
rings. We will see many examples in Section 20.5.
Another nice property of differentials is that they behave well under base change and
localization:

Proposition 20.9. Let k be a ring and let A and B be k -algebras.

(i) (Base change) For the k -algebra C “ A bk B , we have
ΩC{B “ ΩB{A bB C.
(ii) (Localization): For a multiplicative set S Ă A, we have an isomorphism of
S ´1 A-modules
ΩS ´1 A{k “ S ´1 pΩA{k q

Proof See Exercises 20.11.3 and 20.11.4.

Combining Theorem 20.8 and Proposition 20.9, we get the following:

Corollary 20.10. Let k be a ring and let A be a finitely generated k -algebra, or a

localization of such. Then ΩA{k is a finitely generated A-module.

20.4 The sheaf of differentials

The primary motivation for studying the module of differentials ΩB{A is that it provides
an intrinsic sheaf Ω
Ć B{A on Spec B associated to a morphism Spec B Ñ Spec A. In this
section, we globalize this construction to an arbitrary separated morphism X Ñ S .

Definition 20.11. Let X be a separated scheme over a scheme S . Let I be the ideal sheaf
corresponding to the diagonal embedding ∆ : X Ñ X ˆS X . We define the sheaf of
Kähler differentials to be the sheaf
ΩX{S “ ∆˚ pI{I 2 q.
When S “ Spec A, we write ΩX{A “ ΩX{ Spec A .

Concretely, if U “ Spec B Ă X is an affine open that maps into V “ Spec A Ă S , then

the restriction ∆|U : U Ñ U ˆS U “ U ˆV U “ SpecpB bA Bq is a closed embedding

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corresponding to the ideal I “ IpU q in B bA B . Moreover, I{I 2 |U ˆV U “ I{I

Ą2 . Therefore,

ΩX{S |U » Ω
Ć B{A .

Hence the sheaf ΩX{S is built out of the various ΩB{A over affine opens. In particular, ΩX{S
is a quasi-coherent sheaf. If X is of finite type over a ring k , then ΩX{k is also of finite type,
by Corollary 20.10.
The algebraic properties of the Kähler differentials ΩB{k translate into the following
statements for ΩX{S .

Proposition 20.12.
(i) (Base change): Let X Ñ S , and S 1 Ñ S be separated morphisms and let
X 1 “ X ˆS S 1 with projection p : X 1 Ñ X . Then
ΩX 1 {S 1 “ p˚ ΩX{S .
(ii) (Cotangent sequence): Let S be a scheme and f : X Ñ Y a morphism of
separated schemes over S . Then there is an exact sequence of OX -modules
f ˚ pΩY {S q Ñ ΩX{S Ñ ΩX{Y Ñ 0. (20.10)
(iii) (Conormal sequence): Let Y be a closed subscheme of a scheme X over
S . Let IY be the ideal sheaf of Y . Then there is an exact sequence of
OY -modules
IY {IY2 Ñ ΩX{S |Y Ñ ΩY {S Ñ 0. (20.11)

20.5 Examples
Example 20.13 (Affine space). For the affine n-space Ank “ Spec krt1 , . . . , tn s over a ring
k , we have
n
à
ΩAnk {k “ ΩkrtČ
1 ,...,tn s{k
“ OX dti .
i“1

△
Example 20.14. Let k be a field and let hpxq P krxs. Then for A “ krxs{phq, we have
A dx
ΩA{k “ .
h1 pxqdx
Consider the ring map ρ : krus Ñ krxs sending u to hpxq. Writing krxs “ krusrxs{phpxq´
uq, we compute
Ωkrxs{krus “ pkrxsdxq{ph1 pxqdxq
Ωkrus{k bkrus krxs “ krxsdu “ krxsh1 pxqdx
and the cotangent sequence (20.6) takes the form
0 ÝÝÑ krxs du ÝÝÑ krxs dx ÝÝÑ krxs dx{ph1 pxqdxq ÝÝÑ 0

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(Here the left-most map sends du to h1 pxqdx, so the sequence is in fact short exact.)
On the geometric side, assuming h is non constant, ρ defines a finite morphism f : A1k Ñ
1
Ak . The module Ωkrxs{krus is supported on the points where h has multiple roots, or in other
words, where f is not pdeg hq-to-1. △
Example 20.15 (Plane curves). Let k be a field and let X “ Spec A where A “ krx, ys{pf q.
By Theorem 20.8, the module of differentials is given by
ˆ ˙
Bf Bf
ΩA{k “ pA dx ‘ A dyq { dx ` dy .
Bx By
Write fx “ BfBx
and fy “ Bf
By
for the partial derivatives. If X is smooth, that is, V pf, fx , fy q “
H, then X is covered by the two open sets Dpfx q and Dpfy q. The module of differen-
tials ΩA{k will be a locally free A-module of rank 1. Over the open set Dpfx q, we have
pΩA{k qfx “ ΩAfx {k and we may eliminate dx using the relation fx dx ` fy dy “ 0, and
pΩA{k qfx is generated by the element dy{fx . Likewise, over Dpfy q, the module is generated
by ´dx{fy . Note that on the overlap Dpfx q X Dpfy q, these two generators coincide, as
fx dx ` fy dy “ 0 in ΩA{k . This means that we have defined a global section of ΩX{k . △
Example 20.16 (The nodal cubic). Let k be a field and let A “ krx, ys{py 2 ´ x2 px ` 1qq.
Then X “ Spec A is a plane curve which has a singular point at the origin P corresponding
to m “ px, yq, and nonsingular at all other points. The module of differentials is given by
Adx ‘ Ady
ΩA{k “ .
p2y dy ´ p3x2 ` 2xqdxq
In this case ΩA{k has rank 1 for every point px, yq ‰ p0, 0q. Indeed, dx will generate ΩA{k
over the open set Dpyq, and dy will generate over Dpxq (except where 3x ` 2 “ 0, but
these points are covered by the first case, as y ‰ 0 there.)
At the origin, where x “ y “ 0, the relation 2y dy ´ p3x2 ` 2xqdx is identically zero,
and ΩA{k is not locally free there. More precisely, ΩA{k bA kpP q is isomorphic to
A dx ‘ A dy
ΩA{k bA A{px, yq “ “ k dx ‘ k dy
px, y, 2ydy ´ p3x2 ` 2xqdxq
which is a k -vector space of dimension 2. △
Example 20.17 (The cuspidal cubic). Let k be a field of characteristic ‰ 2, 3 and let
A “ krx, ys{py 2 ´ x3 q. The module of differentials is given by
A dx ‘ A dy
ΩA{k “ q
p2y dy ´ 3x2 dx
As in the previous example, ΩA{k is locally free of rank 1 outside the origin px, yq.
In fact, the module ΩA{k contains non-zero torsion elements. Consider for instance the
element η “ 3ydx ´ 2xdy , which is killed by y and x2 :

x2 η “ 3x2 y dx ´ 2x3 dy “ yp3x2 dx ´ 2y dyq “ 0

yη “ 3y 2 dx ´ 2xy dy “ xp3x2 dx ´ 2ydyq “ 0.

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To see that η is non-zero, note that we can view A as an algebra over the polynomial ring
krxs, and
ΩA{krxs “ pA dyq{p2y dyq.
The ‘change of constants map’ ΩA{k Ñ ΩA{krxs , sends dx ÞÑ 0 and dy ÞÑ dy . Therefore,
η “ 3ydx ´ 2xdy maps to 2x dy , which is non-zero, and so η must be non-trivial. △
Example 20.18. Let k be a ring and let X “ P1k . Then ΩX{k » OP1k p´2q. To see this,
we use the standard covering of P1k “ Proj krx0 , x1 s, given by Ui “ D` pxi q. Over U0 ,
Example 20.13 shows that
„ ȷ ˆ ˙
x1 x1
ΩU0 {k » k d
x0 x0
and similarly on U1 . On the intersection D` px0 q X D` px1 q, the two generators are related
by
ˆ ˙ ˆ ˙2 ˆ ˙
x1 x1 x0
d “´ d .
x0 x0 x1
This shows that, up to a sign, ΩA{k is constructed using the same gluing function as OP1k p´2q,
and hence ΩX{k » OX p´2q. △
Example 20.19. In this example we will consider the sheaf of differentials of the hyperelliptic
curves of Example 6.4. Let ppxq be a polynomial of degree d “ 2g ` 1 or d “ 2g ` 2 and
assume that p has distinct roots. In the affine chart U “ Spec A, where A “ krx, ys{py 2 ´
ppxqq, we have
ΩU {k “ A dx ‘ A dy{p2y dy ´ p1 pxq dxq. (20.12)
This is locally free of rank 1: it is generated by dx over the open set Dpyq and by dy
over Dpp1 pxqq, and these two open sets cover U as V pp1 pxq, yq “ V pp1 pxq, ppxqq “ H
by assumption. Over the intersection Dpyq X Dpp1 pxqq, these generators are related by
dy “ p1 pxq{y dx and p1 pxq{y is a unit.
In the other chart, V “ Spec B , where B “ kru, vs{pv 2 ´ u2g`2 ppu´1 qq, we have
ΩV {k “ B du ‘ B dv{p2v dv ´ pp2g ` 2qu2g`1 ppu´1 q ` u2g`2 p1 pu´1 qq duq. (20.13)
A similar analysis as above shows that also ΩV {k is locally free of rank 1 over V .
Over the intersection U X V “ Dpxq “ Dpuq, the module of differentials is given by the
localization of (20.12) and (20.13) in x and u respectively. Here the generators are related by
x “ u´1 and y “ u´g´1 v , which gives
dx “ ´u´2 du, dy “ ´pg ` 1qu´g´2 v dx ` u´g´1 dv.
Consider the element ω “ y ´1 dx P ΩK{k . Over the open sets Dpyq, Dpp1 pxqq, Dpvq,
Dpp1 puqq, this can be represented
dx 2 dy ug´1 du 2 ug´1 dv
ω“ “ 1 “´ “´ 1 .
y p pxq v p puq
From this, it follows that ω in fact defines a global section of the sheaf ΩX{k . The same

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holds for the g differentials y ´1 dx, . . . , xg´1 y ´1 dx. We will see later, in Chapter 21, that
ΓpX, ΩX{k q is g -dimensional, generated by these expressions. △

20.6 The Euler sequence and differentials of Pnk

In this section, we will give a concrete description of the sheaf of differentials on projective
space, suitable for explicit computations. We continue to work over a general ring k .
The description is in terms of the so-called ‘Euler sequence’ in (20.14) below. The name
comes from Euler’sřformula which states that if f is a homogeneous polynomial in x0 , . . . , xn
m Bf
of degree m, then i“0 xi Bx i
“ m ¨ f . (See Exercise 20.11.8). This formula will be used in
the proof below.

Theorem 20.20 (The Euler Sequence). Then there is an exact sequence

Àn
0 ΩPnk {k i“0 OPk p´1q OPnk 0. (20.14)
n

Àn
Proof Write Pnk “ Proj R, where R “ krx0 , . . . , xn s. The module ΩR{k “ i“0 R dxi
is naturally a graded R-module if we give each dxi degree 1.
Next consider the ‘Euler map’ e : ΩR{k Ñ R, defined by dxi ÞÑ xi . This is a map of
graded R-modules. Let M “ Ker e and consider the exact sequence
e
0 M ΩR{k R.

If we apply the tilde functor, we obtain the sequence of sheaves

ẽ
0 M
Ă Ω
Ć R{k OPnk (20.15)
Àn
Note that with the above grading, we have Ω R{k » i“0 OPk p´1q. This will be the middle
Ć n

sheaf in the sequence (20.14). While e is not surjective, we will show that the map ẽ is
surjective, so that (20.15) is a short exact sequence. To prove this, it suffices to show this over
each distinguished open D` pxi q, and for simplicity, we assume i “ 0.
` ˘
Over D` px0 q, the pRx0 q0 -module ΓpD` px0 q, Ω Ć R{k q “ pΩR{k qx0 0 is free of rank n`1,
with basis
dx0 dx1 dxn
, , ..., . (20.16)
x0 x0 x0
We then have
ˆ ˙ ˆ ˙
dx0 dxj xj
e “ 1, e “ for j “ 1, . . . , n
x0 x0 x0
This means that ẽ is surjective over D` px0 q. Moreover, as the modules involved are free
pRx0 q0 -modules, the kernel Mx0 is also free; it is generated by the n elements
ˆ ˙
xj 1 xj
dx0 ´ dxj “ d for j “ 1, . . . , n. (20.17)
x20 x0 x0

Next, we show that M

Ă » ΩPn {k . Note that the universal derivation d : R Ñ ΩR{k preserves
k

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the grading, and so it induces a morphism of OPnk -modules

d˜: OPnk ÝÝÑ Ω

ĆR{k .

Over D` px0 q, it is given by

m´1
ˆ ˙
f xm
0 df ´ f mx0 dx0
d˜ “
xm
0 x2m
0

řn if fBfP Rm is a homogeneous polynomial of degree m, Euler’s formula implies that

Note that
df “ i“0 Bx i
dxi is sent to m ¨ f by the map e. This means that
x0 df ´ f mx0m´1 dx0 m´1
ˆ ˆ ˙˙ ˆ m ˙
f xm0 m ¨ f ´ f mx0 x0
e d˜ m “e “ “ 0 (20.18)
x0 x2m
0 x 2m
0

Note that over D` px0 q, the map d˜ induces the universal derivation pRx0 q0 Ñ ΩpRx0 q0 {k . By
(20.18), this takes values inside pMx0 q0 , and in fact pMx0 q0 “ ΩpRx0 q0 {k as both modules
are generated by the expressions in (20.17). This happens over the other distinguished opens
D` pxi q as well, so we get isomorphisms M Ă|D` pxi q » ΩPn {k |D` pxi q . As these are induced
k
by the universal derivation, they are forced to agree over the overlaps D` pxi xj q as well. This
shows that MĂ » ΩPn { and we get the exact sequence (20.14).
k

Since ΩPnk {k injects into OPnk p´1qn`1 , which has no global sections, we get:

Corollary 20.21. ΓpPnk , ΩPnk {k q “ 0.

Example 20.22. Even on P1k , the Euler sequence is interesting. Using Example 20.18, it
takes the form
0 ÝÝÑ OP1k p´2q ÝÝÑ OP1k p´1q ‘ OP1k p´1q ÝÝÑ OP1k ÝÝÑ 0.
△

20.7 Nonsingularity and smoothness

Recall that a scheme X is said to be nonsingular at a point p P X , if OX,p is a regular local
ring. Moreover, if X is of finite type over a field k , we defined X to be smooth at p if there
is an affine open U “ Spec krx1 , . . . , xn s{pf1 , . . . , fr q containing p, so that the Jacobian
matrix
¨ Bf1 Bf1 ˛
Bx1
¨ ¨ ¨ Bx n

J “ ˝ ... .. ‹ . (20.19)
˚
. ‚
Bfr Bfr
Bx1
¨ ¨ ¨ Bxn
has maximal rank at p, that is rank Jppq “ n ´ dimp X .

Proposition 20.23. Let X be an integral scheme of finite type over a field k . Then X is
smooth over k if and only if ΩX{k is locally free of rank dim X .

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Proof Let us first assume that p P X a closed point. The statements involved are local in
nature, so we reduce to the case where X “ Spec A, where

A “ krt1 , . . . , tn s{pf1 , . . . , fr q.

The conormal sequence takes the form

Jt
Ar ÝÝÑ An ÝÝÑ ΩA{k ÝÝÑ 0

Tensoring this by κppq, we obtain

Jppqt
κppqr ÝÝÝÝÑ κppqn ÝÝÑ ΩA{k bA κppq ÝÝÑ 0. (20.20)

In other words, ΩA{k bA κppq equals the cokernel of the transpose of the Jacobian matrix
Jppq. This shows that
` ˘
dimκppq ΩA{k bA κppq “ n ´ rankpJppqq (20.21)

Now, X is smooth at p if and only if rankpJppqq “ n ´ dim X which by the formula

happens if and only if dimκppq pΩA{k bA κppqq “ dim X . Therefore, if ΩX{k is locally free
of rank dim X in a neighbourhood of p, then X is smooth at p.
For the converse direction, note that ΩX{k is an OX -module of finite type, so there is an
open subset W Ă X for which ΩX{k |W is locally free (see Section 15.3). The rank is in fact
equal to dim X here, because X contains a smooth point, by Corollary 11.22. If we assume
that Jppq has rank n´dim X for every closed point p P X . Then dimκppq pΩA{k bA κppqq “
dim X for every closed point p. As the function p ÞÑ dimκppq ΩX{k ppq can only increase
upon specialization. As this takes the value dim X at the generic point, and every closed
point, Corollary 15.14 shows that ΩX{k is locally free of rank dim X .

Corollary 20.24. If X is a variety over an algebraically closed field k , then the following
are equivalent:
(i) X is nonsingular
(ii) X is smooth
(iii) ΩX{k is locally free of rank dim X .

Example 20.25. For the curve X Ă A2k defined by f “ y 2 ` xp ` t over k “ Fp ptq

from Example ??, the module of differentials is given by ΩA{k “ A dx ‘ A dy{p2yq.
While X is nonsingular, ΩX{k “ Ω
Ć A{k is not locally free in a neighbourhood of the point
p
m “ px ` y, yq. △

Example 20.26. ΩX{k can be locally free (of the wrong rank) even if X is not smooth. For
instance, the scheme X “ Spec Fp rϵs{pϵp q is not smooth over k “ Fp ptq, but ΩX{k “
krϵs{ϵ
Č p dϵ is locally free of rank 1. △

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20.8 The Tangent sheaf

For a scheme X over a field k , we define the tangent sheaf TX , as the dual of the sheaf of
differentials,
TX “ Hom OX pΩX{k , OX q.

This is again a quasi-coherent sheaf on X , and it is of finite type if X is of finite type over k .
If U “ Spec A is an affine open set of X , then TX pU q “ HomA pΩA{k , Aq “ Derk pA, Aq
is the A-module of k -linear derivations of A.
The name comes from the following. If p P X is a k -point, then the sequence (20.20), and
the Jacobian Criterion, shows that there is an isomorphism of k -vector spaces

mp {m2p » ΩOX,p {k bOX,p κppq

Taking duals, we obtain the following isomorphism

pTX qp bOX,p κppq » Homκppq pmp {m2p , κppqq “ Tp X.

Thus TX is a sheaf which collects all the tangent spaces as its fibers.

Example 20.27. For X “ Ank , we get ΩAnk “ TAnk “ OAnnk . △

Example 20.28 (The tangent sheaf of Pnk ). Taking the dual of the Euler sequence, we get the
following exact sequence:

0 ÝÝÑ OPnk ÝÝÑ OPnk p1qn`1 ÝÝÑ TPnk ÝÝÑ 0. (20.22)

Taking the long exact sequence in cohomology, find that ΓpPn , TPnk q is a k -vector space of
dimension pn ` 1q2 ´ 1, and H i pPn , TPnk q “ 0 for all i ą 0. △

20.9 The sheaf of p-forms

If X is a variety of dimension n over a field k , and p is an integer we define the sheaf of
p-forms as the OX -module
Źp
ΩpX{k “ ΩX{k .

On affine space Ank , the `sheaf

˘ of p-forms on affine space is not particularly interesting,
as ΩpAn {k is free of rank np . However, for projective varieties, the sheaves ΩpX{k are of
k
significant interest because they allow us to define important invariants of the variety.
The most important of these is the sheaf ΩnX{k , which is usually called the canonical
sheaf or the canonical bundle of X . If X is smooth over k , then ΩX{k is locally free of rank
n, and so ΩnX{k is an invertible sheaf. The canonical sheaf plays a fundamental role in the
classification of varieties. We will demonstrate this in the setting of curves in Chapter 21.
The following gives a computation of the canonical sheaf of projective space.

Proposition 20.29. ΩnPnk “ OPnk p´n ´ 1q.

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Proof Consider the Euler sequence for the cotangent bundle of Pn

0 ÝÝÑ ΩPn ÝÝÑ OPnk p´1q‘pn`1q ÝÝÑ OPnk ÝÝÑ 0
Taking determinants, we get
` ˘
det OPnk p´1q‘pn`1q “ OPnk p´n ´ 1q “ detpΩPnk q b detpOPnk q “ ΩnPnk .

We can use the Euler sequence to compute the cohomology groups of the sheaves of
p-forms on Pnk as follows. By Exercise 20.11.20, there are exact sequences
n`1
0 ÝÝÑ Ω1Pnk ÝÝÑ OPnk p´1qp 1 q ÝÝÑ O n ÝÝÑ 0.
Pk
n`1
0 ÝÝÑ Ω2Pnk ÝÝÑ OPnk p´2qp 2 q ÝÝÑ Ω1n ÝÝÑ 0.
Pk (20.23)
..
.
n`1
0 ÝÝÑ ΩnPnk ÝÝÑ OPnk p´nqp n q ÝÝÑ Ωn´1
Pn
k
ÝÝÑ 0.

Proposition 20.30. Let k be a field and let n P N. Then

#
k if p “ q and 1 ď p ď n
H q pPnk , ΩpPnk q “ (20.24)
0 otherwise

Proof Recall that the sheaves OPnk p´dq have no higher cohomology on Pnk for d “ 0, . . . , n.
Therefore, the long exact sequence applied to (20.23) shows that for p “ 1, . . . n
H q pX, ΩpPnk q “ H q´1 pX, Ωp´1
Pn
k
q.
The result then follows by induction on p, starting with p “ 0, where the it follows from
Theorem 18.27.
Using the conormal sequence, we can also study differentials on other projective varieties.
Example 20.31 (Hypersurfaces). Let X Ă Pnk be a smooth hypersurface of degree d. The
conormal sequence takes the form
I{I 2 ÝÝÑ ΩPnk {k |X ÝÝÑ ΩX{k ÝÝÑ 0. (20.25)
Note that I{I 2 » OX p´dq is an invertible sheaf. Therefore, the left-most map in the
sequence (20.25) is in fact injective (the map is nonzero because the term in the middle has
rank n ` 1). Hence we get the following exact sequence
0 ÝÝÑ OX p´dq ÝÝÑ ΩPnk {k |X ÝÝÑ ΩX{k ÝÝÑ 0. (20.26)

Taking determinants, we get that ΩnPnk |X » Ωn´1

X b OX p´dq, and hence

Ωn´1
X » OX pd ´ n ´ 1q.
△

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Example 20.32 (Quartic surfaces). Let X Ă P3k be a smooth surface defined by a ho-
mogeneous polynomial of degree 4. We can compute the cohomology groups of Ω1X and
Ω2X » OX using the following three exact sequences

0 ÝÝÑ OP3k p´4q ÝÝÑ OP3k ÝÝÑ ι˚ OX ÝÝÑ 0

0 ÝÝÑ OX p´4q ÝÝÑ Ω1P3k {k |X ÝÝÑ Ω1X{k ÝÝÑ 0
0 ÝÝÑ ΩP3k {k p´4q ÝÝÑ Ω1P3k {k ÝÝÑ Ω1P3k {k |X ÝÝÑ 0

Here the first sequence is the ideal sheaf sequence of X , the second is the conormal sequence,
and the third is obtained by tensoring the first by ΩP3k {k . From the corresponding long exact
sequences in cohomology, we compute the various hp,q “ dimk H q pX, ΩpX q, shown in the
following table (the ’Hodge diamond’ of X ):

h0,0 1
1,0
h h0,1 0 0
h2,0 h1,1 h0,2 “ 1 20 1
2,1 1,2
h h 0 0
h2,2 1

20.10 Application: irrationality of hypersurfaces

One of the main reasons for introducing the sheaves ΩX and ΩpX is to define geometric
invariants. As these are intristically defined, they provide a useful method to distinguish
varieties, i.e., show that two varieties are not isomorphic.
In some cases, these invariants are strong enough to even distinguish varieties up to
birational equivalence. The aim of this section is to prove the following theorem.

Theorem 20.33. Let k be a field and let X Ă Pnk be a nonsingular hypersurface of

degree d. If d ě n ` 1, then X is not rational.

In the theorem, it is essential that the hypersurface is nonsingular. For instance, the curve
defined by

x0 xd´1
1 ´ xd2 “ 0

is rational: it is the image of the morphism P1k Ñ P2k ps : tq ÞÑ pstd´1 : td : sd q

Proposition 20.34. Let f : X 99K Y be a birational map of varieties over a field k and
assume that X is normal. Then the locus of points x P X where f is not defined has
codimension at least 2 in X .

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Theorem 20.35. Let f : X 99K Y be a birational map between smooth projective

varieties over an algebraically closed field. Then, for any integer p ě 0, there is an
isomorphism of global sections of the sheaves of differential forms:
H 0 pX, ΩpX q » H 0 pY, ΩpY q.

Proof Since f : X 99K Y is a birational map, there exists a non-empty open subset U Ă X
such that the map f : U Ñ Y is a morphism and induces an isomorphism between U and its
image f pU q Ă Y . The complement X ´ U has codimension at least 2 in X , because X is
nonsingular, and birational maps between normal varieties are isomorphisms outside subsets
of codimension at least 2.
The morphism f : U Ñ Y induces a pullback map:
f ˚ : Ω1Y ÝÝÑ Ω1U .
Since f is an isomorphism on U , the pullback map f ˚ is an isomorphism over U . Taking the
exterior powers, this induces an isomorphism:
f ˚ : ΩpY ÝÝÑ ΩpU .
Since X is nonsingular, and k is algebraically closed, the sheaf ΩpX is locally free and the
sections of ΩpX over U extend uniquely to sections over X by the Algebraic Hartogs’ theorem
(see Exercise 20.11.24, which applies because X ´ U has codimension at least 2). Thus, the
restriction map:
H 0 pX, ΩpX q ÝÝÑ H 0 pU, ΩpU q
is an isomorphism.
The morphism f : U ÝÝÑ Y is dominant, so the pullback map on global sections
f ˚ : H 0 pY, ΩpY q ÝÝÑ H 0 pU, ΩpU q
is injective (see Exercise 20.11.23). Combining this with the isomorphism H 0 pU, ΩpU q »
H 0 pX, ΩpX q, we obtain an injective map:
H 0 pY, ΩpY q ÝÝÑ H 0 pX, ΩpX q.
By symmetry, the same argument applies to the inverse birational map Y 99K X , which
gives an injective map H 0 pX, ΩpX q Ñ H 0 pY, ΩpY q. Combining these, we conclude that
H 0 pX, ΩpX q » H 0 pY, ΩpY q.
We are now ready to prove the main theorem of this section.
Proof of Theorem 20.33 We apply the previous theorem to the case p “ dim X “ n ´ 1.
Then
ΩX » OX pn ´ d ´ 1q.
When n ě d ` 1, we have H 0 pX, Ωn´1
X q ‰ 0, by the following exact sequence

0 ÝÝÑ OPnk pn ´ 2d ´ 1q ÝÝÑ OPnk pn ´ d ´ 1q ÝÝÑ OX pn ´ d ´ 1q ÝÝÑ 0.

Note that n ě d ` 1 implies that H 1 pPnk , OPnk pmqq “ 0 for all m P Z.

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20.11 Exercises
Exercise 20.11.1. Let k be a field. Compute ΩA{k for the following k -algebras:
a) A “ krx, ys{px2 ` y 2 q. H INT: This will depend on the characteristic of k .
b) A “ krx, yspxyq
c) A “ krx, y, zs{py 2 ` x3 ´ x2 z 2 q
Exercise 20.11.2 (Tangent vectors and derivations). Recall that for a scheme X over a field
k , the tangent space Tp X at a k -point p P X is defined as the k -vector space of linear
functionals ℓ : mp {m2p Ñ k . There is a natural way to identify Tp X with the vector space of
derivations of OX,p into k , that is, Derk pOX,p , kq. Write A “ OX,p and m “ mp .
a) Show that the structure map k Ñ A{m2 induces a splitting A{m2 “ k ‘ m{m2 .
b) For f P A, let Df denote the image of f in m{m2 via the composition A Ñ
A{m2 Ñ m{m2 , i.e., Df “ f ´ f ppq mod m2 . Show that D is k -linear and
satisfies the Leibniz rule.
c) Show that for each k -linear map ℓ : m{m2 Ñ k , the composition D ˝ ℓ is a
k -linear derivation A Ñ k .
d) Show that this defines an isomorphism Homk pm{m2 , kq » Derk pA, kq.
Exercise 20.11.3 (Base change). Show that for the k -algebra C “ A bk B , there is an
isomorphism
ΩC{B “ ΩB{A bB C.
H INT: Show that d b idC is the universal derivation.
Exercise 20.11.4 (Localization). For a multiplicative set S Ă A, show that there is an
isomorphism of S ´1 A-modules ΩS ´1 A{k “ S ´1 pΩA{k q. H INT: Define the universal
derivation by dS ´1 A pa{sq “ psdA a ´ adA sq{s2 .
Exercise 20.11.5 (Products). Let A and B be k -algebras.
a) Show that ΩpAˆBq{k “ ΩA{k b ΩB{k , with universal derivation dAˆB “
dA ˆ dB .
b) Show that ΩAbk B{k “ pΩA{k bk Bq ‘ pΩB{k bk Aq with universal derivation
dpa b bq “ pdA aq b b ` a b pdB bq for a P A, b P B .
Exercise 20.11.6 (The differentials of a tensor product). Let B and C be two A-algebras.
Consider the map
d : B bA C Ñ pΩB{A bA Cq ‘ pB bA ΩC{A q
given by b b c ÞÑ b b dC c ` dB b b c. Show that d is an A-derivation, and show that
ΩBbA C{A » B bA C Ñ pΩB{A bA Cq ‘ pB bA ΩC{A q.
Exercise 20.11.7. Prove the properties of the sheaf of differentials listed in (20.12).
řn Bf
Exercise 20.11.8. Prove Euler’s formula, that i“0 xi Bx i
“ d ¨ f whenever f is a homoge-
neous polynomial of degree d in x0 , . . . , xn . H INT: Prove it first for monomials.
Exercise 20.11.9. Let A Ă B be a ring extension. Show that
ΩBrts{A “ Brtsdt ‘ pΩB{A bB Brtsq.

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Exercise 20.11.10. Let B “ krx, ys{px2 ` y 2 q. Show that if k has characteristic ‰ 2, then
ΩB{k “ pB dx ‘ B dyq {px dx ` y dyq
and if k has characteristic 2, then ΩB{k “ B dx ‘ B dy .
Exercise 20.11.11. Show that for a point x P X , the stalks of ΩX{S are given by
pΩX{S qx » ΩOX,x {OS,f pxq . (20.27)
Exercise 20.11.12. Let F be an OX module. A morphism D : OX Ñ F of OX -modules is
an S -derivation if for all open affine subsets V Ă S and U Ă X with f pU q Ă V , the map
D|U is an OS pV q-derivation of OX pU q with values in F . The set of all such S -derivations is
denoted by DerS pOX , Fq. Show that the sheaf ΩX{S represents the functor of S -derivations,
in the sense that there is a functorial isomorphism
HomOX pΩX{S , ´q » DerS pOX , ´q.
Show that this property defines ΩX{S up to isomorphism.
Exercise 20.11.13. In this chapter, ΩX{S was defined for a separated morphism X Ñ S .
Show that ΩX{S can be defined in general, without the separatedness assumption. H INT:
Show directly that the sheaves Ω
Ć B{A glue for each affine Spec A Ă S and Spec B Ă X
with Spec B mapping into Spec A.
Exercise 20.11.14 (Smooth morphisms). Let f : X Ñ S be a morphism and let p P X be
a point. We say that f is smooth of relative dimension r at p if there exist affine open sets
U “ Spec B Ă X , containing p, and V “ Spec A Ă S with f pU q Ă V and an open
embedding ι : U Ñ Spec Art1 , . . . , tn s{pf1 , . . . , fn´r q such f |U factors as
ι
U ãÝÑ Spec Art1 , . . . , tn s{pf1 , . . . , fn´r q ÝÝÑ Spec A
and such that the Jacobian matrix Jppq of the fi has rank n ´ r at ιppq.
a) Show that AnS Ñ S and PnS Ñ S are smooth for any scheme S .
b) Show that the property of smoothness is stable under base change.
c) Show that if f : X Ñ S is smooth of relative dimension r, then ΩX{S is locally
free of rank r.
Exercise 20.11.15. Consider the morphism f : Spec Zrx, ys{pxy ´3q Ñ Spec Z. Compute
the smooth locus of f , that is, the set of points p P Spec Zrx, ys{pxy ´ 3q so that f is smooth
at p. Find a maximal open set U Ă Spec Z for which f ´1 pU q Ñ U is smooth.
Exercise 20.11.16 (The conormal sheaf). Let X be a smooth scheme over a field and let
Y be a smooth subscheme defined by an ideal sheaf I . The sheaf I{I 2 is naturally an
OY -module via I{I 2 “ I b OX {I “ I b OY . We call I{I 2 the conormal sheaf of Y . Its
dual, NY “ Hom OY pI{I 2 , OY q is the normal sheaf of Y in X .
a) Show that the sheaves I{I 2 and NY are locally free of rank r “ codimpY, Xq.
b) Show that the conormal sequence
0 Ñ I{I 2 Ñ ΩX|k |Y Ñ ΩY |k Ñ 0
is exact.

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c) Show that
0 Ñ TY Ñ TX |Y Ñ NY Ñ 0
is exact.
Exercise 20.11.17. Let k be an algebraically closed field, and let X and Y be two smooth
hypersurfaces in Pnk of different degrees d and e. Show that X and Y are not isomorphic
unless n “ 2 and either pd, eq “ p1, 2q or pd, eq “ p2, 1q.
Exercise 20.11.18. Let k be a field and let X Ă A2k be the subscheme defined by the ideal
I “ px2 , y 2 q. Show that I{I 2 Ñ ΩA2k {k |X is not surjective. What is the kernel?
Exercise 20.11.19. Let 0 Ñ E Ñ F Ñ G Ñ 0 be an exact sequence of locally free sheaves.
a) If E is an invertible sheaf, show that for any p ě 1, there is an exact sequence
0 Ñ E b Symp´1 F Ñ Symp F Ñ Symp G Ñ 0.
b) If G is an invertible sheaf, show that for any p ě 1, there is an exact sequence
0 Ñ Symp E Ñ Symp F Ñ Symp´1 E b G Ñ 0.
Exercise 20.11.20. Let 0 Ñ E Ñ F Ñ G Ñ 0 be an exact sequence of locally free sheaves.
a) If E is an invertible sheaf, show that for any p ě 1, there is an exact sequence
0 Ñ E b ^p´1 F Ñ ^p F Ñ ^p G Ñ 0.
b) If G is an invertible sheaf, show that for any p ě 1, there is an exact sequence
0 Ñ ^p E Ñ ^p F Ñ ^p´1 E b G Ñ 0.
Exercise 20.11.21. Use Exercise 20.11.20 to show that the sequences (20.23) are exact.
Exercise 20.11.22 (Bott vanishing). Let k be a field. Show that H q pPnk , ΩpPnk pdqq “ 0 except
when:
(i) p “ q and d “ 0.
(ii) q “ 0 and d ą p.
(iii) q “ n and d ą p ´ n.
Exercise 20.11.23. Let f : X Ñ Y be a dominant morphism of schemes, and let E be a
locally free sheaf on Y .
a) Show that if Y is integral, then the pullback map
ΓpY, Eq ÝÝÑ ΓpX, f ˚ Eq (20.28)
is injective.
b) Show that (20.28) is not injective when E is the structure sheaf for the morphism
f : Spec k Ñ Spec krϵs{pϵ2 q and
Exercise 20.11.24 (Algebraic Hartogs for locally free sheaves). Let X be a Noetherian
normal scheme and let E be a locally free sheaf of finite rank on X . Show that if U Ă X is
an open set so that X ´ U has codimension at least 2, then the restriction map
ΓpX, Eq ÝÝÑ ΓpU, E|U q
is an isomorphism. H INT: This may be reduced to the usual statement for E “ OX .

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Exercise 20.11.25. Let X Ă P1k ˆ P2k be a nonsingular surface defined by a bihomogeneous

equation f of bidegree pd, eq.
a) For d “ 1, describe X and the two morphisms to P1 and P2 . In particular, show
that X is rational.
b) For e “ 1, show that X is rational.
c) Show that X is not rational if d ě 2 and e ě 3.
d) * Show that X is rational if e “ 2. H INT: One way is to apply Tsen’s theorem
on quadrics over kptq.
Exercise 20.11.26. Let X Ă P2k ˆ P2k be a nonsingular 3-fold defined by a bihomogeneous
equation f of bidegree pd, eq.
a) For d “ 1 or e “ 1, describe X and the two morphisms to P2 . In particular,
show that X is rational.
b) Show that X is not rational if d, e ě 3.

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Curves

421

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Curves

In this chapter, we will study curves. Recall that a curve is a variety of dimension 1 over
a field k . We will mostly be interested in the case when the field k is algebraically closed,
although most of the results in this chapter remain valid over more general fields, for instance,
perfect fields.
When k is algebraically closed, it is quite easy to understand X as a topological space, as
the points are exactly the k -points and the generic point. Moreover, the non-empty closed
sets are either finite sets of k -points or the whole space X . As only the cardinality of k plays
a role, this implies that any two curves over k are homeomorphic (!).
Because of this, a ‘point’ will usually mean ‘closed point’ in this chapter. We will usually
use capital letters P, Q, R, . . . for points in X , to emphasise that closed points are divisors.
We will also focus on nonsingular curves. In other words, each local ring OX,P is a regular
local ring of Krull dimension 1. Equivalently, for each k -point x, the cotangent space mP {m2P
is 1-dimensional as a vector space over k .

Proposition 21.1. Let X be a curve over an algebraically closed field k . Then the
following are equivalent
(i) X is nonsingular
(ii) X is smooth over k
(iii) ΩX{k is an invertible sheaf
(iv) X is normal.

Proof As k is algebraically closed, the equivalence of (i) ô (ii) holds in any dimension,
see Proposition 11.18. The equivalence (i) ô (iv) holds by Corollary 11.48 as each local ring
OX,P has dimension 1.

What makes the study of curves special is that the local rings OX,P at (closed) points are
discrete valuation rings, so they satisfy the statements in Proposition 11.47. In particular, this
means that the maximal ideal mP Ă OX,P is a principal ideal. A generator t P mP for mP is
called a local parameter at P .
Any non-zero rational function f P KpXq can be expressed uniquely f “ u ¨ tn where
n P Z and u is a unit in OX,x . The corresponding valuation map

ordP : KpXqˆ ÝÝÑ Z

is defined by setting ordP pf q “ n. Note that ordP pZě0 q “ OX,P and ordP pZě1 q “ mP .

423

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This fact will be important later on, when we study morphisms from curves to other
varieties and when we study divisors.
Example 21.2. If X is a curve over k , then the normalization X is a nonsingular curve. The
normalization map π : X Ñ X is finite and birational. △
Example 21.3 (Local parameters for plane curves). Let X Ă A2k “ Spec krx, ys be the
affine curve defined by an equation f px, yq “ 0 and let P “ pa, bq be a k -point of X . Then
mP {m2P is generated by x ´ a and y ´ b. If we assume that X is nonsingular at P , the
tangent space is defined by the linear equation
Bf Bf
pP q ¨ px ´ aq ` pP q ¨ py ´ bq “ 0. (21.1)
Bx By
This means that if Bf
By
pP q ‰ 0, then we can express y ´ b in terms of x ´ a in mP {m2P . In
other words: if Bf
By
pP q ‰ 0, then x ´ a is a local parameter at P . △

Example 21.4. Let X be the plane curve defined by the equation x2 ` y 2 “ 1 in A2 “

Spec krx, ys and let P “ p1, 0q. Then the rational function 1 ´ x has valuation 2 at P . To
B
see this, note that y is a local parameter at P , because Bx px2 ` y 2 ´ 1q “ 2x is nonzero at
P . Moreover, in OX,P , we have the relation p1 ´ xqp1 ` xq “ y 2 , and 1 ` x is a unit in
OX,P , and hence ordP p1 ´ xq “ 2. △

21.1 Morphisms between projective curves

Morphisms between curves f : X Ñ Y are rather simple. There are two cases: either f is not
dominant, in which case the image f pXq is a point (being a proper, closed and irreducible
subset of X ); or it is dominant, in which case f pXq is dense in Y , hence equal to an open
set of Y . In the case X and Y are also projective, we can say more.

Proposition 21.5. Let f : X Ñ Y be a dominant morphism of curves over k .

(i) KpXq is a finite field extension of KpY q.
(ii) If X is projective over k , then f is surjective.
(iii) If both X and Y are projective, then f is finite.

Proof (i): As f is dominant, f 7 induces a map of function fields KpY q Ñ KpXq. The two
function fields KpXq and KpY q are both of transcendence degree 1 over k , and so KpXq is
algebraic over KpY q. Note that X is of finite type over Y , since it is of finite type over k .
Therefore KpXq is a finite extension of KpY q.
(ii): If X is projective, then the image f pXq is closed in Y (by Corollary 19.9), hence
equal to Y , as f is dominant.
(iii): By the assumption, the morphism f : X Ñ Y is projective, and so Proposition 19.12
shows that f is finite.
Example 21.6. It is essential that X is projective for the items (ii) and (iii) to hold: the
morphism Spec krx, ys{pxy ´ 1q Ñ Spec krxs is dominant, but it is neither finite nor
surjective. △

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This leads to the notion of the degree of a morphism between curves:

Definition 21.7 (The degree of a finite morphism). Let f : X Ñ Y be a dominant

morphism between curves. The degree of the field exension KpY q Ă KpXq is called
the degree of f and is denoted deg f .

Note that if f : X Ñ Y and g : Y Ñ Z are dominant morphisms between curves, the

composition g ˝ f is dominant and degpg ˝ f q “ deg f ¨ deg g . This follows because
rM : Ks “ rM : LsrL : Ks for a tower of field extensions K Ă L Ă M .

Example 21.8. The ’n-th power map’ f : P1k Ñ P1k of Example 14.34 has degree n. Indeed,
locally it is given by the ring map krus Ñ krxs sending u to xn , and kpxq “ kpuqrxs{pu ´
xn q has dimension n as a kpuq-vector space. △

Example 21.9. If X is a hyperelliptic curve, as discussed in Section 6.4, then the morphism
X Ñ P1k has degree 2. In this case, KpXq is obtained from kpxq by adjoining a square root
y of a polynomial f pxq P krxs, so it is spanned by 1 and y as a kpxq-vector space. △

21.2 Extensions of rational maps

This section presents two fundamental results on nonsingular curves. The first basically says
that any rational map from a nonsingular curve into a a projective variety can be extended to
a morphism. Combining this with Theorem 10.3 and the fact that every curve is birationally
equivalent to a nonsingular curve, we obtain the second result, which says that the category
of projective nonsingular curves over k with dominant maps is equivalent to the category of
finitely generated field extensions of k of transcendence degree 1.

Proposition 21.10. Let X be an irreducible curve over k and let P P Xpkq be a

nonsingular point. Then any morphism f : X ´ tP u Ñ Y to a projective variety Y has
a unique extension to a morphism f : X Ñ Y .

Proof It suffices to treat the case where Y “ Pnk . Let t be a local parameter of OX,P ,
and denote by K the function field of X . Restricting to the generic point, the morphism f
gives a K -point Spec K Ñ PnK . By Proposition ??, any such a morphism is described by
homogenous coordinates pa0 tν0 : ¨ ¨ ¨ : an tνn q where the ai are units in OX,P and the νi ’s
are integers. After multiplying by t´ min νi we may assume that νi ě 0 for all i and at νi0 “ 0
for at least one i0 . Now the ai are induced by non-vanishing sections of OX over some some
open neighbourhood U of P . Also, after shrinking U if neccesary, we may assume that t
will also be a section of OX over U with P as the only zero. Therefore, the ai tνi define a
map U Ñ Pnk . Over the open set U ´ tP u, this morphism coincides with f , so we have the
desired extension.
Finally, the extension is unique because X is separated over k .

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Corollary 21.11. Any rational map between two nonsingular projective curves can be
extended to a morphism. In particular, if two nonsingular projective curves are birational,
then they are isomorphic.

Proof Any rational map X 99K Y is a morphism outside a finite number of points
x1 , . . . , xr . Therefore, the first statement follows by applying Proposition 21.10 finitely
many times.
If two curves X and Y are birationally equivalent, then there are open subsets U Ă X
and V Ă Y and an isomorphism f : U Ñ V . If X and Y are nonsingular, both f and f ´1
extend respectively to morphisms g : X Ñ Y and h : Y Ñ X , and since h ˝ g|U “ idU and
g ˝ h|V “ idV , it follows that h ˝ g “ idX and g ˝ h “ idY (two morphisms that agree on
an open dense set of a separated scheme are equal, by Proposition 9.50).
Example 21.12. Let X Ă P2k be the projective curve defined by the equation x0 x22 “
x31 ` x0 x21 . The projection px0 : x1 : x2 q ÞÑ px0 : x1 q induces a morphism
f : X ´ tp0 : 0 : 1qu Ñ P1k .
To see that f extends over p0 : 0 : 1q, following Proposition 21.10, consider the affine
open set D` px2 q “ A2k with affine coordinates u “ x0 {x2 and v “ x1 {x2 . In this chart, the
equation of X becomes u “ v 3 ` uv 2 and p0 : 0 : 1q corresponds to pu, vq “ p0, 0q. The
morphism f is given by pu, vq ÞÑ px : yq.
Since u “ v 3 ` uv 2 , we can express u in terms of v as u “ v 3 p1 ´ v 2 q´1 in OX,p0,0q
(note that 1 ´ v 2 is a unit in OX,p0,0q ). This shows that t “ v is a local parameter at p0, 0q.
In terms of t, the morphism takes the form
ˆ 3 ˙
t
pu, vq ÞÑ : t .
1 ´ t2
Dividing by t, shows that the extension is given by
ˆ 2 ˙
t `2 2
˘
pu, vq ÞÑ : 1 “ t : 1 ´ t .
1 ´ t2
Hence the extension of f sends p0 : 0 : 1q to p0 : 1q.
Note that while f extends over p0 : 0 : 1q the original map P2 ´ tp0 : 0 : 1qu Ñ P1k does
not. △

Proposition 21.13. Any nonsingular curve is quasi-projective.

Proof Let X be a nonsingular curve and let U1 , . . . , Us be an affine cover of X . For

each i “ 1, . . . , s, choose an affine embedding Ui Ñ Ank i , and embed Ank i Ă Pnk as a
distinguished open set. Let Ui denote the closure of Ui in Pnk i . As X is nonsingular and Ui is
projective, we may extend the embedding Ui Ñ Ui to a morphism ϕi : X Ñ Ui .
Next we consider the product of all of the ϕi
ϕ : X ÝÝÑ U1 ˆ ¨ ¨ ¨ ˆ Us Ă Pnk 1 ˆ ¨ ¨ ¨ ˆ Pnk s .
Let V be the closure of ϕpXq. Then V is a projective variety (by the Segre embedding).

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Note that if we restrict to Ui Ă X , the composition of ϕ with the projection pi : V Ñ Ui

to the i-th factor coincides with the inclusion Ui Ñ Ui . Therefore, the induced morphism
ϕ : X Ñ V is not constant, so V is a projective curve. Moreover, the image W “ ϕpXq
is an open set of Y . We claim that ϕ : X Ñ W is an isomorphism; this will prove the
proposition.
To see this, let x P X , and choose an index i such that x P Ui . As the composition
ϕi pi
Ui ÝÑ W Ý Ñ Ui coincides with the inclusion Ui Ă Ui , the induced composition of maps of
local rings
p7i,ϕpxq ϕ7i,x
OUi ,ϕi pxq ÝÝÝÝÑ OV,ϕpxq ÝÝÑ OX,x
is an isomorphism. Note that both pi and ϕi are dominant, so both ring maps here are injective,
and hence ϕ7i,x is an isomorphism. Moreover, V is nonsingular, hence normal, at every point
of the open set W “ ϕpXq. As ϕ : X Ñ W is a birational morphism of normal curves, it is
an isomorphism (by the universal property of normalization).

Corollary 21.14. Let X be a projective curve over a field k . Then the normalization X
is also projective.

Proof By the previous proposition, the nonsingular curve X is in any case quasi-projective,
so we may regard X Ă Pnk as a locally closed subset of projective space. We claim that X is
closed in Pnk . Let V denote the closure of X and let σ : V Ñ V denote the normalization of
V . As V and X are birational, there is a birational map V 99K X . As X is projective, this
extends to a morphism g : V Ñ X . As V is normal, this factors via a morphism h : V Ñ X
as in the diagram below:

V
h
σ

X X V

From the diagram, we have σpV q Ă X . However, σ is finite, hence surjective by Proposition
9.27. This gives a contradiction unless X “ V . This completes the proof.

Theorem 21.15. There is an equivalence of categories between the following categories:

(i) The category of nonsingular projective curves over k and dominant mor-
phisms.
(ii) The category of finitely generated field extensions of k of transcendence
degree 1 and maps of k -algebras (with the arrows reversed).

Proof If X and Y are two nonsingular projective curves, any rational map X 99K Y
extends to a morphism. This, combined with Theorem 10.24 on page 208, shows that the
functor X ÞÑ KpXq is fully faithful.
Next, to show that it is essentially surjective, we need to show that every finitely generated
field K of transcendence degree 1 over k is of the form KpXq for some nonsingular
projective curve X . If a1 , . . . , ar is a transcendence basis for K over k , then the k -subalgebra

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A “ kra1 , . . . , ar s will be of dimension 1, by Theorem 10.6 on page 201. The affine scheme
X “ Spec A is naturally a closed subvariety of the affine space Ark , and taking the Zariski
closure in Prk , yields a (possibly singular) projective curve Y which is birational to X . Taking
the normalization of Y , we finally get a nonsingular projective curve which has K as its
function field (by Corollary 21.14).
The above theorem is not true with ‘projective’ replaced by ‘affine’. A1k and A1k ´ t0u are
two affine curves which isomorphic function fields but which are not isomorphic.
Example 21.16 (Morphisms to P1k ). If X is a nonsingular curve, then any element g P KpXq,
defines a morphism to P1k ,
G : X Ñ P1k . (21.2)
We define G as follows. Let U Ă P1k be the maximal open set where g is regular, that is,
g P OP1k pU q. Likewise, let V Ă P1k be the maximal open set where 1{g is regular. Note that
U X V is non-empty (it contains the generic point), and U Y V “ X (for each p P P1k , the
local ring OX,P is a DVR, hence either g or 1{g belongs to it). Now g P OX pU q defines a
morphism U Ñ D` pt0 q » A1k . Likewise, 1{g defines a morphism U Ñ D` pt1 q » A1k , and
these clearly glue to a morphism G : X Ñ P1k .
Note that G is a constant map if and only if g is constant. This has the following conse-
quence:

Proposition 21.17. For a nonsingular curve X over k , there is a one-to-one correspon-

dence between the elements of KpXq ´ k and dominant morphisms of k -varieties
X Ñ P1k .

21.3 Sheaves on curves

Recall that an element of an A-module is called a torsion element if it is killed by a nonzero-
divisor of A, and a module is a torsion module if all elements are torsion. On the other hand,
a module is torsion free if no non-zero element is torsion. The sum of two torsion elements is
clearly torsion, so the subset of a module M formed by the torsion elements, is a submodule
T . It has the property that M {T is torsion free. If the base ring A is a PID, a torsion free
A-module is in fact free. This follows from the Structure Theorem for Modules over a PID,
see Corollary A.60.
This generalizes to sheaves on curves as follows. If X is a curve, then any OX -module of
finite type contains a torsion subsheaf T , whose sections over an open set U Ă X equals
the subgroup of FpU q of elements annihilated by some nonzerodivisor of OX pU q (see
Exercise 14.12.40 on page 295). The quotient E “ F{T is torsion free in the sense that
on open affine subsets U EpU q (which equals FpU q{T pU q) is a torsion free module over
OX pU q. Note that for x P X , the stalk Ex is a torsion free module over OX,x which is a PID.
Hence every stalk Ex is free, and so E is locally free by Proposition ??.
As X is a curve, the torsion part T is a sheaf supported on finitely
Àr many points p1 , . . . , pr .
Moreover, T is the direct sum of its stalks at these points: T “ i“1 Tpi .

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Theorem 21.18. Let X be a nonsingular curve and let F be an OX -module of finite

type. Then there is a decomposition
F “E ‘T
where T Ă F is the torsion subsheaf and E is locally free.

Proof What remains to be seen is that the splitting F “ E ‘ T . We claim that the exact
sequence
0 T F E 0 (21.3)
is split exact. Let U be an affine neighbourhood about pi . By Exercise ??, E|U is the tilde
of a projective module, and so the sequence (21.3) splits when restricted to U . Hence there
is a map ϕi : F|U Ñ Tpi splitting the inclusion Tpi Ñ F . This map extended by zero
gives a map of sheaves ϕi : F Ñ Tpi . Then, taking the direct sum of the ϕi , we get a map
ϕi : F Ñ T which splits the inclusion T Ñ F for the entire torsion sheaf.

Proposition 21.19. Let f : X Ñ Y be a finite morphism of projective nonsingular curves

over a field k . Then f˚ OX is a locally free sheaf of rank deg f . More generally, if E is a
locally free sheaf of finite rank on X , then f˚ E is locally free of rank pdeg f qprank Eq.

Proof Note first that f˚ OX and f˚ E are coherent, because f is finite (see Exercise 14.12.35).
Let U “ Spec B Ă X and V “ Spec A Ă Y be affine open subsets so that f pU q Ă V
and f |U is induced by a ring map ϕ : A Ñ B . Then ϕ is injective, because X and Y are
integral and f is dominant.
Then f˚ OX |V “ f˚ A r “ B ĂA . Now, as B is an integral domain, and ϕ : A Ñ B is
injective, B is torsion-free as an A-module, and hence f˚ OX is torsion-free when restricted
to V . Therefore, f˚ OX is locally free by Theorem 21.18. To compute the rank, note that the
stalk of f˚ OX at the generic point is given by the fraction field of B , i.e., KpXq, and this
has rank rKpXq : KpY qs “ deg f as a vector space over KpY q.
The statement for f˚ E follows from this, because as X is a curve, E is trivial over a
covering consisting of open sets of the form f ´1 pU q where U is affine.

21.4 Divisors on curves

Let X be a nonsingular curve over an algebraically closed field k . The codimension 1-subsets
of a curve are precisely the closed points, so that a Weil divisor is a finite formal combination
ÿ
D“ ni Pi
of closed points P1 , . . . , Pr in X , where the coefficients are integers. We define the degree as
ÿ
deg D “ ni
If D is given by the Cartier data pUi , fi q (see Definition 17.32), then
ÿ
deg D “ ordP pfi q
P PX

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where we for each P choose an i so that P P Ui .

Pullbacks of divisors
If f : X Ñ Y is a dominant morphism of curves, we can pull back invertible sheaves from
Y to X , as well as sections of these. By the correspondence between divisors and invertible
sheaves, this gives us a way of pulling back divisors from Y to X . In the context of projective
curves, we can make this a little bit more explicit.
We assume that the morphism f : X Ñ Y is dominant, hence finite, it is then surjective,
and f 7 induces an inclusion of function fields KpY q Ă KpXq.
If Q P Y is a closed point, choose a local parameter t P OY,Q . Then we define
ÿ
f ˚ pQq “ ordP pf 7 tqP,
P Pf ´1 pQq

where as usual ordP is the order of vanishing at P . The number ordP pf 7 tq is called the
ramification index of f at P , and it is usually written eP . Multiplying t by a unit in OY,Q does
not alter ordP pf 7 tq because a unit in OY,Q stays a unit in OX,P . Therefore, the ramification
index eP is independent of the choice of local parameter t. Extending f ˚ by linearity, we
obtain a well-defined map of groups

f ˚ : Div X ÝÝÑ Div Y.

We can also understand this map on the level of Cartier divisors: if D is a Cartier di-
visor on Y given by the data tpUi , gi qu, where gi P KpY qˆ , we can consider the data
tpf ´1 Ui , f 7 gi qu, which defines a Cartier divisor on X .
Note that f ˚ pQq is a divisor supported on the inverse image f ´1 pQq. In fact, divisor
f ˚ pQq has the same multiplicities as the scheme-theoretic fiber XQ . This follows because
f ˚ pQq is locally defined by the equations f 7 pgi q and these form local generators for the ideal
sheaf of the closed subscheme XQ .

Example 21.20 (Principal divisors). Let g P KpXqˆ be a rational function. Then if G : X Ñ

P1k denote the induced morphism, (from Example 21.16 on page 428), we have

div g “ G˚ pp0 : 1q ´ p1 : 0qq.

This follows because G7 pt0 {t1 q “ g and G7 pt1 {t0 q “ 1{g and and t0 {t1 and t1 {t0 are local
parameters at p0 : 1q and p1 : 0q respectively. △

Lemma 21.21. If f : X Ñ Y is finite and D is a divisor on Y , we have

deg f ˚ D “ deg f ¨ deg D.

Proof It suffices to treat the case when D “ Q is a point. Let Spec A be an affine
neighbourhood of Q and Spec B the inverse image of Spec A. As we saw in Proposition
21.19, B is a torsion free A-algebra which is locally free of rank equal to rKpXq : KpY qs “
deg f . If t is a local parameter at y the value ordP pf 7 tq is the ramification index of f at P .

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Then
ÿ ÿ
deg f ˚ Q “ ordP pf 7 tq “ eP “ deg f.
f pP q“Q f pP q“Q

Lemma 21.22. For a non-zero g P KpY q and a morphism f : X Ñ Y , we have

f ˚ div g “ div g ˝ f.

Proof Let G : Y Ñ P1 be the extension of g then G ˝ f is the extension of g ˝ f and so

according to Example 21.20 above, it holds that
f ˚ div g “ f ˚ pG˚ pp0 : 1q ´ p1 : 0qq “ pG ˝ f q˚ pp0 : 1q ´ p1 : 0qq “ div g ˝ f.

Corollary 21.23. For a non-zero rational function g P KpXq, we have deg div g “ 0.
Hence the degree map descends to a well-defined map
deg : ClpXq Ñ Z.
In other words, linearly equivalent divisors have the same degree.

Proof This is clear if g is a constant. If not, g defines a dominant morphism G : X Ñ P1k

so that
div g “ G˚ pp1 : 0q ´ p0 : 1qq.
and we are done by the above lemma.

Corollary 21.24. If D is a divisor with degpDq ă 0, then ΓpX, OX pDqq “ 0.

Proof The group ΓpX, OX pDqq consists of rational functions f P KpXq so that divpf q `
D is effective. However, if f is non-zero, then divpf q ` D has negative degree, whereas
effective divisors have non-negative degree.
Example 21.25. Assume that k is a field of characteristic 0. and consider the curve X Ă
A2k “ Spec kru, vs given by the equation
v 2 “ u3 ` u2 ` 1 (21.4)
which is a nonsingular curve. Consider the rational function g “ v ` 1 on X . What is div g ?
Note that g is regular, so there are no points P for which ordP pgq ă 0. Rewriting (21.4) as
pv ´ 1qpv ` 1q “ u2 pu ` 1q, (21.5)
we see that the zeros of v `1 are the points x “ p0, ´1q and y “ p´1, ´1q. Near x “ p0, 1q
both pv ` 1q and u ` 1 are invertible, and the equality
v ´ 1 “ u2 pu ` 1qpv ` 1q´1 (21.6)
shows that u is local parameter there (the maximal ideal mP is generated by v ´ 1 and u). In

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the same vein, near y “ p´1, ´1q both v ` 1 and u are invertible, and we infer from (21.6)
that u ` 1 is a local parameter. It follows that
div g “ ordP pu2 qP ` ordy pu ` 1qQ “ 2P ` Q. (21.7)
△
Example 21.26. Consider the curve Y Ă P2k “ Proj t0 , t1 , t2 given by the equation
t22 t0 “ t31 ` t21 t0 ` t30
Note that the curve in the previous example equals X X Dpt0 q, where we use coordinates
u “ t1 {t0 , v “ t2 {t0 . Let us compute div g for the same rational function g “ t2 {t0 ` 1 as
before, but this time on Y . For this, we only need to consider the points where t0 “ 0. From
the equation, we see that there is a single point in Y XV pt0 q, namely the point z “ p0 : 0 : 1q.
To compute ordP pgq here, we use the chart Dpt1 q. Then Y X Dpx2 q is isomorphic to the
plane curve given by the equation
u “ v 3 ` v 2 u ` u3 (21.8)
where now u “ t0 {t2 and v “ t1 {t2 . The point z is then the origin pu, vq “ p0, 0q in
D` pt2 q. Note that g “ u´1 ` 1. Rewriting (21.8) as
v 3 “ up1 ´ v 2 u ´ u3 q,
we see that v is a uniformzer at z and that ordP puq “ 3. Hence we find we also see that
ordz puq “ 3, and so

ordP pgq “ ordP pu´1 ` 1q “ ordP pu ` 1q{uq “ ordP pu ` 1q ´ ordP u “ ´3

Finally, we concude thats
div g “ 2p1 : 0 : ´1q ` p1 : ´1 : ´1q ´ 3p0 : 0 : 1q. (21.9)
Note that, since Y is projective we may use Corollary 21.23 and conclude that deg div g “ 0,
which immediately yields 21.9. △

The canonical divisor

Also in this section, we work with a curve X defined over an algebraically closed field k . As
k is algebraically closed, the module of differentials ΩX{k is an invertible sheaf.

Definition 21.27 (The canonical divisor). Let X be a nonsingular curve over k . A

canonical divisor is a divisor KX such that
OX pKX q » ΩX{k . (21.10)

Because of (21.10), any two canonical divisors will be linearly equivalent. It is therefore
the class of KX in the class group ClpXq which is ‘canonical’, rather than the divisor itself.
In our setting, one can describe the set of canonical divisors as follows. We will consider

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ΩKpXq{k , which is are called rational differential forms. Note that ΩKpXq{k is 1-dimensional
as a KpXq-vector space, being the stalk of the invertible sheaf ΩX{k at the generic point.
If η is a local generator ΩX{k at a point P P X , it defines a generator for ΩKpXq{k as a
KpXq-vector space as well, and every rational differential form ω is of the form ω “ gη for
some rational function g P KpXq. In fact, ΩKpXq{k “ ΩOX,P {k bOX,P KpXq.
Now let ω P ΩKpXq{k be a nonzero element. For each point P P X choose a generator ηP
for ΩOX,P {k and write ω “ gP ηP with gP P KpXq. Then define
ÿ
div ω “ ordP pgP qP. (21.11)
P PX

The expression on the right in (21.11) is independent of the choice of local generators. This is
because two generators ηP1 and ηP will be related by ηP “ uηP1 with u a unit in OX,P , and
hence ω “ gP ηP “ ugP ηP1 , and ordP pugP q “ ordP pgP q. Note also that the sum in fact is
finite, because any local generator ηP extends to a generator for ΩU {k in some neighbourhood
U of P , and the corresponding rational function gP has only finitely many zeroes and poles
(and X ´ U is a finite set as well).
If ω and ω 1 are two rational differentials, then ω 1 “ hω for some h P KpXq. Therefore,
for each P P X we have ordP phgP q “ ordP phq ` ordP pgP q, and by the formula (21.11),
we get
divphωq “ div h ` div ω.

What we have done so far is valid over any field as long as ΩX{k is invertible. When the
ground field is algebraically closed, there is a local description of the rational differentials in
terms of local parameters that make calculations easier.

Lemma 21.28. Assume that X is nonsingular at the closed point P P X and that t is
local parameter at P . Then each element of ΩKpXq{k is of the form g dt with g P KpXq;
in other words, ΩKpXq{k is of rank 1 over KpXq with dt as a basis.

Proof The Zariski cotangent space mP {m2P at P is always generated by the class of a local
parameter, and by Proposition ??, it follows that dt generates ΩOX,P {k when X is nonsingular
at P .

Note that gdt defines a section of ΩU {k over some open set U Ă X . In other words, ω is a
rational section of ΩX{k . Using Proposition 17.37, we obtain

Proposition 21.29. The invertible sheaf associated to div ω is isomorphic to ΩX .

Example 21.30. Let us find the canonical divisor of P1k “ Proj krx0 , x1 s. Write x for the
affine coordinate t “ x1 {x0 on D0 px0 q. We will consider the differential

ω “ dt

which is an element in Ωkptq{t . Note that ω “ 1 ¨ dt “ 1 ¨ dpt ´ aq for any a P k . Thererefore,

as t ´ a forms a local parameter at a, we see that the divisor of ω is 0 on D` px0 q. It remains

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to see what happens at the point p0 : 1q. Here u “ x´1 is a local parameter, and
dx “ u´2 du
Therefore, divpωq “ ´2p0 : 1q. As this divisor has negative degree, we again compute that
ΓpP1k , ΩP1k q “ 0. △
Example 21.31. Assume that k is of characteristic different from 2. Let X Ă A2k be the
elliptic curve given by the equation
v 2 “ u3 ´ u,
and consider the differential ω “ du. At a point p “ pa, bq where b ‰ 0, the coordinate u is
a local parameter, and so du “ dpu ´ aq has zero valuation at p. When b “ 0, the curve has
three points: p1 “ p0, 0q, p2 “ p´1, 0q, and p3 “ p1, 0q.
At these points, v will be a local parameter, and since 2vdv “ p3u2 ´ 1qdu, it holds that
du “ 2v{p3u2 ´ 1qdv.
Hence ordpi pduq “ 1 for all three. Summing up, we conclude that
div ω “ p1 ` p2 ` p3 .
△
Example 21.32. We consider the projectivization X Ă P2k of the previous example, i.e. the
curve whose homogeneous equation is
x21 x2 “ x30 ´ x0 x22 .
Consider again the rational differential ω “ dpx0 {x2 q. We know the behaviour of ω on
the distinguished open set D` px2 q, so what remains to compute the divisor of ω , is the
valuation ordP pωq for each point in X X V px2 q, but this intersection has just one single
point x “ p0 : 1 : 0q.
Dehomogenizing the chart D` px1 q by setting u “ x0 {x1 and v “ x2 {x1 , the equation of
X in D` px1 q becomes
v “ u3 ´ uv 2 .
Since 1 ` uv is invertible near x, this shows that u is a local parameter at x and that
ordP pvq “ 3. Our differential ω takes the form ω “ dpx0 {x1 ¨ x1 {x2 q “ dpu{vq “
pudv ´ vduq{v 2 . We find
dv “ 3u2 ´ v 2 ´ 2uvv 1 qdv,
which yields
udv ´ vdu “ p3u3 ´ uv 2 ´ 2u2 vv 1 ´ vqdu
(21.12)
“ p2u3 ´ 2uv 2 ´ 2u2 vv 1 qdu
The terms uv 2 , 2u2 vv 1 vanish to order at least 5 at x, and the dominating term in (21.12) is
2u3 , which means that ordP pωq “ ordP pu3 q ´ 2 ordP pvq “ ´3. We conclude that
div ω “ p0 : 0 : 1q ` p´1 : 0 : 0q ` p1 : 0 : 0q ´ 3p0 : 1 : 0q
△

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21.5 The genus of a curve

Let X be a projective nonsingular curve over a field k . We define the arithmetic genus of X
is defined as the integer
pa pXq “ 1 ´ χpOX q “ 1 ´ dimk H 0 pX, OX q ` dimk H 1 pX, OX q.
If we assume that X is geometrically irreducible, that is, Xk̄ is irreducible, then H 0 pX, OX q “
k , and so pa pXq “ dimk H 1 pX, OX q.
The geometric genus of X is defined as
pg pXq “ dimk H 0 pX, ΩX q.
Note that χpOX q “ dim H 0 pX, OX q ´ dim H 1 pX, OX q, so we get
pa pXq “ 1 ´ χpOX q.
These numbers are defined using different sheaves, and there is no a priori reason to expect
that they should have anything to do with each other. However, we shall see later in the
chapter that there is a strong relation between them: they are equal whenever X is nonsingular.
For the time being we will still refer to the arithmetic genus pa as the genus of X .
Example 21.33. When X “ P1 , we have H 1 pP1 , OX q “ 0 so the arithmetic genus is 0.
Likewise, we have that H 0 pP1 , ΩP1 q “ H 0 pP1 , OP1 p´2qq “ 0, so also pg “ 0. △
Example 21.34. Let X Ă P2 be a plane curve, defined by a homogeneous polynomial
d´1
f px0 ,1 , x2 q of degree d. In Chapter ??, we computed that H 1 pX, OX q » k p 2 q . Hence the
genus of X is equal to pd´1qpd´2q
2
. △
Example 21.35. Let X “ P1k with the usual covering U0 “ Spec krts and U1 “ Spec krt´1 s.
The differential form dt is an element of ΩKpXq{k , which generates ΩX{k |U0 . This means
that ordx pdtq “ 0 for every x P U0 . For the remaining point p1 : 0q at infinity, note
that t´1 is the local parameter there, p1 : 0q corresponding to the origin in U1 . We have
dpt´1 q “ ´t´2 dt, and hence dt “ ´pt´1 q´2 dpt´1 q. This means that ordp1:0q dt “ ´2, so
that div dt “ ´2p1 : 0q.
As a Cartier divisor, the corresponding divisor is given by pU0 , 1q, pU1 , t2 q. This shows
that ΩX » OP1 p´2q. △
Example 21.36. The arithmetic genus can show unexpected behaviour over non-algebraically
closed fields. For instance, one can regard X “ P1C as a nonsingular curve Spec R. Then
dimR H 0 pX, OX q “ 2, and dimR H 1 pX, OX q “ 0, and hence X has arithmetic genus
pa pXq “ ´1! The issue is that X is not geometrically irreducible, as the base change XC
has two components. △

21.6 Hyperelliptic curves

Let X be a hyperelliptic curve defined by the affine equation
y 2 “ xn ` an´1 xn´1 ` ¨ ¨ ¨ ` a0
Let λ1 , . . . , λn be the roots of ppxq. We assume that X is smooth, so the λi are distinct.

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Example 21.37 (Canonical divisors). We can compute a canonical divisor of X by choosing

KX “ divpωq, where ω “ y ´1 dx (the appearance of the term y ´1 will be explained later.)
In the affine chart U “ Spec krx, ys{py 2 ´ ppxqq, with coordinates x and y , we have

KX |U “ 0. (21.13)

Indeed, for any point pa, bq P U , where b ‰ 0, x ´ a defines a local parameter, as B

By
py 2 ´
ppxqq “ 2y is nonzero at P . Moreover,
ω “ y ´1 dx “ y ´1 dpx ´ aq.
so ordP pωq “ 0 in this case, because y is a unit in OX,P .
For the points of the form pλi , 0q, y forms a local parameter (as y 2 equals px ´ λi q times
a unit in the local ring). Here we have
dx 2 dy
“ 1
y p pxq
As p as no repeated roots, p1 pxq is a unit in OX,P , so ordP pωq “ 0 also in this case. This
means that KX |U “ 0.
The remaining points lie in the closed set V puq in Spec kru, vs{pv 2 ´ u2g`2 ppu´1 qq. If
we assume n is even, then these are two points P1 “ p0, 1q and P2 “ p0, ´1q. For these
points, u defines a local parameter. Moreover,
dx 2 dy ug´1 du
ω“ “ 1 “´ .
y p pxq v
As v is invertible at P1 and P2 , we get

divpωq “ pg ´ 1qP1 ` pg ´ 1qP2 . (21.14)

If n is odd, then V puq consists of a single point, P “ p0, 0q, and v 2 “ u¨(unit) in OX,P .
Therefore, v is a local parameter at P and ordP puq “ 2. We furthermore get that 2vdv “
du¨(unit), and hence by (21.14), we get
divpωq “ pg ´ 1q ordP puqP “ p2g ´ 2qP.
△

Proposition 21.38. Let X be a hyperelliptic curve of genus g defined by the affine

equation y 2 “ ppxq where p has degree 2g ` 1 or 2g ` 2. Then
dx x dx xg´1 dx
H 0 pX, ΩX q “ k ‘k ‘ ¨¨¨ ‘ k
y y y

Let us consider some examples of divisors on these hyperelliptic curves. For simplicitly,
we will assume that n “ 2g ` 1 is odd, and X is defined using the affine equation

y 2 “ ppxq “ x2g`1 ` x
In other words, X is glued together by the affine schemes U “ Spec A and V “ Spec B ,

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where
krx, ys kru, vs
A“ and B “
p´y 2 `x 2g`1 ` xq p´v ` u ` u2g`1 q
2

As before, we glue Dpxq to Dpuq using the identifications u “ x´1 and v “ x´g´1 y .
We will compute the groups ΓpX, OX pmP qq, where P is the ‘point at infinity’ of X , that
is, the point corresponding to the maximal ideal m “ pu, vq in V . In other words, we look
for rational functions f P KpXq which are regular on X ´ P “ U and have a pole of order
at most m at p.
To control the pole of f at P , we first compute the orders of vanishing of x, y, u, v at P .
In the local ring OX,P “ Bm , we can use the defining relation to express u as

u “ v 2 p1 ` u2g q´1 “ v 2 punitq. (21.15)

Therefore, v is a local parameter at P . Using this, the orders of v, u, x, y are given by

ordP pvq “ 1, ordP puq “ 2 ordP pxq “ ordP pu´1 q “ ´2

ordP pyq “ vpxg`1 vq “ pg ` 1qp´2q ` 1 “ ´2g ´ 1. (21.16)

Let us identify KpXq with the fraction field of A, so that f is expressed in terms of x and
y . Note P does not lie in U , and in fact X ´ P “ U , as V puq “ V pu, vq “ tP u. As f is
regular on U “ Spec A, this means that f can be identified with a polynomial in x and y .
If we view A as a module over krxs, we can write A “ krxs ‘ krxsy and express f as
apxq ` bpxqy where apxq and bpxq are polynomials. By (21.16), we have
ordP papxqq “ ordP papu´1 qq “ ´2 deg a
ordP pbpxqyq “ ordP pbpu´1 qyq “ ´ deg b ´ 2g ´ 1.
Therefore, as we assume g ě 1, we have ordP pf q ď ´2 for any non-constant rational
function with a pole at P . Hence ΓpX, OX pP qq “ k consists of only the constants.
On the other hand, for the divisor 2P , we obtain a new section, namely the rational function
x. As ´ deg b ´ 2g ´ 1 ď ´3, we find that there are additional functions with bpxq ‰ 0,
and hence
ΓpX, OX p2P qq “ k ‘ k x.
The invertible sheaf L “ OX p2P q is in fact globally generated by these two sections: the
section ‘1’ generates L over the open set X ´ p, and x generates L in a neighbourhood of p.
This is of course no big surprise: the morphism defined by 1, x is exactly the double cover
morphism X Ñ P1k .
For the divisor D “ 3P , we allow poles of order 3 at P . Note that we have ordP papxq `
bpxqyq ď ´4 unless g “ 1 and b is constant. This means that ΓpX, OX p3pqq “ k‘kx‘ky
if g “ 1, and ΓpX, OX p3pqq “ k ‘ kx for g ě 2.
Let us investigate the two cases g “ 1 and g “ 2 in more detail.

Example 21.39 (g “ 1). When g “ 1, the sections s0 “ 1, s1 “ x, s2 “ y define a closed

embedding
ϕ : X ÝÝÑ P2k .

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To see this, write t0 , t1 , t2 for the homogeneous coordinates on P2k , so that ϕ˚ ti “ si for
i “ 0, 1, 2. Over U “ Dps0 q “ ϕ´1 D` pt0 q, the morphism ϕ is given by
ϕ|U : U ÝÝÑ D` pt0 q » A2k ; px, yq ÞÑ p1 : x : yq (21.17)
which can be identified with the closed embedding
Spec krx, ys{p´y 2 ` x3 ` xq ÝÝÑ Spec krx, ys.
In particular, ϕ is a closed embedding over ϕ´1 D` pt0 q. Likewise, over V “ Dpxq “
ϕ´1 D` pt1 q, we rewrite the sections in terms of u and v :
p1 : x : yq “ px´1 : 1 : x´1 yq “ pu : 1 : vq,
so the morphism takes the form
ϕ|V : V ÝÝÑ Dpt1 q Ă P2k ; pu, vq ÞÑ pu : 1 : vq. (21.18)
This shows that ϕ is a closed embedding over ϕ´1 D` pt1 q as well. As these two open
sets cover the image ϕpXq Ă P2k , the morphism ϕ is a closed embedding. The relation
y 2 “ x3 ` x gives the following defining equation for X in P2k :
t22 t0 “ t31 ` t20 t1 .
That is, X embeds as a plane cubic curve. △
Example 21.40 (g “ 2). When g “ 2, the divisor 3P does not give a projective embedding,
as there is only the two global sections 1 and x. For the divisor 4P , we have ΓpX, OX p4P qq
is spanned by the three sections 1, x, x2 . Geometrically, the induced morphism X Ñ P2k
maps onto the degree 2 curve t21 “ t0 t2 . It is not a projective embedding because degree 2
curves have genus 0. More directly, X Ñ P2k factors as X Ñ P1k Ñ P2k , where the first map
is the double cover and P1k Ñ P2k is the second Veronese embedding.
On the other hand, for the divisor 5P , we obtain
ΓpX, OX p5P qq “ k ‘ kx ‘ kx2 ‘ ky.
Moreover, the sections s0 “ 1, s1 “ x, s2 “ x2 , and s3 “ y globally generate OX p5pq,
and define a morphism
ϕ : X ÝÝÑ P3k
Over the open set U “ Dps0 q, ϕ is given by the map
ϕ|U : U ÝÝÑ A3k ; px, yq ÞÑ p1 : x : x2 : yq
Over V “ Dps1 q, we rewrite the sections in terms of u, v as 1, u´1 , u´2 , u´3 v , and so
multiplying by the unit u3 , the morphism is given by
ϕ|V : V ÝÝÑ A3k ; pu, vq ÞÑ pu3 : u2 : u : vq
This shows that ϕ is a closed embedding.
From the defining sections, we see that s21 ´ s0 s2 “ 0, so the image of X lies on the
quadric surface t21 ´ t0 t2 “ 0. There are additional relations among the sections coming
from the relation y 2 “ x5 ` x, for instance, t23 t0 ´ t1 t22 ´ t1 t20 “ 0. Computing the primary

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decomposition of the ideal generated by these two polynomials, we find that X is embedded
in P3k as the curve defined by the prime ideal
I “ pt21 ´ t0 t2 , t20 t2 ` t32 ´ t1 t23 , t20 t1 ` t1 t22 ´ t0 t23 q.
Let us use the ideal I to verify that the genus of X is equal to 2. A free resolution of I is
given by
T
0 Ñ Rp´4q2 ‘ Ý
Ñ Rp´2q ‘ Rp´3q2 Ñ I Ñ 0
where T is the matrix ¨2 ˛
t0 ` t22 ´t23
T “ ˝ ´t1 t2 ‚
t0 ´t1
Applying tilde, we get the following sequence of sheaves on P3k
T
0 Ñ OP3k p´4q2 Ý
Ñ OP3k p´2q ‘ OP3k p´3q2 Ñ I Ñ 0
Taking Euler characteristics, we find that χpX, OX q equals
ˆ ˆ ˙ ˆ ˙ ˆ ˙˙
´4 ` 3 ´2 ` 3 ´3 ` 3
χpOP3 q ´ χpIq “ 1 ´ ´2 ` `2 “ ´1.
3 3 3
As h0 pX, OX q “ 1, we find that h1 pX, OX q “ 2, as expected. △

21.7 Exercises
Exercise 21.7.1. Let X be a variety over an algebraically closed field k and let x, y P X be
two k -points. Show that if OX,x Ă OX,y and mx Ă my , then x “ y .
Exercise 21.7.2. Let f : X Ñ Y be a birational morphism between curves, where Y
is nonsingular. Then f pXq is open in Y , and f induces an isomorphism X » f pXq.
H INT: The complement of f pXq is a finite set of points. For x P X , show that OY,f pxq
contains OX,x inside KpXq and deduce that OY,f pxq “ OX,x by the properties of discrete
valuation rings.
Exercise 21.7.3. Find the singularities of the curve in P2k whose equation is x2 y 2 ` x2 z 2 `
y 2 z 2 “ 0.
Exercise 21.7.4. a) Show that the pushforward of a torsion sheaf is a torsion
sheaf.
b) Show that a sheaf F is torsion iff it is supported on a proper closed subset
c) Show that if F is a torsion sheaf on a curve X then H 0 pX, Fq “ 0 if and only
if F “ 0.
Exercise 21.7.5. Let R “ krx0 , . . . , xn s{a and suppose that each of the ideals xi are
prime. Consider X “ Proj R as a closed subscheme of Pnk with ideal sheaf IX . Show that
H 1 pPnk , IX pdqq “ 0 for every d P Z. H INT: Use Proposition 16.19.

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The Riemann–Roch theorem

A fundamental problem in algebraic geometry, is how to compute the spaces of global

sections H 0 pX, OX pDqq for some divisor D. The sections of OX pDq allow us to define
morphisms and rational maps to projective space Pnk , and in good cases, we even get a
projective embedding. Subsequently, we we can ask about geometric information coming
from the embedding, e.g., the degree, or the equations defining the image.
In this chapter we will develop the most powerful tool to attack these questions, the
Riemann–Roch theorem. The theorem finds its origins in Bernhard Riemann’s 1857 work on
meromorphic functions on Riemann surfaces, and Gustav Roch’s subsequent refinement from
1865. Since then, the theorem has been generalized in many ways, including the Hirzebruch–
Riemann–Roch theorem, which holds in all dimensions, and the vast generalization found in
the Grothendieck–Riemann–Roch theorem.

22.1 The Riemann–Roch formula

In this chapter, X will denote a nonsingular projective curve over an algebraically closed
field k . When we say ‘point’, we will always mean a closed point.
The cohomology groups H i pX, Fq are finite-dimensional k -vector spaces. For notational
purposes, we define
hi pX, Fq :“ dimk H i pX, Fq
Since X is a curve, the interesting cohomology groups are H 0 pX, Fq and H 1 pX, Fq; the
other cohomology groups are zero by Theorem 18.24. We will mostly be interested in the
case when F “ OX pDq for some divisor D. In this case, we will occasionally write hi pDq
for hi pX, OX pDqq, provided the context is clear.
Our most basic tool for studying the cohomology groups H 0 pX, OX pDqq is the ideal
sheaf sequence of a point p P X , which takes the form
0 ÝÝÑ OX p´pq ÝÝÑ OX ÝÝÑ κppq ÝÝÑ 0 (22.1)
The first map is the inclusion and the second is evaluation at p. Here we have identified
the ideal sheaf mp Ă OX by the invertible sheaf OX p´pq, and the sheaf i˚ Op with the
skyscraper sheaf with value κppq at p. If L is an invertible sheaf, we can tensor (22.1) by L
and get the exact sequence
0 ÝÝÑ Lp´pq ÝÝÑ L ÝÝÑ κppq ÝÝÑ 0, (22.2)
where Lp´pq is the invertible sheaf of sections of L vanishing at p, and we also identify

440

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L b κppq » κppq (note that L b κppq is also the constant sheaf with value k at p). In
particular, taking L “ OX pD ` pq in (22.2) we get
0 ÝÝÑ OX pDq ÝÝÑ OX pD ` pq ÝÝÑ κppq ÝÝÑ 0 (22.3)
This leads to the following basic bound:

Lemma 22.1. Let X be a nonsingular projective curve, and let D be a divisor on X .

Then
(i) h0 pX, OX pD ` pqq ď h0 pX, OX pDqq ` 1 for each p P X .
(ii) h0 pX, OX pDqq ď deg D ` 1.

Proof The first item follows by taking global sections in the řsequence (22.3). To see the
second inequality, it suffices to consider the case when D “ np p is effective (otherwise
the left-hand side is 0). In that case, the inequality follows by applying the first inequality
deg D times.
Recall that the Euler characteristic χpFq is defined as the alternating sum of the dimensions
hi pX, Fq. Applying χ to (22.3), and recalling that χ is additive on exact sequences, we get
χpOX pD ` pqq “ χpOX pDqq ` χpκppqq “ χpOX pDqq ` 1. (22.4)
The next result is a very useful formula to compute χpOX pDqq for any divisor D:

Theorem 22.2 (Easy Riemann–Roch). Let X be a smooth projective curve of genus

g “ h1 pX, OX q and let D be a divisor on X . Then
χpOX pDqq “ h0 pOX pDqq ´ h1 pX, OX pDqq “ deg D ` 1 ´ g (22.5)

Proof Let p P X be a point. Consider the formula (22.5). By (22.4), the left-hand side
increases by 1 if we replace D by D ` p. On the other hand, also the right-hand side of the
equation above increases by 1 by adding p to D, because degpD ` pq “ deg D ` 1. This
means that the theorem holds for a divisor D if and only if it holds for D ` p for any closed
point p. So by adding and subtracting points, we can reduce to the case when D “ 0. But in
that case the left-hand side of the formula is simply
dimk H 0 pX, OX q ´ dimk H 1 pX, OX q “ 1 ´ g
which equals the right-hand side, by the definition of g .
As a first basic corollary, we prove that the divisor of a rational function always has degree
0. This explains the observations that the ‘number of zeros’ and ‘number of poles’ matched
up in the examples in Chapter 21.

Corollary 22.3. If f P KpXq ´ 0, then degpdivpf qq “ 0.

Proof Let D “ divpf q, then OX pDq » OX (Proposition 17.33), so degpdivpf qq “ 0 by

the formula (22.5).
Note that the right-hand side of 22.5 is easy to compute, as it only involves the degree of

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D and the genus. Of course, the number we are really after is the number h0 pX, OX pDqq,
since this is the dimension of global sections of OX pDq. So if we, for some reason, could
argue that H 1 pX, OX pDqq “ 0 we would have a formula for the dimension of the space of
global sections of OX pDq.
In any case, we can certainly say that h1 pX, OX pDqq ě 0, and this leads to the following
lower bound on h0 pX, OX pDqq, which is often enough in applications.

Corollary 22.4. h0 pX, OX pDqq ě deg D ` 1 ´ g .

Example 22.5. That the Riemann–Roch formula holds for X “ P1 follows from the
computations in Chapter 18. Indeed, as ClpP1k q “ Z, it suffices to check that the formula
holds for divisors of the form D “ dP where P P P1 is a point and d P Z. In this case, the
right-hand-side of the fomula equals deg D ` 1 ´ 0 “ d ` 1.
If d ě 0, H 0 pP1 , OX pdP qq is identified with the space of homogenous degree d polyno-
mials in x0 , x1 , while H 1 pP1 , OX pdP qq “ 0. Hence h0 pX, Dq “ d ` 1. If d ă 0, we have
h0 pX, Dq “ 0 and h1 pX, Dq “ ´d ´ 1. △
Example 22.6. Consider again the case where X is a hyperelliptic curve of genus 2, as in Ex-
ample 21.40. We have the following table of the various cohomology groups H i pX, OX pnpqq
for the point p “ pu, vq:

D 0 1p 2p 3p 4p 5p 6p 7p
H 0 pX, OX pDqq 1 1 2 2 3 4 5 6
H 1 pX, OX pDqq 2 1 1 0 0 0 0 0
χpOX pDqq -1 0 1 2 3 4 5 6

△
As the example shows, the cohomology groups H 1 pX, OX pDqq vanish provided that the
degree deg D is large enough. This is a general fact for all divisors on all projective curves,
and it will be a consequence of the following theorem, known as ’Serre duality’. Although
we will only prove this when X is a curve, we include the general statement, which holds in
any dimension:

Theorem 22.7 (Serre duality). Let X be a smooth projective variety of dimension n

over an algebraically closed field k . Then there is an isomorphism H n pX, ΩnX q » k , and
for any locally free sheaf E on X , a perfect pairing
H 0 pX, Eq ˆ H n pX, E _ b ΩnX q ÝÝÑ H n pX, ΩnX q » k (22.6)

The theorem allows us to view the space of global sections H 0 pX, Eq as the dual of
the top cohomology group H n pX, E _ b ΩnX q via a certain k -linear pairing. The important
consequence of this is that H 0 pX, Eq and H n pX, E _ b ΩnX q have the same dimension.
In the case where X is a curve, we choose a canonical divisor KX so that ΩX “ OX pKX q.
Applying the theorem to E “ OX pKX ´ Dq, we get an equality

h1 pX, OX pDqq “ h0 pX, OX pKX ´ Dqq,

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and the previous Riemann–Roch formula takes the following form:

Theorem 22.8 (Riemann–Roch). Let X be a nonsingular projective curve of genus g

and let D be a divisor on X . Then
h0 pX, OX pDqq ´ h0 pX, OX pKX ´ Dqq “ deg D ` 1 ´ g

This result is significantly more powerful than the Riemann–Roch formula we saw earlier.
It gives stronger conclusions, because the group H 0 pX, OX pKX ´ Dqq is easier to interpret:
it is the space of global sections of the sheaf associated to the divisor KX ´ D. Often, we
can readily argue that no such global sections exist, due to simple reasons. For instance, in
the case deg D ą dim KX then KX ´ D cannot be effective, as effective divisors have
non-negative degree.
In fact, we can use the Riemann–Roch formula to compute the degree of the canonical
divisor KX :

Corollary 22.9. Let X be a nonsingular projective curve of genus g “ h1 pOX q. Then

(i) deg KX “ 2g ´ 2
(ii) h0 pX, OX pKX qq “ g .

Proof From Serre duality, we get that H 0 pX, OX pKX qq and H 1 pX, OX q have the same
dimension. Therefore, applying the Riemann–Roch formula to D “ KX , we get

g ´ 1 “ dimk H 0 pX, OX pKX qq ´ dimk H 0 pX, OX pKX ´ KX qq “ deg K ` 1 ´ g

which gives that deg KX “ 2g ´ 2.

Combining this with the previous paragraph, we obtain:

Corollary 22.10. Let D be a divisor of degree ě 2g ´ 1. Then

(i) H 1 pX, OX pDqq “ 0.
(ii) h0 pX, OX pDqq “ deg D ` 1 ´ g.
Moreover, if deg D “ 2g ´ 2, then H 1 pX, OX pDqq ‰ 0 if and only if D „ KX .

Proof For the last part, note that h0 pX, OX pKX ´ Dqq ‰ 0 if and only if KX ´ D is
effective. But the only effective divisor of degree 0 is the divisor 0, so this happens if and
only if KX and D are linearly equivalent.

Therefore, in our path to computing h0 pX, OX pDqq for a divisor D, we are left with
the ‘intermediate cases’ where D has degree between 0 and 2g ´ 2. In this region, the
computations become more subtle, and in particular, the rank of H 0 pX, OX pDqq does not
depend on deg D and g alone.

deg D ă0 0 1 ... 2g ´ 2 ě 2g ´ 1
h0 pX, Dq ? ? ? ... ? deg D ` 1 ´ g
h1 pX, Dq ´pdeg D ` 1 ´ gq ? ? ... ? 0

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22.2 Proof of Serre duality

The aim of the next few sections is to prove the following:

Theorem 22.11 (Serre duality). Let X be a projective curve over an algebraically closed
field k . Then there is an OX -module ωX of finite type, together with an isomorphism
H 1 pX, ωX q » k , such that for any locally free sheaf F on X , there is a perfect pairing
H 0 pX, Fq ˆ H 1 pX, ωX b F _ q ÝÝÑ H 1 pX, ωX q » k (22.7)
In particular, H 0 pX, Fq and H 1 pX, ωX b F _ q have the same dimension.

The sheaf ωX is called a dualizing sheaf. The existence of ωX is usually not enough for
applications or explicit computations. The cruicial point is that, when X is nonsingular, the
dualizing sheaf is isomorphic to the cotangent sheaf:

Theorem 22.12. If X is a nonsingular, projective curve, the dualizing sheaf ωX is

isomorphic to the cotangent sheaf ΩX .

These results are fundamental in algebraic geometry and one can find several proofs in the
literature (see for instance, Hartshorne (2013), ?, Serre (1955),Serre (2013), Kempf (1993),
?). The proof presented here uses very little machinery, and it is inspired by the proofs found
in Kempf (1993) and ?. The approach here is however more ad hoc and much less conceptual
than the standard proofs, and it gives essentially no information about the isomorphism
H 1 pX, ΩX q » k .
We will prove the two theorems in three steps:
(i) We note that both theorems hold for X “ P1k , in which ωX “ OP1k p´2q serves
as a dualizing sheaf (and we know this coincides with ΩP1k ).
(ii) Then we prove existence of ωX for a general curve, using a finite map f : X Ñ
P1k . The sheaf ωX is constructed just to satisfy the formal properties of Serre
duality.
(iii) We finally prove that ωX » ΩX .
The first step, Serre duality for P1k , is the easiest:

Lemma 22.13 (Serre duality for P1k ). Serre duality holds for P1k with ωP1k “ OP1k p´2q.

Proof We showed in Chapter 18 that Serre duality holds for any projective space for the
sheaves F of the form OP1k pdq. For P1k specifically, we identify H 0 pX, OP1k pdqq with the
k -vector space of homogeneous polynomials of degree d and H 1 pX, OP1k p´2 ´ dqq with
the k -vector space spanned by Laurent monomials x´u ´v
0 x1 with u ` v “ d ` 2, u, v ě 1.
The multiplication map
#
x´1 ´1
0 x1 if pu, vq “ pa ´ 1, b ´ 1q
xa0 xb1 ˆ x´u
0 x ´v
1 :“
0 otherwise

defines a perfect pairing, which induces (22.7) for F “ Opdq with d ě 0. For d ă 0, all
groups are zero.

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Recalling that any locally free sheaf F on P1k is a direct sum of invertible sheaves, we get
the pairing (22.7) in general.

22.3 The dualizing sheaf

Let ϕ : A Ñ B be a map of rings, making B into a finite A-module. For an A-module N , we
will consider the ‘dual module’ HomA pB, N q, which is a priori an A-module. The crucial
observation is that this can also be viewed as a B -module, via the rule
b ¨ ϕpyq :“ ϕpb ¨ yq, yPB
for each A-linear map ϕ : B Ñ N . Note that HomA pB, N q is finitely generated as a
B -module if N is finitely generated as an A-module.
The main property we need is the following:

Lemma 22.14. For a B -module L, there is a natural isomorphism of A-modules

HomA pLA , N q ÝÝÑ HomB pL, HomA pB, N qqA (22.8)

Proof The map (22.8) is defined by sending ψ : L Ñ N to ϕ : L Ñ HomA pB, N q defined

by ϕpℓq “ pb ÞÑ ψpb¨ℓqq. The inverse is defined by the map sending ϕ : L Ñ HomA pB, N q
to ℓ ÞÑ ϕpℓqp1q.

The operation N ÞÑ HomA pB, N q defines a functor from A-modules to B -modules. It

behaves well with respect to localization: if S Ă A is a multiplicative set, then as B is finitely
generated, there is a natural isomorphism of S ´1 B -modules
S ´1 HomA pB, N q “ HomS ´1 B pS ´1 B, S ´1 N q. (22.9)
Because of this, the construction generalizes well to a version for sheaves. For a finite
morphism f : X Ñ Y of Noetherian schemes and a quasi-coherent OY -module G , we define
f ! G by the formula
Γpf ´1 U, f ! Gq “ HomOY pU q pOX pf ´1 U q, GpU qq (22.10)
for each affine U “ Spec A. Note that f ! G is quasi-coherent, as it coincides with the tilde of
HomA pB, N q over affines. As f is finite, f ! G is moreover of finite type if G is.
By (22.10), we have f˚ f ! G “ Hom OY pf˚ OX , Gq. More generally, if F is an OX -module
of finite type, we have a natural isomorphism
f˚ Hom OX pF, f ! Gq » Hom OP1 pf˚ F, Gq. (22.11)
k

This follows because also the isomorphisms in the map (22.8) are compatible with localiza-
tions.

Definition 22.15. Let X be a nonsingular projective curve over k , and let f : X Ñ P1k
be a finite morphism. We define the dualizing sheaf of X to be the OX -module
ωX “ f ! ωP1k .

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So far we haven’t used the fact that X is nonsingular; any projective curve admits a
dualizing sheaf ωX . In the nonsingular case, we will prove in Section 22.4 that ωX » ΩX .
As a first step towards this, we next show that ωX is invertible.

Proposition 22.16. Let X be a nonsingular projective curve. Then ωX is an invertible

sheaf.

Proof Since X is a nonsingular curve, ωX is locally free if and only if it is torsion free.
Let T denote the torsion subsheaf and E is the torsion free part, so that ωX “ E ‘ T by
Theorem 21.18. Applying f˚ , we get

f˚ ωX “ f˚ E ‘ f˚ T .

Applying the formula (22.11), shows that f˚ ωX “ Hompf˚ OX , ωP1k q. As f is finite and
surjective, f˚ OX is locally free (Proposition 21.19). Since ωP1k “ OP1k p´2q is also invertible,
we find that f˚ ωX is also locally free. Note that f˚ T is again a torsion sheaf on P1k . As
f˚ T is a direct summand of a locally free sheaf, we must have f˚ T “ 0. This implies that
ΓpX, T q “ ΓpP1k , f˚ T q “ 0. On a curve, the only torsion sheaf with no global sections is
the zero sheaf, so T “ 0 as well. Therefore, ωX is locally free.
We next show that ωX has rank 1. To show this, it suffices to compute the stalk at the
generic point η “ Spec KpXq. We can compute this using the affine open sets Spec B Ă X
and Spec A Ă P1k . Then with S “ B ´ 0, the stalk ωX,η is given by

S ´1 HomA pB, Lq “ HomKpP1k q pKpXq, KpP1k qq

which is a KpP1k q-vector space of dimension equal to the degree of the field extension
KpXq{KpP1k q. Hence, as a KpXq-vector space it has dimension 1. Therefore, ωX is an
invertible sheaf.

Having established this fact, we can finish the proof of Serre duality on X . For any locally
free sheaf F on X , we have a a chain of natural bijections:

H 1 pX, F _ bOX ωX q “ H 1 pP1k , f˚ pF _ bOX ωX qq (22.12)

“ H 1 pP1k , f˚ HompF, ωX qq (22.13)
“ H 1 pP1k , Hompf˚ F, ωP1k qq (22.14)
1
“H pP1k , pf˚ Fq_ bOP1 ωP1k q (22.15)
k

“ H 0 pP1k , f˚ Fq_ (22.16)

“ H 0 pX, Fq_ . (22.17)

Many of the steps here are non-trivial: (22.12) and (22.17) use Lemma 18.22. (22.13) and
(22.15) follow by Proposition 15.17 and the fact that F , f˚ F , ωX and ωP1 are locally free.
(22.14) uses the adjoint property (22.11) and finally (22.16) is Serre duality on P1k .

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22.4 The dualizing sheaf is the canonical sheaf

The goal of this section is to show that the dualizing sheaf ωX is isomorphic to the cotangent
sheaf ΩX . Note that both of these sheaves are locally free: the first by Corollary 22.16, and
ΩX because X is nonsingular.
The main idea of the proof is to use the following two basic properties of the dualizing
sheaf:
(i) H 1 pX, ωX q ‰ 0.
(ii) H 1 pX, ωX b OX pP qq “ 0 for every point P P X .
In fact, we will see that these properties characterize ωX uniquely, and it is this which in turn
allows us to conclude that ΩX “ ωX .
We will now work with the fiber product X ˆ X (taken over k ) with the two projections
p, q : X ˆ X Ñ X . Consider the diagonal ∆ Ă X ˆ X (which we view as a closed
subscheme, because X is separated over k ). Since X ˆ X is a nonsingular surface, ∆ is a
divisor on X ˆ X , and the corresponding ideal sheaf I is the invertible sheaf OXˆX p´∆q.
Consider the ideal sheaf sequence

0 ÝÝÑ OXˆX p´∆q ÝÝÑ OXˆX ÝÝÑ i˚ OX ÝÝÑ 0

where i : X Ñ X ˆ X is the closed diagonal embedding.

Tensoring the sequence by OXˆX p∆q, we obtain

0 ÝÝÑ OXˆX ÝÝÑ OXˆX p∆q ÝÝÑ OXˆX p∆q bOXˆX i˚ OX ÝÝÑ 0

Now the point is that OXˆX p´∆q bO O{I » I{I 2 . Therefore, by the projection formula,
we have
OXˆX p´∆q bO i˚ OX “ i˚ pi˚ pI{I 2 qq “ i˚ pΩX q

Dually, this means that the sheaf on the right-hand side is isomorphic to i˚ TX , and we have
an exact sequence

0 ÝÝÑ OXˆX ÝÝÑ OXˆX p∆q ÝÝÑ i˚ TX ÝÝÑ 0. (22.18)

We now tensor this sequence by q ˚ ωX , to get a sequence

0 ÝÝÑ q ˚ ωX ÝÝÑ q ˚ ωX p∆q ÝÝÑ i˚ pωX bOX TX q ÝÝÑ 0 (22.19)

Here we have used the projection formula again:

i˚ TX bOXˆX q ˚ ωX “ i˚ pTX bOX i˚ q ˚ ωX q “ i˚ pTX bOX ωX q.

If we restrict the sequence (22.19) to the open set V ˆ X , where V is affine, and take the
long exact sequence in cohomology, we get

ΓpV ˆ X, i˚ pωX bOX TX qq ÝÝÑ H 1 pV ˆ X, q ˚ ωX q ÝÝÑ H 1 pV ˆ X, q ˚ ωX p∆qq

(22.20)
By Lemma 18.22, we may identify the first group with ΓpV, ωX bOX TX q. The next lemma
identifies the middle group.

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Lemma 22.17. Let V and X be varieties over k with V affine. Let q : V ˆ X Ñ X

denote the second projection. Then for each quasi-coherent sheaf F on X , there are
natural isomorphisms
H 0 pV ˆ X, q ˚ Fq “ OV pV q bk H 0 pX, Fq
H 1 pV ˆ X, q ˚ Fq “ OV pV q bk H 1 pX, Fq. (22.21)

Proof Recall the exact sequence

d0
0 Ñ H 0 pX, Fq Ñ FpU0 q ˆ FpU1 q ÝÝÑ FpU0 X U1 q Ñ H 1 pX, Fq Ñ 0. (22.22)
Tensoring by OV pV q over k preserves exactness, so we get an exact sequence

0 Ñ OV pV q bk H 0 pX, Fq Ñ OV pV q bk FpU0 q ˆ OV pV q bk FpU1 q

Ñ OV pV q bk FpU0 X U1 q Ñ OV pV q bk H 1 pX, Fq Ñ 0.
Note that the map in the middle is exactly the Cech complex with of the sheaf q ˚ F with
respect to the two affine open sets V ˆ U0 and V ˆ U1 . By exactness, we get the desired
isomorphisms.
By the lemma, we may identity the middle cohomology group in (22.20) with ΓpV, OX q bk
H 1 pX, ωX q. Below, we will show that
H 1 pV ˆ X, q ˚ ωX p∆qq “ 0. (22.23)
Therefore, the sequence (22.20) can be written
ΓpV, ωX b TX q ÝÝÑ H 1 pX, OX q bk OX pV q ÝÝÑ 0. (22.24)
These identifications are compatible with restriction maps, i.e., if V 1 Ă V is a smaller affine
open set, the following diagram commutes
δ
ΓpV, ωX b TX q H 1 pV ˆ X, q ˚ ωX q H 1 pX, ωX q bk OX pV q

δ
ΓpV 1 , ωX b TX q H 1 pV 1 ˆ X, q ˚ ωX q H 1 pX, ωX q bk OX pV 1 q

Therefore, these maps glue to define a map of sheaves

ωX b TX ÝÝÑ OX .
By 22.24, this map is surjective. As both sheaves are locally free, this must be an isomorphism
(the kernel is locally free of rank 0), so ωX b TX » OX . But this means that ωX » ΩX .
To conclude, we need to show that (22.24) holds. Let y P V be a k -point and consider the
subscheme ι : y ˆ X Ñ V ˆ X . Then identifying y ˆ X with X , we have
ι˚ pq ˚ ωX q “ ωX
and
ι˚ pOXˆX p∆q|V ˆX q “ OX pyq

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Hence ι˚ pq ˚ ωX p∆qq “ ωX pyq. This sheaf has an H 1 which is zero, by the dualizing
property:
H 1 pX, ωX pyqq “ H 0 pX, OX p´yqq_ “ 0
Indeed, ΓpX, OX p´yqq is the group of global regular functions vanishing at y , but all regular
functions are constants, as X is projective.
Applying the lemma below (to Y “ V ˆ X , S “ V , f “ p, B “ A{m and F “
q ˚ ωX p∆q), we find that the A-module M “ H 1 pV ˆX, q ˚ ωX p∆qq satisfies M bA A{m “
0 for every maximal ideal m in A. Therefore, M “ 0 by Nakayama’s lemma. This concludes
the proof.

Lemma 22.18 (H 1 and base change). Let f : Y Ñ S be a a projective morphism of

Noetherian schemes, where S “ Spec A is affine, and assume that Y can be covered
by two affine open sets U0 and U1 . Let h : YB Ñ Y be the morphism defined by a base
change Spec B Ñ Spec A. Then for any OY -module of finite type F , we have a natural
isomorphism of B -modules
H 1 pY ˆA B, h˚ Fq “ H 1 pY, Fq bA B. (22.25)

Proof Consider the exact sequence

d
Ý FpU0 X U1 q Ñ H 1 pY, Fq Ñ 0.
FpU0 q ˆ FpU1 q Ñ (22.26)
coming from the Cech complex on U0 and U1 . Tensoring this complex by B gives the Cech
complex of the pullback h˚ F . Therefore, H 1 pY ˆA B, h˚ Fq is the cokernel of d b idB
and so since the tensor product is right exact, we get the isomorphism (22.25).

22.5 Exercises
Exercise 22.5.1. Let X be a nonsingular quasi-projective curve over an algebraically closed
field k . Show that if ΓpX, OX q ‰ k , then X is affine. H INT: Embed X in a projective
nonsingular curve X and consider divisors supported on the closed subset X ´ X .
Exercise 22.5.2. Let f : X Ñ Y be a non-constant morphism of curves. If D is a divisor on
Y , show that h0 pX, f ˚ OY pDqq ě h0 pY, OY pDqq.
Exercise 22.5.3 (Riemann–Roch for higher rank). Let X be a nonsingular projective curve
and let E be a locally free sheaf of rank r. Show that
χpX, Eq “ degpdet Eq ` rχpOX q (22.27)
Exercise 22.5.4 (Lüroth’s theorem). Let X be a nonsingular proper curve over an alge-
braically closed field k and let f : P1k Ñ X be a non-constant morphism. Show that X is
isomorphic to P1k .
Exercise 22.5.5. Find an example of an OX -module F of finite type so that the equality in
(22.25) is not true if we replace H 1 by H 0 .
Exercise 22.5.6. Let X Ă P2k be a plane curve of degree ď 3. Show that the automorphism
group of X is infinite.

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Exercise 22.5.7. Let X Ă P2k be a plane curve of degree 4. Show that any automorphism
ϕ : X Ñ X is induced by a linear transformation P2k Ñ P2k . Deduce that the general plane
quartic has trivial automorphism group.
Exercise 22.5.8 (Curves of genus 5). a) Let X be a complete intersection of 3
quadric hypersurfaces in P4k . Show that X is non-hyperelliptic of genus 5.
b) Conversely, show that any non-hyperelliptic curve of genus 5 is contained in
the intersection of 3 quadric hypersurfaces Q0 X Q1 X Q2 in P4k .
c) * Show that there exist non-hyperelliptic genus 5 curves which are not complete
intersections of 3 quadrics (it can happen that Q0 X Q1 X Q2 has dimension
ě 2).
Exercise 22.5.9 (Curves of genus 6). Show that the following varieties define genus 6 curves:
a) A degree 5 curve in P2k
b) A p2, 7q-divisor in P1k ˆ P1k .
c) A p3, 4q-divisor in P1k ˆ P1k .
d) * Show that the classes of curves constructed in a), b), c) are disjoint.
Exercise 22.5.10 (Curves of degree 3). Let X Ă Pnk be a nonsingular curve of degree 3.
Show that X has genus ď 1 and is either a twisted cubic or a plane cubic.
Exercise 22.5.11 (Curves of degree 4). Let X Ă Pnk be a nonsingular curve of degree 4.
Show that X has genus 0, 1 or 3. Find a projective models in each case.
Exercise 22.5.12 (Hyperelliptic curves). Let X be a hyperelliptic curve of genus g , that is,
admitting a base-point free divisor D of degree 2.
a) Show that if g ě 2, then D is unique up to linear equivalence, and that
KX „ pg ´ 1qD. P OSSIBLE HINT: If f, g : X Ñ P1k are two degree 2
morphisms, consider a point P P X such that f ´1 pf pP qq “ P ` Q and
g ´1 pgpP qq “ P ` Q for P, Q, R P X distinct. Then consider the divisor
P ` Q ` R.
b) Show that if g “ 1, then X contains infinitely many such divisors D which are
non-linearly equivalent.
Exercise 22.5.13. Let X be a curve over a field k and let F be a sheaf on X . Show that
H i pXk , q ˚ Fq » H i pX, Fq bk k for all i, where q : Xk Ñ X is the base change morphism.
Exercise 22.5.14 (Trace map). Let f : X Ñ Y be a finite morphism of nonsingular projective
curves. Show that there is a natural map Tr : f˚ OX Ñ OY which splits the usual map
f 7 : OY Ñ f˚ OX . Deduce that
f˚ OX » OY ‘ E
for some locally free sheaf E .
Exercise 22.5.15. a) Let f : X Ñ Y be a domninant morphism of nonsingular
projective curves. Show that gpXq ě gpY q H INT: Apply Exercise 23.10.14.
b) (Lüroth’s theorem): Let f : P1k Ñ X be a dominant morphism of nonsingular
projective curves. Show that X » P1k .

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Exercise 22.5.16 (Fermat’s Last Theorem for polynomials). As an application of Exercise

23.10.15, show that if n ě 3, there are no non-constant polynomials P, Q, R P krts such
that
P n ` Qn “ Rn .
What happens when n “ 1 or 2?
H INT: Construct a morphism to a Fermat curve in P2k .

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Applications of the Riemann–Roch theorem

23.1 A criterion for being a closed embedding

If f : X Ñ Y is a morphism of schemes, it can easily happen that f is a homeomorphism
without being an isomorphism, even when X and Y are varieties. (See Example 23.3 below).
In light of this, it is natural to ask if there is a sufficient extra condition to guarrantee that
f is a closed embedding. As it turns out, it turns out that it is enough that f induces an
isomorphism on cotangent spaces:

Proposition 23.1. Let f : X Ñ Y be a finite map of schemes of finite type over an

algebraically closed field k . Then f is a closed embedding if and only if
(i) f is injective on k -points; and
(ii) for each k -point x P Xpkq, the induced map on cotangent spaces
f ˚ : mf pxq {m2f pxq ÝÝÑ mx {m2x (23.1)
is surjective.

Proof The conditions (i) and (ii) are easily seen to be satisfied for a closed embedding, so
we need to prove the converse direction.
The question is local on Y , so we may assume that Y is affine, say Y “ Spec A. As f is
finite, X is also affine, say X “ Spec B , for some finite A-module B , and f is induced by a
ring map ϕ : A Ñ B . We need to show that ϕ is surjective.
Viewing ϕ as a map of A-modules, it is enough to show that ϕm : Am Ñ Bm is surjective
for every maximal ideal m in A. Let y P Y be the k -point corresponding to m. Recall that
the scheme-theoretic fiber Xy of y is given by SpecpB{mBq. If y is not in the image of f ,
then B{mB “ 0. In this case Bm “ 0 because mB is the unit ideal in B , and ϕm is trivially
surjective. If y is in the image, then assumption (i) shows that y “ f pxq for a unique k -point
x P X , corresponding to a maximal ideal n in B .
We claim that n “ mB . Consider the ideal b “ ϕpmqB in B . As x maps to y , we have in
any case an inclusion b Ă n. By the hypothesis (ii), the composition m Ñ m{m2 Ñ n{n2
is surjective, so b Ñ n{n2 is surjective, and hence b ` n2 “ n. Consider the B -module
M “ n{b. Then b ` n2 “ n implies that nM “ M and so M “ 0 by Nakayama’s lemma.
Therefore b “ n.
From this it follows that A{m “ B{ϕpmqB “ B{n “ k . Moreover, the map ϕ b A{m,
which is the map A{m Ñ B{n, is surjective. As B is a finite A-module, it follows that ϕ is
surjective as well (by Nakayama’s lemma again).

452

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Note that the condition 23.1 is equivalent to the induced map on tangent spaces
dfx : Tx X ÝÝÑ Tf pxq Y (23.2)
is injective. Yet another way of rephrasing it is that f is a closed embedding if and only if the
induced maps Xpkq Ñ Y pkq and XpRq Ñ Y pRq are injective, where R “ krϵs{ϵ2 is the
ring of dual numbers.
Example 23.2. The assumption that k is algebraically closed is important in the above
proposition. For instance, the scheme map f : Spec C Ñ Spec R satisfies (i) and (ii), but it
is not a closed embedding. △
Example 23.3. We saw in Example ?? that the morphism
f : Spec Crxs ÝÝÑ Spec Crx, ys{py 2 ´ x3 q
is a homeomorphism. f satisfies the condition (i), but not (ii), as the map on tangent spaces
at the origin is the zero map. △

23.2 Ample invertible sheaves and Serre’s theorems

If L is a globally generated invertible sheaf, and X is Noetherian, we can find finitely many
sections s0 , . . . , sn that generate L and define a morphism to projective space ϕ : X Ñ Pn
so that L “ ϕ˚ OPn p1q. It is natural to ask when ϕ is a closed embedding. If this is the case,
then we can view X as a closed subscheme of Pn and L is simply the restriction OPn p1q|X
to X . Hence the closed embedding is completely determined by the sections s0 , . . . , sn and
the polynomial relations between them.

Definition 23.4. Let X be a scheme over a Noetherian ring A. We say that an invertible
sheaf L is very ample if there is a closed embedding ι : X Ñ PnA such that L » ι˚ OPnS p1q.
L is said to be ample if LbN is very ample for some N ą 0.

Example 23.5. If X “ Spec A is affine, then OX is very ample. The closed embedding in
question is simply the identity morphism Spec A Ñ P0A “ Spec A. △
Example 23.6. If R is a graded ring which is finitely generated in degree 1 over a field k ,
then OProj R pdq is very ample for every d ą 0. If x0 , . . . , xn are generators for R, then the
homogeneous polynomials in x0 , . . . , xn of degree d define a morphism to projective space
Proj R Ñ PN n
k which is a composition of the closed embedding X Ñ Pk and the Veronese
embedding vd : Pnk Ñ PN k . △
We will see several examples of ample, but non-very ample invertible sheaves in Chapter
21 (Example 21.40).
Exercise 23.2.1 (Serre’s theorem). Let X Ă PnA be a projective scheme over a Noetherian
ring A and let L be an ample invertible sheaf. Let F be an OX -module of finite type. Then
there is an integer m0 such that F b Lm is globally generated (by a finite set of global
sections) for all m ě m0 . H INT: Reduce to the case X “ PnA and L “ Op1q. Then apply
Corollary ??.

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In this chapter, we give a few of the (many) consequences of the Riemann–Roch formula.
We begin by translating the results of Chapter ?? into concrete numerical criteria for a divisor
D to be base point free or very ample. Then we use these results to classify all curves of
genus ď 4.

23.3 Base-point freeness and very ampleness on curves

Let X be a nonsingular projective curve over an algebraically closed field k and let D be a
divisor on X . We say that a point P P X is a base point of D (or perhaps more precisely, the
linear system |D|) if every section of H 0 pX, OX pDqq vanishes at P . If there are no base
points, we say that D is base-point free.
In light of the isomorphism OX pDqbOX mP » OX pD´P q, we see that P is a base-point
if and only if the inclusion H 0 pX, OX pD ´ P qq Ă H 0 pX, OX pDqq is an isomorphism. In
general, the dimension of H 0 pX, OX pD ´ P qq can be at most 1 less than H 0 pX, OX pDqq,
by the exact sequence
ev
0 ÝÝÑ H 0 pX, OX pD ´ P qq ÝÝÑ H 0 pX, OX pDqq ÝÝÝ
P
Ñ κppq (23.3)

Therefore, D is base-point free if and only if for every point P P X ,

h0 pD ´ P q “ h0 pDq ´ 1.
If this condition is satisfied, we obtain a morphism f : X Ñ Pnk , by Theorem 16.41.
We now turn to the question whether f is a closed embedding. For this, we use the two
conditions appearing in Proposition 23.1. That is, f should be injective on k -points, and the
map f ˚ on cotangent spaces should be surjective.
Let us choose a basis s0 , . . . , sn for H 0 pX, OX pDqq. Then, over the open set Dps0 q, f
is induced by the ring map
„ ȷ
x1 xn
k ,..., ÝÝÑ ΓpDps0 q, OX q (23.4)
x0 x0
sending xi {x0 to si {s0 . Here we recall that si {s0 is viewed as regular function on Dps0 q as
in Section 16.8.
To show that f is injective, we want to show that for any two points
ř P, Q P Xpkq, we have
f pP q ‰ f pQq. This happens if we are able to find a section s “ ař i si such that spP q “ 0
and spQq ‰ 0. Because in that case, f pP q lies in the hyperplane V p ai xi q Ă Pnk , whereas
f pQq does not.
Therefore, we want to show that there is a section s which lies in H 0 pX, OX pD ´ P qq,
but not in H 0 pX, OX pD ´ P ´ Qqq. It is therefore enough to show that for any P and Q,
we have a strict inequality h0 pD ´ P ´ Qq ă h0 pD ´ P q. As h0 pD ´ P q “ h0 pDq ´ 1,
by the above, this is equivalent to the condition

h0 pD ´ P ´ Qq “ h0 pDq ´ 2 for every pair P, Q P X. (23.5)

Next we would like to know whether the pullback map on cotangent spaces

f ˚ : mf pP q {m2f pP q ÝÝÑ mP {m2P

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is surjective. As X is a nonsingular curve, mP {m2P is 1-dimensional vector space, so we just

need f ˚ to be non-zero.
Let us for simplicitly assume that P maps to the point p1 : 0 : ¨ ¨ ¨ : 0q P Pnk (we may
acheive this by a linear change of coordinates). This means that the morphism is given locally
by Dps0 q Ñ Dpx0 q as in (23.4). More precisely, if yi “ xi {x0 i “ 1, . . . , n denote the
affine coordinates on Dpx0 q, we have f 7 pyi q “ si {s0 P ΓpDps0 q, OX q. In these coordinates,
mf pP q {m2f pP q is the k -vector space generated by the classes y1 , . . . , yn , and f ˚ maps yi to
si {s0 in mP {m2P . It is therefore enough to show that some linear combination of the yi pulls
back
ř sto an element in mP which vanishes to ř order 1 at P . Note that a linear combination
ai s0i vanishes to order 1 at P if and only if ai si vanishes to order 1 at P , that is, lies in
H 0 pX, OX pD ´ P qq but not in H 0 pX, OX pD ´ 2P qq.
Therefore, to show that such linear forms exist, we need to show that H 0 pX, OX pD´2P qq
is strictly smaller than H 0 pX, OX pD ´ P qq. As above, the dimension can drop by at most
1, so this happens if and only if
h0 pD ´ 2P q “ h0 pD ´ P q ´ 1 “ h0 pDq ´ 2
To summarize, we have proved the first two items in the following theorem:

Theorem 23.7 (Base-point free and very ample divisors on curves).

Let X be a nonsingular projective curve over an algebraically closed field k and let D be
a divisor on X . Then
(i) D is base point free if and only if
h0 pD ´ P q “ h0 pDq ´ 1 for every point P P Xpkq.
(ii) D is very ample if and only if
h0 pD ´ P ´ Qq “ h0 pDq ´ 2 for every two points P, Q P Xpkq
(including the case P “ Q)
(iii) If deg D ě 2g , then D is base point free and h1 pX, Dq “ 0.
(iv) If deg D ě 2g ` 1, then D is very ample.

As OX pDq|U » OU over U “ X ´ SupppDq, it is enough to check the condition (i) for

all P in the support of D.
Proof The items (i) and (i) follow from Proposition 23.1 and the discussion above.
For (iii), note that H 1 pD ´ P q and H 1 pDq are both zero, by Corollary 22.10. Therefore,
the exact sequence (23.3) is in fact a short exact sequence, and hence h0 pD´pq “ h0 pDq´1.
Since this holds for any P P X , we conclude by (i)
Similarly, for (iv), we get h1 pD´P ´Qq “ 0, which implies h0 pD´P ´Qq “ h0 pDq´2,
and this allows us to apply (ii).
In particular, the theorem implies that a divisor D on a curve is ample if and only if
deg D ą 0.
Example 23.8. On X “ P1 a divisor D is base point free if and only if deg ě 0 and very
ample if and only if deg D ą 0. △

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Example 23.9. If X is a curve of genus 1, a divisor D is base point free if deg D ě 2. We

will see later that, if D “ P for a k -point P , we have h0 pX, OX pDqq “ 1, so D cannot be
base point free (a generator s P H 0 pX, OX pDqq satisfies divpsq “ P , so it vanishes at P ).
A divisor D on X is very ample if deg D ě 3. △

23.4 Curves of genus 0

The results of the previous sections are particularly strong when the genus is small. In the
case X is a curve of genus 0, a divisor D is base-point free if and only if deg D ě 0 and
very ample if and only if deg D ą 0.
If we apply this observation to a divisor of the form D “ P for a k -point P , we see that
h0 pDq “ 1 ` 1 ´ 0 “ 2 and D defines a closed embedding X Ñ P1k . Therefore, we have
proved the following remarkable theorem:

Theorem 23.10. Let X be a nonsingular curve of genus 0 over an algebraically closed

field k . Then X is isomorphic to P1k .

Example 23.11. Consider the plane curve X Ă P2k defined by the homogeneous equation
x20 ` x21 ´ x22 “ 0.
Then X as a k -point P “ p1 : 0 : 1q. Let us find a basis for H 0 pX, OX pP qq.
We are looking for rational functions g which have a pole of order 1 at P and no poles
at any other point. The linear form x0 ´ x2 vanishes at P , but it has multiplicity 2 there.
To make up for this, we can consider a quotient ℓ{px0 ´ x2 q where ℓ is a linear form in
x0 , x1 , x2 vanishing with order 1 at P . For instance, we can take ℓ “ x1 . And indeed, the
quotient
x1
g“
x0 ´ x1
satisfies
divpgq “ p1 : 0 : 1q ` p1 : 0 : ´1q ´ 2p1 : 0 : 1q “ p1 : 0 : ´1q ´ p1 : 0 : 1q.
and so g P H 0 pX, OX pP qq. As P has degree 1, we see that H 0 pX, OX pP qq “ k ‘ k g .
The corresponding morphism
G : X ÝÝÑ P1k
px0 : x1 : x2 q ÞÑ px1 : x0 ´ x2 q
can be viewed as the projection from P . △
Example 23.12 (Non-algebraically closed fields). For non-algebraically closed fields k , there
can be curves of genus 0 which are not isomorphic to P1k . For instance, the conic curve
X “ V px20 ` x21 ` x22 q Ă P2R has genus 0, but it has no R-points at all. Similarly, the conic
X “ V px20 ` x21 ´ 3x22 q has infinitely many R-points, but no Q-points. In these examples,
X cannot be isomorphic to P1k , but the base change to the algebraic closure Xk̄ “ X ˆk k
becomes isomorphic to P1k .
That being said, curves of genus 0 are well understand, even over general fields k . Here

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is a brief explaination. If we assume that H 0 pX, OX q “ k and H 1 pX, OX q “ 0 and

Xpkq ‰ H, then X » P1Q . Indeed, if X admits a k -point P , then one shows that the
invertible sheaf OX pP q is very ample by pulling back to Xk̄ , where one can apply Theorem
23.10. Therefore, we again find that X » P1k .
A nice and surprising fact, which holds for any field k , is that any nonsingular projective
curve is isomorphic to plane curve of degree 2, X Ă P2k , provided that H 0 pX, OX q “ k and
H 1 pX, OX q “ 0. To prove this, one considers the anticanonical divisor D “ ´KX (which
always exists, regardless of whether X has a k -point or not). By working on Xk̄ , one shows
that this is very ample, and gives a closed embedding X Ñ P2k . △

We conclude by another characterisation of P1 , which will be important for studying curves

of higher genus as well.

Proposition 23.13. Let X be a nonsingular projective curve over an algebraically closed

field k and D any divisor on X . Then
dimk H 0 pX, OX pDqq ď deg D ` 1
with equality if and only if X » P1k .

Proof We may assume that D is effective, i.e., D “ P1 ` ¨ ¨ ¨ ` Pd for some k -points

P1 , . . . , Pk P X (possibly equal) (otherwise replace D by some other effective divisor
D1 „ D). The proof is by induction on deg D “ d.
First suppose d “ 1. There is an exact sequence

0 ÝÝÑ OX ÝÝÑ OX pP q ÝÝÑ κpP q ÝÝÑ 0.

Now h0 pX, OX q “ 1 and h0 pκpP qq “ 1 therefore h0 pOX pP qq ď 2. If h0 pOX pP qq “ 2,

then |P | cannot have any base points, so we obtain a finite morphism X Ñ P1k . As X is
nonsingular and deg P “ 1, this must be an isomorphism, and so X » P1k .
Next suppose D “ P1 ` ¨ ¨ ¨ ` Pd . Let D1 “ P1 ` ¨ ¨ ¨ ` Pd´1 . There is an exact sequence

0 ÝÝÑ OX pD1 q ÝÝÑ OX pDq ÝÝÑ κpPd q ÝÝÑ 0.

Now h0 pOX pD1 qq ď d by induction and h0 pκpPd qq “ 1 so h0 pOX pDqq ď d`1. Moreover,
h0 pOX pDqq ď d ` 1 with equality if and only if h0 pOX pD1 qq “ d which by induction
happens if and only if X » P1k .

Corollary 23.14. A nonsingular curve X over an algebraically closed field is isomorphic

to P1k if and only if ClpXq » Z.

Proof By Proposition 17.21, the Picard group of any projective space Pnk is isomorphic to
Z. Conversely, suppose X is a curve with ClpXq » Z. Let p, q be two distinct points on
X . By assumption, the two divisors p and q are linearly equivalent, so the invertible sheaf
L “ OX ppq has at least 2 linearly independent global sections (one vanishing at p and one
vanishing at q ). Therefore, X » P1k by Proposition 23.13.

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23.5 Curves of genus 1

A plane curve X Ă P2k of degree 3 has genus 1. This follows by taking cohomology of the
exact sequence
0 ÝÝÑ OP2k p´3q ÝÝÑ OP2k ÝÝÑ ι˚ OX ÝÝÑ 0.
In this section, we show that in fact every curve of genus 1 arises this way:

Theorem 23.15. Any projective curve X of genus 1 over an algebraically closed field
can be embedded as a plane cubic curve in P2k .

Proof Pick a point P P X and consider the divisor D “ 3P . Then D has degree 3 ě
2g ` 1, so it is very ample. Furthermore, by Riemann–Roch, h0 p3P q “ 3, so there is a
projective embedding ϕ : X Ñ P2k . The image ϕpXq is a smooth curve of degree equal to
deg ϕ˚ OP2k p1q “ deg D “ 3.
We can make this a little bit more explicit, as follows. For a point P P X , consider the
multiples D “ mP where m “ 1, . . . , 6. By the Riemann–Roch formula, we can compute
the spaces of global sections as follows
h0 basis
OX pP q 1 1
OX p2P q 2 1, x
OX p3P q 3 1, x, y
OX p4P q 4 1, x, y, x2
OX p5P q 5 1, x, y, x2 , xy
OX p6P q 6 1, x, y, x2 , xy, x3 , y 2
As H 0 pX, OX p6P qq is 6-dimensional, there must be a linear relation between the sections
1, x, y, x2 , xy, x3 , y 2 . It is not hard to see that, after a coordinate change, we can take it to
be the following cubic
y 2 ` a1 xy ` a2 y “ x3 ` a3 x2 ` a4 x ` a5
where a1 , a2 , a3 , a4 , a5 P k . When the characteristic of k is not 2 or 3, the equation can be
further simplified, to
y 2 “ x3 ` ax ` b.
This is known as the Weierstrass form of the curve X . Any such equation defines a nonsingular
genus 1 curve, provided that 4a3 ` 27b2 ‰ 0 (Exercise 11.6.4).
In contrast to the genus 0 case, there are many non-isomorphic genus 1 curves. For instance,
among the curves Xt “ V px21 x2 ` x30 ` x0 x22 ` tx32 q Ă P2k , where t P k , a given curve Xa
is isomorphic to at most a finite number of other Xt ’s.

The group law on X

One of the most important facts about genus 1 curves is that the set of k -points Xpkq form an
abelian group. That is, there is a natural way of ‘adding’ k -points on X . This only happens
for curves of genus 1.

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To make the discussion a bit more concrete, we consider the curve X Ă P2k defined by the
equation
y 2 z “ x3 ´ xz 2
The key point in understanding the group structure is to consider divisors on X .
If L Ă P2k is a line, we get a divisor L|X on X , supported on the finite set L X X . More
formally, we take a section s P OP2k p1q defining L and restrict it to X . The restricted divisor
on X consists of three points P, Q, R (counted with multiplicity). In particular, since any
two lines are linearly equivalent on P2k , we get for every pair of lines L, L1 and corresponding
triples P, Q, R, a relation
P ` Q ` R „ P 1 ` Q1 ` R1
where „ denotes linear equivalence.
Let us consider the point O “ p0 : 1 : 0q on X . This is a special point on X : it is an
inflection point, in the sense that there is a line L “ V pzq Ă P2k which intersects X with
multiplicity three at O. That means that L restricts to 3O on X . The point O will serve as
the identity in the group Xpkq. It has the property that any three collinear points P, Q, R in
X satisfy
P ` Q ` R „ 3O
0
Next, consider the subgroup Cl pXq Ă ClpXq consisting of degree 0. This fits into the exact
sequence
deg
0 ÝÝÑ Cl0 pXq ÝÝÑ ClpXq ÝÝÝÑ Z ÝÝÑ 0
Let us define the following map:
ξ : Xpkq ÝÝÑ Cl0 pXq
P ÞÑ rP ´ Os

Lemma 23.16. ξ is a bijection.

Proof ξ is injective: ξpP q “ ξpQq implies that P „ Q. Then P “ Q (otherwise X would

be rational by Proposition 23.14). (Alternatively, it follows because h0 pX, OX pP qq “ 1 by
Riemann–Roch). ř
ξ is surjective: Take a divisor D “ ni Pi P DivpXq of degree 0. Then D1 “ D ` O
has degree 1, so by Riemann–Roch, H 0 pX, OC pD1 qq is 1-dimensional. Hence there exists
an effective divisor of degree 1 in |D1 |, which must then be of the form D1 “ Q for some
k -point Q. But this means that D ` O „ Q, or in other words, D „ Q ´ O “ ξpQq, as we
wanted to show.
Using this bijection, we can put a group structure on the set Xpkq by transporting the
group structure of Cl0 pXq. We have shown:

Theorem 23.17. The set of k -points Xpkq on a genus 1 form an abelian group.

The group law has the following famous geometric interpretation. Given two points

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R
L
Q
x

P `Q

Figure 23.1 The group law on X

P, Q P X , we draw the line L they span (see Figure 23.1). This intersects X in one more
point, say R. In the group Cl0 pXq we have
P ` Q ` R “ 3O
To define the ‘sum’ P ` Q (which should be a new k -point of X ), we then look for a point
S so that
S ´ O “ pP ´ Oq ` pQ ´ Oq
or in other words, S ` O “ P ` Q. In light of the above, this becomes S ` O “ 3O ´ R
or, R ` S ` O “ 3O. This tells us that we should define S as follows: We draw the line L1
from O to R (shown as the dotted line in the figure), and define S to be the third intersection
point of L1 with X . By construction, we get pP ´ Oq ` pQ ´ Oq “ pS ´ Oq in Cl0 pXq.
We then define P ` Q to be the k -point S .
Given the equation of X in P2k , and coordinates for the points P and Q, we can of course
write down explicit formulas for the coordinates of S , and they are rational functions in the
coordinates of P and Q. This construction has many applications in cryptography.

23.6 Hyperelliptic curves

A nonsingular curve X of genus g ě 2 is said to be hyperelliptic if there is a finite degree
2 map X Ñ P1k . There was already a notion of ‘hyperelliptic curves’ in Section ??, so we
should start by showing that the two definitions are equivalent.
Let X be curve admitting a degree 2 morphism f : X Ñ P1k . Then the function field
KpXq is a degree 2 extension of KpP1k q “ kpxq. Let 1, y be a basis for KpXq as a kpxq-
vector space. Then the elements 1, y, y 2 are kpxq-linearly dependent, so there is a relation of
the form Ay 2 ` By ` C “ 0 for some elements A, B, C P kpxq. Multiplying this relation
by a suitable polynomial, we may assume that A, B, C P krxs. Furthermore, multiplying
the relation by A, and replacing y by Ay , we see that we may take the relation to be of the
form y 2 ` By ` C where B, C P krxs. If the characteristic of k is not equal to 2, we may
furthermore complete the square to reduce to y 2 “ ppxq for some ppxq P krxs. Finally, by
performing variable changes of the form y “ px ´ aq ¨ v , and we may assume that ppxq has
no repeated roots.
This shows that X has the same function field as the nonsingular curve defined by y 2 “

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ppxq in A2k . Hence X is birational, hence isomorphic, to one of the hyperelliptic curves of
Section ??.
If the field k has characteristic 2, most of the steps in the above parapgraph remain valid,
except that the defining equation has the form y 2 ` qpxqy “ ppxq for some qpxq P krxs of
degree ď g .
The construction in Section ?? shows that there are hyperelliptic curves of any genus g .
Here is another construction:
Example 23.18. Let X Ă P1k ˆ P1k be a smooth divisor of bidegree p2, g ` 1q. Then by the
exact sequence
0 ÝÝÑ OP1k ˆP1k p´2, ´g ´ 1q ÝÝÑ OP1k ˆP1k ÝÝÑ ι˚ OX ÝÝÑ 0
and by Exercise 18.19.7,
H 2 pP1k ˆ P1k , Op´2, ´g ´ 1qq “ H 1 pP1k , Op´2qq b H 1 pP1k , Op´g ´ 1qq » k g ,
so X has genus g .
Explicitly, X is defined by a bihomogeneous equation of the form
f px0 , x1 ; y0 , y1 q “ Apy0 , y1 qx20 ` Bpy0 , y1 qx0 x1 ` Cpy0 , y1 qx21
where A, B, C P kry0 , y1 s are homogeneous of degree g`1. Therefore, the second projection
p2 : X Ñ P1k is finite of degree 2, so X is hyperelliptic of genus g .
In fact, any hyperelliptic curve can be embedded in P1k ˆk P1k as a divisor of bidegree
p2, g `1q, at least when char k ‰ 2. To see this, note that we may write ppxq “ qpxq2 `rpxq
where q and r are polynomials of degree ď g ` 1. Then, if we make the substitution u “ x
and v “ y ` qpxq, we see that the curve is birational to the plane curve with affine equation
v 2 ´ 2vqpuq ´ rpuq “ 0 in Spec kru, vs. A projective model in P1k ˆk P1k is given by the
bihomogeneous equation
x20 y0g`1 ´ 2x0 x1 qpy1 {y0 qy0g`1 ´ rpy1 {y0 qy0g`1 x21 “ 0.
△
The following theorem summarizes the various ways to define a hyperelliptic curve.

Theorem 23.19. Let X be nonsingular curve of genus g ě 2 over an algebraically closed

field with char k ‰ 2. Then the following are equivalent:
(i) X admits a finite degree 2 map f : X Ñ P1k .
(ii) X is birational to a plane curve of the form y 2 “ ppxq, where p is a
polynomial with distinct roots.
(iii) X admits a divisor D of degree 2 and h0 pDq “ 2.
(iv) X is isomorphic to a divisor in P1k ˆk P1k of bidegree p2, g ` 1q.

Proof The discussion in the beginning of this section showed that (i)ô (ii). Example 23.18
showed that (iv)ô(i). Hence we need only address the item (iii).
(i)ñ(iii): Take D “ f ˚ Q for a point Q P P1 pkq. Then D has degree 2, and two sections,
because the morphism is non-constant.
(iii)ñ(i): Given such a divisor D, let us show that D is base-point free. By Theorem

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23.7, it suffices to show that h0 pD ´ P q “ 1 for every point P P X . Since h0 pDq “ 2,

h0 pD ´ P q can drop by at most 1 by the exact sequence (23.3), so h0 pDq ď 1. On the other
hand, D ´ P is a degree 1 divisor, hence h0 pD ´ P q ď 1 by Proposition 23.13.

23.7 Curves of genus 2

Let X be a nonsingular projective curve of genus 2. We have seen examples of such curves
already, namely the hyperelliptic curve obtained by gluing two copies of the affine curve
y 2 “ ppxq where ppxq is a polynomial of degree 5 or 6. However, we show the following,
which tells us that any genus 2 curve arisises from the construction in Section 6.4.
In fact, if X is a genus 2 curve, the canonical divisor KX is base-point free, and gives a
morphism
f : X ÝÝÑ P1k
As KX has degree 2g ´ 2 “ 2, this proves the following:

Proposition 23.20. Any curve of genus 2 is isomorphic to a hyperelliptic curve.

We saw in Proposition 18.39 that a genus 2 curve cannot be embedded in the projective
plane P2k . However, in light of Theorem 23.19, it can be embedded as a p2, 3q-divisor in
P1k ˆk P1k .

23.8 Curves of genus 3

We next classify curves of genus 3. We have seen a few examples of such curves already:
Example 23.21. A plane curve X Ă P2 of degree d “ 4 has genus 12 pd ´ 1qpd ´ 2q “ 3.
Note that
ΩX “ OP2 pd ´ 3q|X “ OX p1q
so KX is very ample, since it is the restriction of the very ample invertible sheaf OP2 p1q. △
Example 23.22. The X on the quadric surface Q » P1k ˆ P1k in P3k of type (2,4) is hyperellip-
tic. It is a curve of degree 6 and genus 3 in P3k . As we showed in Section 23.7, the canonical
divisor KX is not very ample (in this case, it defines a degree 2 morphism X Ñ P2k onto a
nonsingular conic). △
These two examples are a bit different. The curves in the first example have very ample
canonical divisor KX , whereas in the second case KX defines the degree 2 map X Ñ P1k .
We show that this distinction is a general phenomenon for curves of genus 3:

Proposition 23.23. Let X be a curve of genus 3. Then there are two possibilities:
(i) KX is very ample. Then X embeds as a plane curve of degree 4.
(ii) KX is not very ample. Then X is a hyperelliptic curve, and it embeds as a
p2, 4q divisor in P1k ˆ P1k .

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Proof of Proposition 23.23 We still need to check the last part of the above theorem, namely
that every hyperelliptic curve arises as a curve of type p2, 4q on the quadric surface P1k ˆ P1k Ă
P3k .
Write F for the divisor inducing the degree 2 morphism X Ñ P1k . We first claim that
KX „ 2F . Since both divisors have degree 4, it suffices to show that KX ´ 2F is effective.
Note that in any case h0 pX, 2F q ě 3, as the sections x20 , x0 x1 , x21 are linearly independent
(they are pullbacks of independent sections from ΓpP1k , OP1k p2qq and the pullback map is
injective by Exercise ??). Then applying Riemann–Roch to the divisor D “ 2F , we get

h0 p2F q ´ h0 pKX ´ 2F q “ 4 ` 1 ´ 3 “ 2,
from which we find that h0 pKX ´ 2F q ‰ 0, as desired.
We next look for a base-point divisor D of degree 4. This is the divisor which will induce
the ‘first projection’ X Ñ P1k , once we have established that X is a p2, 4q-divisor. We will
find points P1 , . . . , P4 so that the divisor D “ P1 ` ¨ ¨ ¨ ` P4 satisfies
(i) h0 pDq “ 2
(ii) D is base-point free.
(iii) D ´ F is not effective.
In fact, there is a Zariski open set U Ă Xpkq ˆ Xpkq ˆ Xpkq ˆ Xpkq so that these
conditions are satisfied for every pP1 , P2 , P3 , P4 q P U .
Let us start with the condition (i). Note that by Riemann–Roch, we have

h0 pDq ´ h0 pKX ´ Dq “ 4 ` 1 ´ 3 “ 2
Note that KX ´ D has degree 2g ´ 2 ´ 4 “ 0, so KX ´ D is a divisor of degree 0. This is
effective if and only if KX „ D. Note that in selecting the points P1 , P2 , P3 , P4 , we have 4
dimensions of freedom, whereas there is only a 2-dimensional space of canonical divisors
|KX |, therefore most divisors P1 ` P2 ` P3 ` P4 will not be a canonical divisor.
To argue a little bit more rigorously, note that KX is base-point free, so h0 pKX ´ P1 q will
be be equal to 3 ´ 1 “ 2 for all points P1 . The divisor LX ´ P1 has at most finitely many
base points, so h0 pKX ´ P1 ´ P2 q “ 1 for most points P2 . Continuing subtracting points
like this, we see that KX ´ P1 ´ P2 ´ P3 ´ P4 is not effective, and hence the claim (i) holds.
We can use a similar argument to show (ii). We need to show that

h0 pD ´ P q “ h0 pDq ´ 1 “ deg D ` 1 ´ 3 ´ 1 “ 1
for every point P . Suppose this is not the case, and let P be a base point of D. Since
D “ P1 ` P2 ` P3 ` P4 , we may suppose that P “ P4 . By Riemann–Roch, we are done if
we can show that h0 pKX ´ D ` P q “ 0. Note that KX ´ D ` P “ KX ´ P1 ´ P2 ´ P3 .
There is a 3-dimensional space of effective divisors of the form P1 ` P2 ` P3 for points
Pi P X , but only a 2-dimensional linear system of effective canonical divisors |KX |. Hence
KX ´ D ` P is not effective.
Finally, to see (iii), write F “ Q1 ` Q2 for the degree 2 divisor on X . When defining
the divisor D, we may assume that the points Pi are choosen so that Q1 will not be a base-
point of D and Q2 will not be a base-point of D ´ Q1 . Therefore h0 pD ´ Q1 q “ 1 and
h0 pD ´ Q1 ´ Q2 q “ 0, and so D ´ F is not effective.
We therefore have two morphisms: f : X Ñ P1k and g : X Ñ P1k (induced by D). By the

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universal property of the fiber product, this gives a morphism

ϕ “ pf ˆ gq : X ÝÝÑ P1k ˆk P1k

We claim that this is a closed embedding.
Let F “ P ` Q be the divisor which induces f .
Next, we claim that D`F is very ample. This divisor is base-point free of degree 2`4 “ 6
and h0 pD ` F q “ 6 ` 1 ´ 3 “ 4 (there is no h1 because 6 ą 2g ´ 2).
We can see the sections of H 0 pX, OX pD ` F qq as follows. Write H 0 pF q “ xx0 , x1 y
and H 0 pDq “ xy0 , y1 y. As F is base-point free, the Koszul complex on x0 , x1 gives the
following exact sequence:

0 Ñ OX pD ´ F q Ñ OX pDq2 Ñ OX pD ` F q Ñ 0 (23.6)

Note that H 0 pD ´ F q “ 0 by item (iii). Then H 1 pD ´ F q “ 0 as well, by Riemann–Roch.

This implies that the multiplication map H 0 pOX pDqq2 Ñ H 0 pOX pD ` F qq which sends
ps, tq to x0 s ` x1 t, is an isomorphism. Hence
H 0 pD ` F q “ xx0 y0 , x0 y1 , x1 y0 , x1 y1 y. (23.7)

Now, to show that D ` F is very ample, we need to show that

h0 pX, D ` F ´ p ´ qq “ h0 pD ` F q ´ 2 “ 2
for any pair of points p, q P X . By Riemann–Roch again, we can conclude if we know that
h0 pKX ´ D ´ F ` p ` qq “ 0. But since KX „ 2F , we have
KX ´ D ´ F ` p ` q „ F ´ D ` p ` q
These are divisors of degree 0, so if this is effective, we must have D ´ F „ p ` q . But this
contradicts item (iii) above.
Therefore, D ` F is very ample, and embeds X into P3k . By (23.7), the image of X lies
on the quadric surface u0 u3 ´ u1 u2 “ 0. Hence X Ñ P3k factors via the Segre embedding
X Ñ P1k ˆk P1k Ñ P3k . Hence we may view X as a divisor X Ă P1k ˆk P1k . By construction,
the two projections to P1k are given by the sections of D and F . This means that X has
bidegree p2, 4q as a divisor on P1k ˆk P1k .

23.9 Curves of genus 4

Recall that curves of genus g ě 2 split up into two disjoint classes, the hyperelliptic curves,
which admit a degree 2 morphism to P1k ; and the canonical curves, where KX is very ample.
Here are two examples of genus 4 curves in P1k ˆk P1k .

Example 23.24. Consider a type p2, 5q curve C on Q Ă P3k . Then C has degree 7 “ 2 ` 5
and C is hyperelliptic (because of the degree 2 map coming from projection onto the first
fact p1 : Q Ñ P1 ). A type p3, 3q curve on Q is also of genus 4. It is a degree 6 complete
intersection of Q and a cubic surface. △

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Let us try to classify curves X of genus 4. If X is hyperelliptic, then a very similar

argument to the one in the previous section shows that X embeds as a p2, 5q-divisor in
P1k ˆk P1k .
So we may assume that KX is very ample. This means that KX defines an embedding
ϕ : X Ñ Pg´1 k “ P3k and the degree of the embedded curve is deg KX “ 2g ´ 2 “ 6.
What are the equations of X in P3k ? To answer this, we use a very powerful technique in
curve theory, namely combining Riemann–Roch with the sheaf cohomology of projective
space.
Twisting the ideal sheaf sequence of X by OP3k p2q and taking cohomology gives the exact
sequence

0 Ñ H 0 pP3k , IX p2qq Ñ H 0 pP3k , OP3k p2qq Ñ H 0 pX, OX p2qq Ñ ¨ ¨ ¨

Note that OP3 p1q “ OX pKX q. Applying Riemann-Roch to the divisor 2KX , we get

h0 pOP3 p2q|X q “ deg 2KX ` 1 ´ g ` h1 pOX p2KX qq “ 12 ` 1 ´ 4 “ 9.

(Note that h1 pOP3 p2q|X q “ h1 p2KX q “ h0 p´KX q “ 0, as ´KX has negative degree).
Since h0 pP3k , OP3 p2qq “ 10, it follows that the map H 0 pOP3 p2qq Ñ H 0 pOX p2qq must
have a nontrivial kernel, and so h0 pP3k , IX p2qq ą 0.
The conclusion is that X lies in some surface of degree 2, say Q Ă P3k is defined by
q P krx0 , x1 , x2 , x3 s. Note that X does not lie in a plane (then X could not have genus
4). This implies that q must be irreducible. So X lies on either a singular quadric cone
V px0 x2 ´ x21 q or the nonsingular quadric surface V px0 x3 ´ x1 x2 q. The quadric surface Q
is moreover unique, as if X Ă Q X Q1 , and so X would have degree ď 4.
What about cubics in IX ? Of course x0 q, . . . , x3 q give a 4-dimensional space in IX “
H 0 pP3k , IX p3qq. We next consider the sequence

0 Ñ H 0 pOX p3qq Ñ H 0 pOP3 p3qq Ñ H 0 pOX p3qq Ñ ¨ ¨ ¨

We have h0 pP3k , OP3 p3qq “ 20, and h0 pX, OX p3qq “ 15 by Riemann–Roch. Hence
h0 pOX p3qq ě 5, and we find that there is a cubic c P krx0 , x1 , x2 , x3 s which is not in
the span of x0 q, . . . , x3 q , hence c R pqq. Note that V pq, cq is a 1-dimensional subscheme of
degree 6 containing X as a component. As X is also of degree 6, we must have X “ V pq, cq.
This proves the following theorem:

Theorem 23.25. Let X be a nonsingular curve of genus 4. Then either

(i) X is hyperelliptic (in which case X embeds as a p2, 5q-divisor in P1 ˆ P1 q;
or
(ii) X “ V pQq X V pCq is the complete intersection of a quadric surface and a
cubic surface in P3k .

There are further classical results about the canonical embedding for genus ě 3.

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Theorem 23.26 (Enriques–Petri). Let X be a nonsingular non-hyperelliptic curve of

genus ě 3. Then the image of the canonical emedding X Ñ Pg´1 is defined by quadratic
equations unless:
(i) X is trigonal, i.e., birational to an affine curve of the form y 3 “ ppxq.
(ii) X is isomorphic to a curve of degree 5 in P2k .

23.10 Exercises
Exercise 23.10.1. Let X be a nonsingular quasi-projective curve over an algebraically closed
field k . Show that if ΓpX, OX q ‰ k , then X is affine. H INT: Embed X in a projective
nonsingular curve X and consider divisors supported on the closed subset X ´ X .
Exercise 23.10.2. Let f : X Ñ Y be a non-constant morphism of curves. If D is a divisor
on Y , show that h0 pX, f ˚ OY pDqq ě h0 pY, OY pDqq.
Exercise 23.10.3 (Riemann–Roch for higher rank). Let X be a nonsingular projective curve
and let E be a locally free sheaf of rank r. Show that
χpX, Eq “ degpdet Eq ` rχpOX q (23.8)
Exercise 23.10.4 (Lüroth’s theorem). Let X be a nonsingular proper curve over an alge-
braically closed field k and let f : P1k Ñ X be a non-constant morphism. Show that X is
isomorphic to P1k .
Exercise 23.10.5. Find an example of an OX -module F of finite type so that the equality in
(22.25) is not true if we replace H 1 by H 0 .
Exercise 23.10.6. Let X Ă P2k be a plane curve of degree ď 3. Show that the automorphism
group of X is infinite.
Exercise 23.10.7. Let X Ă P2k be a plane curve of degree 4. Show that any automorphism
ϕ : X Ñ X is induced by a linear transformation P2k Ñ P2k . Deduce that the general plane
quartic has trivial automorphism group.
Exercise 23.10.8 (Curves of genus 5). a) Let X be a complete intersection of 3
quadric hypersurfaces in P4k . Show that X is non-hyperelliptic of genus 5.
b) Conversely, show that any non-hyperelliptic curve of genus 5 is contained in
the intersection of 3 quadric hypersurfaces Q0 X Q1 X Q2 in P4k .
c) * Show that there exist non-hyperelliptic genus 5 curves which are not complete
intersections of 3 quadrics (it can happen that Q0 X Q1 X Q2 has dimension
ě 2).
Exercise 23.10.9 (Curves of genus 6). Show that the following varieties define genus 6
curves:
a) A degree 5 curve in P2k
b) A p2, 7q-divisor in P1k ˆ P1k .
c) A p3, 4q-divisor in P1k ˆ P1k .
d) * Show that the classes of curves constructed in a), b), c) are disjoint.

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Exercise 23.10.10 (Curves of degree 3). Let X Ă Pnk be a nonsingular curve of degree 3.
Show that X has genus ď 1 and is either a twisted cubic or a plane cubic.
Exercise 23.10.11 (Curves of degree 4). Let X Ă Pnk be a nonsingular curve of degree 4.
Show that X has genus 0, 1 or 3. Find a projective models in each case.
Exercise 23.10.12 (Hyperelliptic curves). Let X be a hyperelliptic curve of genus g , that is,
admitting a base-point free divisor D of degree 2.
a) Show that if g ě 2, then D is unique up to linear equivalence, and that
KX „ pg ´ 1qD. P OSSIBLE HINT: If f, g : X Ñ P1k are two degree 2
morphisms, consider a point P P X such that f ´1 pf pP qq “ P ` Q and
g ´1 pgpP qq “ P ` Q for P, Q, R P X distinct. Then consider the divisor
P ` Q ` R.
b) Show that if g “ 1, then X contains infinitely many such divisors D which are
non-linearly equivalent.
Exercise 23.10.13. Let X be a curve over a field k and let F be a sheaf on X . Show that
H i pXk , q ˚ Fq » H i pX, Fq bk k for all i, where q : Xk Ñ X is the base change morphism.
Exercise 23.10.14 (Trace map). Let f : X Ñ Y be a finite morphism of nonsingular
projective curves. Show that there is a natural map Tr : f˚ OX Ñ OY which splits the usual
map f 7 : OY Ñ f˚ OX . Deduce that
f˚ OX » OY ‘ E
for some locally free sheaf E .
Exercise 23.10.15. a) Let f : X Ñ Y be a domninant morphism of nonsingular
projective curves. Show that gpXq ě gpY q H INT: Apply Exercise 23.10.14.
b) (Lüroth’s theorem): Let f : P1k Ñ X be a dominant morphism of nonsingular
projective curves. Show that X » P1k .
Exercise 23.10.16 (Fermat’s Last Theorem for polynomials). As an application of Exercise
23.10.15, show that if n ě 3, there are no non-constant polynomials P, Q, R P krts such
that
P n ` Qn “ Rn .
What happens when n “ 1 or 2?
H INT: Construct a morphism to a Fermat curve in P2k .

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Some results from Commutative Algebra

469

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A.1 Operations on ideals

If A is a ring and S Ă A is subset, the ideal generated by S is defined as the ideal

ta1 s1 ` ¨ ¨ ¨ ` ar sr | r P N, ai P A, si P Su
ř
For a collection of ideals tai uiPI , the sum iPI ai is defined as the ideal generated by finite
sums ai1 ` ¨ ¨ ¨ ` air where aij P aij . This is the smallest ideal containing all the ai . In
particular, for two ideals a and b, the sum a ` b consists of sums a ` b where a P a, b P b.
If a1 , . . . , ar are ideals, the product ideal a1 ¨ ¨ ¨ ar is the ideal generated by products
a1 ¨ ¨ ¨ ar , where ai P ai . There is always an inclusion

a1 ¨ ¨ ¨ ar Ă a1 X ¨ ¨ ¨ X ar .

By the next result, this is an equality if the ideals are pairwise coprime, that is, ai ` aj “ p1q
for each i ‰ j .

Theorem A.1 (Chinese remainder theorem). Let a1 , . . . , an Ă A be aś

collection of
n
ideals which are pairwise coprime. Then the homomorphism φ : A Ñ i“1 A{ai is
surjective, and
Ker φ “ a1 X ¨ ¨ ¨ X an “ a1 ¨ ¨ ¨ an ,
giving an isomorphism
č ź
A{ ai » A{ai .

The radical of an ideal a is defined as

?
a “ t a | an P a for some n P N u

If a and b are two ideals, we define the ideal quotient as the ideal

pa : bq “ t a P A | ab Ă a u.

If x P A, the ideal p0 : xq is sometimes called the annihilator of x.

An element a P A is called a zero divisor if there exists a nonzero b P A such that ab “ 0.
It is a unit if there exists a b with ab “ 1. a is nilpotent if an “ 0 for some n P N. a is
irreducible if it is not a unit and a “ xy implies that x or y is a unit.
An A-algebra is a ring B equipped with a ring map ϕ : A Ñ B . For instance, B “
Arx1 , . . . , xn s is an A-algebra. B is said to be finitely generated if it is isomorphic to one of
the form Arx1 , . . . , xn s{a for some ideal a Ă Arx1 , . . . , xn s.
The ring map ϕ allow us to multiply elements of B with elements of A, by defining
ab “ ϕpaqb. For this reason, it is common to suppress ϕ from the notation.

A.1.1 Prime ideals. An ideal p is prime if xy P p implies either x P p or y P p. This is

equivalent to saying that A{p is an integral domain.

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A.1.2 Ideals in quotients. For an ideal a of A, there is a one-to-one correspondence

tIdeals in A{au ÐÑ tIdeals in A containing au (A.1)
Let π : A Ñ A{a be the quotient map. If b Ă A is an ideal containing a, then πpbq “ b{a
is an ideal of A{a. Conversely, if J is an ideal of A{a, then π ´1 pJq is an ideal containing
a. These operations are inverse to each other, as πpπ ´1 pJqq “ J and π ´1 pπpbqq “ ta P
Auπpaq P πpbq “ b ` a “ b.
This correspondence preserves inclusions: if a Ă b Ă c, then b{a Ă c{a, and if I Ă J Ă
A{a, then π ´1 pIq Ă π ´1 pJq Ă A.

A.1.3 Prime ideals in quotients. The correspondence (A.1) also gives a bijection between
the prime ideals of A{a and the prime ideals containing a. This follows because b is prime if
and only if A{a is an integral domain, and A{b “ pA{aq{pb{aq is an integral domain if and
only if b{a is prime in A{a.

A.2 Some field theory

A field extension K{k is a ring map k Ñ K between two fields. The degree of the field
extension is defined as the dimension of K as a k -vector space; this number is denoted by
rK : ks. If k Ñ K Ñ L is a sequence of field extensions, then rL : ks “ rK : ksrL : Ks.
We say K{k is called finite if K is finite-dimensional as a k -vector space. It is called
fintely generated if there exist elements a1 , . . . , an P K so that K “ kpa1 , . . . , an q. Here
kpa1 , . . . , an q denotes the field extension generated by the ai , i.e., the field of fraction of the
ring kra1 , . . . , an s.
An element a P K is called algebraic over k if it satisafies a polynomial equation
P paq “ 0, where P has coefficients in k . The field extension K{k is said to be algebraic, if
every element of K is algebraic over k . K{k is called transcendental if it is not algebraic.
K{k is purely transcendental if there exists an algebraically independent collection of
elements txi uiPI such that K “ kppxi qiPI q. An extension is finite if and only if it is algebraic
and finitely generated.
A field k is called algebraically closed if k Ă L algebraic implies that k “ L. If k is a
field, we call a field extension K{k an algebraic closure if it is algebraically closed, and for
any other extension k Ñ L where L is algebraically closed, there is a map of k -algebras
K Ñ L. The algebraic closure is unique up to isomorphism (of k -algebras).

A.2.1 Transcendence degree. If K{k is a field extension, a set of elements a1 , . . . , ar

from K is said to algebraically independent over k if there are no polynomial relations
between them, i.e., P pa1 , . . . , ar q ‰ 0 for every polynomial P P krt1 , . . . , tn s. Equivalently,
the map ti ÞÑ ai defines a k -algebra isomorphism krt1 , . . . , tr s » kra1 , . . . , ar s. Likewise,
one says that a possibly infinite subset S Ă K is algebraically independent if any finite
collection of (distinct) elements of S is algebraically independent.
A transcendence basis for K over k is a maximal algebraically independent set S Ă K . If
a1 , . . . , an are algebraically independent, they form a transcendence basis if and only if the
field extension kpa1 , . . . , an q Ă K is algebraic. One can prove that all transcendence bases
have the same cardinality. This common cardinality is called the transcendence degree of K

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over k and is denoted by trdegk K . In general, the transcendence degree may be infinite,
but for finitely generated field extensions, it will always be finite. Note that if A Ă B is an
extension of integral domains with B of finite type over A, then the associated extension
KpAq Ă KpBq will be a finitely generated field extension with a finite transcendence
degree.
If k Ă K Ă L are fields, and tai uiPI is a transcendence basis for K{k and tbj ujPJ is
a transcendence basis for L{K , then tai , bj uiPI,jPJ is a transcendence basis for L{k . In
particular,
trdegk pLq “ trdegk pKq ` trdegK pLq.

A.2.2 Separable extensions. A field extension K{k is called separable if every algebraic
element K over k is a root of a separable polynomial over k , that is, a polynomial whose
roots are distinct in an algebraic closure of k . Equivalently, K{k is separated if for every
field extension k Ñ L, the tensor product K bk L is reduced.

‚ A field extension of characteristic 0 is always separable.

‚ In characteristic p ą 0, an extension is separable if and only if the formal derivative of
every irreducible polynomial is non-zero.
‚ Every purely transcendental extension is separable.

A field k is said to be perfect if every irreducible polynomial over k has no repeated root in
any field extension K{k . Any field of characteristic 0, such as Q, R, or C, is perfect. If k has
characteristic p, k is perfect if and only if every element of k is a p-th power.

Example A.2. Examples of perfect fields include: Any field of characteristic 0, such as Q, R,
or C; finite fields, since any finite field of characteristic p is closed under taking p-th powers;
and algebraically closed fields, by virtue of having no proper algebraic extensions. △

As in the definition of an algebraic closure, we define the separable closure and perfect
closure of a field. These are again unique up to isomorphism.

A.3 Modules
À
An A-module M is said to be free if M » AI , where I is an index set and AI “ iPI A.
The rank of M is defined as the cardinality of I .

A.3.1 Finiteness conditions. M is said to be finite if M » An for some n ě 0.

M is finitely generated, or of finite type if there there are finitely many elements m1 , . . . , mn
which generate it as an A-module, that is, there is a surjective map of A-modules An Ñ M .
M is of finite presentation if there is an exact sequence

Am Ñ An Ñ M Ñ 0

for some integers m, n ě 0.

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A.3.2 Composition series. An A-module M is called simple if it is nonzero and its only
submodules are 0 and M itself. Equivalently, M » A{m for some maximal ideal m.
If M is an A-module, then a compostion series is a chain of submodules 0 “ M0 Ă
M1 Ă M2 Ă ¨ ¨ ¨ Ă Ms “ M such that Mi`1 {Mi is simple for any i “ 0, . . . , s ´ 1. Given
a compsition series, the quotients Mi`1 {Mi are uniquely determined up to isomorphism. We
define thelength of M , lengthpM q to be the length s of a composition series.
The length is additive on exact sequences.

A.3.3 Torsion. If M is an A-module, we define the torsion submodule of M as the sub-

module Mtors “ t m P M | am “ 0 for some a P A ´ 0 u. M is said to be torsion free if
the torsion submodule is zero.

A.4 Localization
A nonempty subset S of a commutative ring A is called a multiplicative set if it is closed
under multiplication and contains the identity element of A.
The localization of A with respect to a multiplicative set S , denoted S ´1 A, is the set of
fractions a{s with a P A and s P S . There is a well-defined addition and multiplication
making S ´1 A into a ring. Formally, S ´1 A is constructed by definiing an equivalence relation
on A ˆ S by pa, sq „ pa1 , s1 q if there exists an element t P S such that tpas1 ´ a1 sq “ 0 in
A. The elements of S ´1 A are denoted by a{s or as .
There is a canonical localization map
ρ : A Ñ S ´1 A, x ÞÑ x{1
which makes S ´1 A into an A-module. The kernel of ρ consists of those elements which are
annihilated by some element of S , i.e., sx “ 0 for some s P S . In particular, the map ρ is
injective if A contains no zerodivisors.
The localization S ´1 A is the zero ring if and only if 0 P S (if 0 P S , then a{s “ 0{1 by
definition).
If M is an A-module, one also defines a localization S ´1 M as the set of fractions m{s,
for m P M , s P S , using the equivalence relation pm, sq „ pm1 , s1 q if tpms1 ´ m1 sq “ 0
in M . As above, there is a canonical localization map ρ : M Ñ S ´1 M . Also, S ´1 M is
naturally an S ´1 A-module.
The localization S ´1 A and the map ρ satisfies the following universal property: For every
A-module N and any ring map f : A Ñ B such that f psq P B is invertible for all s P S ,
there exists a unique map of rings g : S ´1 A Ñ B making the following diagram commute:
f
A B
φ (A.2)
D!g

S ´1 A
The map g is uniquely determined by gpm{sq “ f pmq ¨ f psq´1 for all m P M and s P S .
Example A.3. The first prototype example is when S “ A ´ p for a prime ideal p. In this
case the localization S ´1 A is denoted Ap . The ring Ap is a local ring with maximal ideal

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pAp . The elements in A ´ p become units in Ap , and every non-unit in Ap is in the maximal
ideal pAp . △
Example A.4. The second prototype example is when S “ t1, f, f 2 , . . . u for some f P A.
´1
In this case the localization Sř A is denoted . Elements of Af are of the form a{f n where
ř Af´n
n
a P A and n ą 0. The map an t ÞÑ an f induces an isomorphism Arts{p1 ´ f tq »
Af . △
The functor M ÞÑ S ´1 M has excellent properties. It is exact, and commutes with direct
sums, direct limits, and tensor products.
If p is a prime ideal, and f R p, then there is a canonical localization map Af Ñ Ap . The
localizations Af form a directed system as f runs over the set of elements in A ´ p and there
is an isomorphism
lim
ÝÑ Af ÝÝÑ Ap .
f PA´p

A.4.1 Ideals in localizations. Let S Ă A be a multiplicative set. If a Ă A is an ideal

such that S X a ‰ H, then S ´1 a “ p1q in S ´1 A. Hence the ideals that intersect S map to
non-proper ideals in S ´1 A.
For prime ideals, there is a one-to-one correspondence
tPrime ideals P in S ´1 Au ÐÑ tPrime ideals p in A with p X S “ Hu (A.3)
´1 ´1 ´1
The correspondence sends P Ă S A to ρ pP q, and the inverse is given by p ÞÑ S p.
These maps are mutual inverses. For a prime ideal P in S ´1 A, we have S ´1 ρ´1 pP q “ P .
Indeed, a{s P P ô a{1 P P ô a P ρ´1 pP q ô a{s P S ´1 pρ´1 pP qq. On the other hand,
if p is a prime ideal in A such that p X S “ H, then ρ´1 pS ´1 pq “ p. The ‘Ą’-inclusion is
clear. Conversely, if a P ρ´1 pS ´1 pq, then a{1 P S ´1 p, so there exists t P S such that ta P p.
Since p X S “ H, t R a, therefore a P a.
A.4.2 Radicals and intersections. For an ideal a Ă A, we have
? č
a“ p. (A.4)
aĂp
?
Indeed, if f P a, then f r P a for some r P N. Therefore, f r P p for every p that contains a,
which implies f P p.
Conversely, if f r R a for every r P N, then Af ‰ 0 and the localized ideal af is a proper
ideal in Af . Taking the preimage of any maximal ideal in Af containing af , we get a prime
ideal p Ă A such that p Ą a and p X t1, f, f 2 , . . . u “ H. But then f does not lie in the
intersection of the right-hand side of (A.4).

A.4.3 Local rings. A ring A is called local if there is exactly one maximal ideal in A.
If A is a ring and m is an ideal, the following are equivalent:
(i) A is a local ring and m is the maximal ideal
(ii) All elements in A ´ m are units
(iii) m is maximal, and every element of the form 1 ` x where x P m is a unit.
Fields are local rings (with maximal ideal p0q). Other examples include discrete valuation
rings and localizations Ap (with maximal ideal pAp .)

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A.4.4 Integral domains. If A is an integral domain, we let K “ KpAq be the fraction

field, i.e., the localization S ´1 A where S “ A ´ t0u. Then A Ă K and via the map
a ÞÑ a{1. Moreover, the various localizations Af , Ap and Am lie as subrings of K .

Lemma A.5. Let A be an integral domain. Then

č
A“ Am
m

where the intersection is taken over all maximal ideals of A.

Ş
Proof The containment A Ď m Am is clear. Conversely, let f P K be an element which
lies in all the localizations Am . Consider the ideal I “ t a P A | af P A u. If I “ p1q, then
1 P I , i.e., f P A, so we are done. If I is a proper ideal, it is contained in a maximal ideal m.
But then f cannot be contained in Am , because otherwise there would exist some s R m such
that sf P A and hence A P I , which contradicts the maximality of m.

A.5 Tensor products

Let M and N be two A-modles. We define the tensor product M bA N as the quotient of
the free module AM ˆN with basis pem,n q modulo the relations
em1 `m2 ,n ´ em1 ,n ´ em2 ,n “ 0
em,n1 `n2 ´ em,n1 ´ em,n2 “ 0
eam,n ´ em,an “ aem,n ´ eam,n “ 0
for all m P M , n P N , a P A. We write m b n for the class of em,n in M bA N .
The assignment pm, nq ÞÑ m bA n defines map γ : M ˆ N Ñ M bA N which is bilinar
as a map of A-modules. It satisfies the following universal property: for any bilinear map
ϕ : M ˆ N Ñ P , there is a unique map ϕ : M bA N Ñ P so that ϕ “ ϕ ˝ γ .
Example A.6. If A and B are k -algebras over some field k , then A bk B is a ring via the
multiplication rule pa b bq ¨ pa1 b B 1 q “ paa1 b bb1 q. As such, it is naturally a k -algebra
via the map k Ñ A bk B sending x ÞÑ xp1 b 1q. If R is another k -algebra, the universal
property says that there is a bijection
Homk pA bk B, Rq “ Homk pA, Rq ˆ Homk pB, Rq.
△

A.6 Basic formulas

In the formulas below, M, N, L denote A-modules; S Ă A is a multiplicative set; and p is
a prime ideal of A. Each equality ‘=’ between two modules means that there is a canonical
isomorphism between them.
A.6.1 Localization identities.
(i) S ´1 pM {N q “ S ´1 M {S ´1 N
(ii) If M is finitely presented: S ´1 HomA pM, N q “ HomS ´1 A pS ´1 M, S ´1 N q.

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A.6.2 Tensor product identities.

(i) A bA M “ M
(ii) M bA N “ N bA M
(iii) MÀbA pN bA P q “ pM À bA N q b A P
(iv) p iPI Mi q bA N “ p iPI Mi bA N q
(v) M bA A{a “ M {aM for every ideal a Ă A.
(vi) If A Ñ B is a ring map, and P is a B -module, then there is a canonical
isomorphism of B -modules

M bA pN bB P q “ pM bA N q bB P.
(vii) HomA pM bA N, P q “ HomA pM, HompN, P qq.
(viii) S ´1 M “ M bA S ´1 A.
(ix) S ´1 pM bA N q “ S ´1 M bS ´1 A S ´1 N .
(x) If A “ k is a field, dimk pM bk N q “ pdimk M q ¨ pdimk N q.
(xi) If f : A Ñ B is a ring map and M , N are B -modules, then M bB N ”
M bA N .

A.6.3 Exactness properties. If 0 Ñ M 1 Ñ M Ñ M 2 Ñ 0 is an exact sequence, then

0 ÝÝÑ S ´1 M 1 ÝÝÑ S ´1 M ÝÝÑ S ´1 M 2 Ñ 0

is exact for every multiplicative set S ;

0 Ñ HompL, M 1 q Ñ HompL, M q Ñ HompL, M 2 q

0 Ñ HompM 2 , Lq Ñ HompM, Lq Ñ HompM 1 , Lq
are exact for every A-module L; and

M 1 bA N Ñ M bA N Ñ M 2 bA N Ñ 0
is exact for any A-module N .

Proposition A.7.
(i) For an A-module M , M “ 0 if and only if Mm “ 0 for every maximal
ideal m.
(ii) A sequence of A-modules 0 Ñ M 1 Ñ M Ñ M 2 Ñ 0 is exact if and only
if the induced sequence 0 Ñ Mm1 Ñ Mm Ñ Mm2 Ñ 0 is exact for every
maximal ideal m.
(iii) A map of A-modules ϕ : M Ñ N is injective (resp. surjective) if and only
if ϕm : Mm Ñ Nm is injective (resp. surjective) for every maximal ideal m.

A.6.4 Finitely presented modules and Hom. We say that a module M is finitely presented
if there is an exact sequence
Aq Ñ Ap Ñ M Ñ 0. (A.5)

If A is Noetherian, then any finitely generated module finitely presented.

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Proposition A.8. If M is finitely presented, then HomA pM, ´q commutes with local-
ization. That is, there is an isomorphism of S ´1 A-modules
S ´1 HomA pM, N q ÝÝÑ HomS ´1 A pS ´1 M, S ´1 N q (A.6)

Proof The isomorphism is defined as follows: sending ϕ : M Ñ N to the S ´1 A-linear

map m{s ÞÑ ϕpmq{s, defines a map of A-modules

HomA pM, N q ÝÝÑ HomS ´1 A pS ´1 M, S ´1 N q

Since multiplication by an element s P S is an isomorphism on the A-module on the right,

the map extends to a map (A.6).
For M “ A, we have HomA pA, N q “ N and (A.6) is the isomorphism S ´1 HomA pA, N q “
§´1 N Ñ HomS ´1 A pS ´1 A, S ´1 N q “ S ´1 N . More generally, it is an isomorphism for any
M “ AN , N P N.
The map (A.6) is functorial in M . Applying HomA pM, ´q to (A.5), we get the following
commutative diagram:

0 S ´1 HomA pM, N q S ´1 HomA pAp , N q S ´1 HomA pAq , N q

0 HomS ´1 A pS ´1 M, S ´1 N q HomS ´1 A pS ´1 Ap , S ´1 N q HomS ´1 A pS ´1 Aq , S ´1 N q

The two right-most vertical maps are isomorphisms (as shown above). By the five lemma, the
left-most vertical map is an isomorphism as well.

A.7 Change of rings

For a ring map ϕ : A Ñ B , there are three natural operations on modules. Here M is an
A-module, and N is a B -module:

‚ ϕ! M “ M bA B is the induced B -module.

‚ ϕ˚ M “ HomA pB, M q is the coinduced module.
‚ ϕ˚ N “ NA is the module formed by restriction of scalars, i.e., N but viewed as an
A-module.

These form adjoint functions, in the sense that there are natural bijections

HomA pϕ! M, P q Ñ HomB pM, ϕ˚ P q (A.7)

defined by sending ℓ : M bA B Ñ P to m ÞÑ ℓpm b 1q; and

HomA pϕ˚ P, M q “ HomB pP, ϕ˚ M q (A.8)

sending ℓ : PA Ñ M to h : P Ñ HomA pB, M q defined by hppq “ pb ÞÑ ℓpb ¨ pqq.

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A.8 Nakayama’s Lemma

Lemma A.9 (Nakayama’s lemma, local version). Let A be a local ring with maximal
ideal m, and let M be a finitely generated module. If mM “ M , then M “ 0.

Proof Assume M ‰ 0, and let m1 , . . . , mr , r ě 1, be a minimal set of generators for M .

As M “ mM , we may write mr as a linear combination of the form mr “ a1 m1 ` ¨ ¨ ¨ `
ar mr for some ai P m. This means that p1 ´ ar qmr “ a1 m1 ` ¨ ¨ ¨ ` ar´1 mr´1 . However,
1 ´ ar is a unit in A, so this relation allows us to express mr in terms of m1 , . . . , mr´1 ,
contradicting the minimality assumption on m1 , . . . , mr .

Corollary A.10. Let A be a local ring, and let φ : M Ñ N be map of finitely generated
A-modules, such that M {mM Ñ N {mN is surjective. Then φ is surjective.

Proof If M {mM Ñ N {mN is surjective, then N “ φpM q ` mN . Hence

mpN {φpM qq “ pmN ` φpM qq{φpM q “ N {φpM q.
Applying Nakayama’s Lemma to N {φpM q, we get that N {φpM q “ 0, so φ is surjective.

The above result can be helpful also in the case A is not local, as by Proposition A.7, a
map of A-modules ϕ : M Ñ N is surjective if and only if the localization ϕm : Mm Ñ Nm
is surjective for each maximal ideal m.

A.9 Cayley–Hamilton Theorem

Theorem A.11 (Cayley-Hamilton). Let A be a ring, a Ă A an ideal and let M be a

finitely generated A-module. Let φ : M Ñ M be an A-module homomorphism with
φpM q Ă aM . Then there exists a monic polynomial
xn ` an´1 xn´1 ` ¨ ¨ ¨ ` a0 P Arxs
with a0 , . . . , an´1 P a such that
φn ` an´1 φn´1 ` ¨ ¨ ¨ ` a0 id “ 0 P HomA pM, M q,
Here φi denotes the composition of φ with itself i times.

Let T “ pbij q1ďi,jďn be an pn ˆ nq-matrix with entries in A, and let Tij be the pn ˆ nq-
matrix obtained by deleting row i and column j . The adjugate of T is the matrix adjpT q “
pcij q1ďi,jďn , where
cij “ p´1qi`j detpTji q.
The adjoint matrix has the property that T ¨ adjpT q “ adjpT q ¨ T “ detpT qIn , where In is
the pn ˆ nq identity matrix. This follows by the cofactor expansion of the determinant.
Proof Let m1 , . . . , mn be generators for M . By the assumption on ϕ, we have φpmi q P

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aM for all i “ 1, . . . , n. Therefore, for each i, we may express

n
ÿ
φpmi q “ bij mj
j“1

for some elements bij P a. Next, consider M as a module over the ring Arxs by setting
xi ¨ m :“ φi pxq for all m P M . If I denotes the n ˆ n identity matrix, and T “ pbij q, then
the above equation can be written as
¨ ˛
m1
˚ .. ‹
pxI ´ T q ¨ ˝ . ‚ “ 0
mn
If we multiply both sides by the adjoint matrix of pxI ´ T q, we obtain
¨ ˛
m1
˚ .. ‹
detpxI ´ T q ¨ I ¨ ˝ . ‚ “ 0
mn
Let P pxq be the monic polynomial detpxI ´ T q. Then P pxq anihilates all the mi , and hence
P pϕq acts as the zero endomorphism on M . By the Laplace expansion of the determinant,
we also see that the coefficient of xj of P belongs to aj .

A.10 Integral Extensions

Let A Ă B be rings, and let b P B . We say b is integral over A if it satisfies a monic
polynomial relation
bn ` an´1 bn´1 ` ¨ ¨ ¨ ` a1 b ` a0 “ 0, ai P A.

Theorem A.12. Let A Ă B , and let b P B . The following are equivalent:

(i) b is integral over A.
(ii) Arbs is a finitely generated A-module.
(iii) There is a ring C with Arbs Ă C Ă B such that C is a finitely generated
A-module.

Proof (i) ñ (ii): If b is integral, we can write bk “ ´pak´1 bk´1 ` ¨ ¨ ¨ ` a0 q, and therefore
we may express every higher power
bk`m “ pak´1 bk`m´1 ` ¨ ¨ ¨ ` a0 bm q.
in terms of bi for i ă k ` m. This gives that 1, b, . . . , bk´1 generate Arbs.
(ii) ñ (iii): Take C “ Arbs.
(iii) ñ (i): Applying the Cayley–Hamilton theorem to M “ C , a “ A and ϕ : M Ñ M
given by multiplication by b, we get that ϕ satisfies an integral relation with coefficients in A.
Applying this relation to 1 P C , we get that b is integral over A.

Corollary A.13. Any finite ring extension A Ă B is integral.

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Proof Taking C “ B in (iii) in the proposition above, we see that any element b P B is
integral over A.
For a ring extension A Ă B , we define integral closure of A in B as the subring
C “ tb P B | b integral over Au
If A is equal to its integral closure, we say it is integrally closed in B .
Using Theorem A.12 on can show that The integral closure C of A in B is itself integrally
closed in B .
An integral domain A is said to be normal if it is integrally closed in its fraction field
K “ kpAq. In other words, any element z P K which satisfies a monic equation with
coefficients in A, is already contained in A.
Example A.14. The ring Z is normal. More generally, every unique factorisation domain is
integrally closed. △

Proposition A.15. Let A be an integral domain. Then the following are equivalent:
(i) A is integrally closed.
(ii) Ap is integrally closed for all prime ideals p Ă A.
(iii) Am is integrally closed for all maximal ideals m Ă A.

Proposition A.16. Let A Ă B be rings, and let C Ă B be the integral closure of A in

B . Let S Ă A be a multiplicatively closed subset. Then S ´1 C Ă S ´1 B is the integral
closure of S ´1 A in S ´1 B .

Proposition A.17. Let A Ă B be an integral extension of integral domains. Then A is a

field if and only if B is a field.

There is a collection of results, the Cohen–Seidenberg Theorems, about prime ideals in

integral extensions. We collect the ones we need in the following theorem.

Theorem A.18. Let A Ă B be an integral extension of rings.

(i) (Lying-Over) If p prime ideal in A, there is prime ideal q in B so that
q X A “ p.
(ii) If q Ă q1 are prime ideals in B such that q X A “ q1 X A, then q “ q1 .
(iii) (Going–Up) If p Ă p1 are two prime ideals in A and q P Spec B with
q X A “ p, there is a q1 P Spec B with q1 X A “ q.
(iv) (Going–Down) Assume that A is integrally closed and that p1 Ă p are
two prime ideals. If q P Spec B is such that q X A “ p, then there is a
q1 P Spec B such that q1 X A “ p1 .

Proposition A.19. Let A Ă B be an integral ring extension. Then dim A “ dim B .

Proof Let p0 Ĺ p1 Ĺ ¨ ¨ ¨ Ĺ ps be a chain of prime ideals in A. By the Lying-Over

Theorem, there is a prime ideal q0 Ă B such that q0 X A “ p0 . Now B{q0 is integral over

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A{p0 and p1 {p0 is a prime ideal of A{p0 . Applying the Going-Up Theorem again, we see
that there is a prime ideal q1 in B{q0 such that q1 X pA{p0 q “ p1 {p0 . This ideal must have
the form q1 “ q1 {q0 for some prime ideal q1 Ą q0 in B . The inclusion q1 Ą q0 must be
strict, because otherwise we would have p1 “ p0 . Furthermore, q1 X A “ p1 . Proceeding in
this manner, we can produce a chain of prime ideals in B of the same length l, and hence
dim A ď dim B .
For the reverse inequality, suppose that q0 Ĺ q1 Ĺ ¨ ¨ ¨ Ĺ qs is a chain of prime ideals
in B . Define pi “ qi X A. Then p0 Ď ¨ ¨ ¨ Ď ps is a chain of prime ideals in A. We claim
that each inclusion pi Ă pi`1 is strict. Given this, we can conclude that dim A ď dim B ,
complete the proof. Suppose for a contradiction that pi “ pi`1 . It suffices to show that
qi Ě qi`1 . Let y P qi`1 . Since B{qi is integral over A{pi , we have an integral relation
y n ` an´1 y n´1 ` ¨ ¨ ¨ ` a1 y ` a0 “ 0, and we may assume that n is minimal with this
property. Note that a0 P A{pi “ A{pi`1 and a0 P qi`1 {qi , and so a0 “ 0. But this means
that ypy n´1 ` ¨ ¨ ¨ ` y 1 q “ 0. As B{qi is an integral domain, and y n´1 ` ¨ ¨ ¨ ` a1 is non-zero
(by the minimality of n), we must have y “ 0, and hence y P qi .

The following is a non-trivial result from commutative algebra about the integral closure:

Theorem A.20 (Finite generation of integral closure). Let A be an integral domain,

K “ KpAq its fraction field, and let K Ă L be a finite separable field extension. Let B
be the integral closure of A in L (that is, the elements of L which are integral over A).
Then
(i) If A is integrally closed, then B is a finitely generated A-module
(ii) If A is finitely generated as a k -algebra, then B is a finitely generated
A-module.

The second part does not hold in general: there are non-noetherian rings where the integral
closure is not finitely generated.

A.11 Noetherian rings

A ring A is called Noetherian if every ascending chain of ideals I1 Ă I2 Ă ¨ ¨ ¨ stabilizes,
meaning that there exists n P N such that In “ In`1 “ ¨ ¨ ¨ . Equivalently, A is Noetherian if
every ideal of A is finitely generated.

Theorem A.21 (Hilbert’s Basis Theorem). If A is a Noetherian ring, then the polyno-
mial ring Arx1 , . . . , xn s is also Noetherian.

The ring A is called Artinian if it satisfies the descending chain condition on ideals, which
means that every descending chain of ideals I1 Ą I2 Ą ¨ ¨ ¨ eventually stabilizes, with there
existing some n P N such that In “ In`1 “ ¨ ¨ ¨ .
If A is an Artinian ring, then A is also Noetherian and there are only finitely many prime
ś Moreover, A is the product of local Artinian
ideals. In particular, its has Krull dimension zero.
rings. More precisely, the natural map A » p Ap is an isomorphism.

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Theorem A.22 (Krull’s IntersectionŞTheorem). Let A be a Noetherian local integral

domain with maximal ideal m. Then jě1 mj “ p0q.
Ş
Proof 1 Write J for the ideal jě1 mj , and let a P J . We claim that a P mJ , so that
J “ mJ . Given this, Nakayama’s Lemma implies that J “ 0.
Choose generators x1 , . . . , xn for the maximal ideal m. Then the ideal md consists of
elements of the form P px1 , . . . , xn q where P P ArX1 , . . . , Xn s is a homogeneous poly-
nomial of degree d. As a P md for every d, we may choose homogeneous polynomials
P1 , P2 , ¨ ¨ ¨ P ArX1 , . . . , Xn s so that a “ Pd px1 , . . . , xn q for every d. Consider the ideal
pP1 , P2 , . . . q generated by all the Pi . As ArX1 , . . . , Xn s is Noetherian, this ideal is finitely
generated, say by homogeneous elements P1 , . . . , Pr of degrees d1 , . . . , dr . In particular, we
may express
Pr`1 “ c1 P1 ` ¨ ¨ ¨ ` cr Pr
for some ci P ArX1 , . . . , Xn s. In this relation, we may choose the ci to be homogeneous of
degree deg Pr`1 ´ di “ d ` 1 ´ di ą 0. This means that ci px1 , . . . , xn q P m and hence
ÿ č
a “ Pr`1 px1 , . . . , xr q “ ci px1 , . . . , xn qPi px1 , . . . , xn q P m ¨ md .
i dě1

In particular, this implies that in a Noetherian ring no nonzero element a P A can lie in all
powers of all maximal ideals.

A.12 Primary decomposition

An ideal q is called primary if xy P q implies that either x P a or y n P a for some n P N. If
?
q is a primary ideal, then the radical p “ q is a prime ideal; it is the smallest prime ideal
containing q. We say that q is p-primary. If q, q1 are p-primary, then the intersection q X q1 is
also p-primary.
If p is a prime ideal, then any power pl is primary. q “ px2 , yq is a primary ideal which is
not the power of any prime ideal.

Theorem A.23 (Lasker–Noether). In a Noetherian ring A, any ideal a can be expressed

as a finite intersection of primary ideals:
a “ q1 X ¨ ¨ ¨ X ar (A.9)
?
The set of prime ideals pi “ qi are uniquely determined by a.
?
A decomposition (A.9) is called
Ş minimal if the prime ideals qi are different, and the
decomposition is minimal, i.e., i‰j qi Ć qj for each j .

Example A.24. Consider the ideal a “ px2 , xz, xy, yzq of the polynomial ring A “
1 This proof follows ”Hervé Perdry. An elementary proof of Krull’s intersection theorem”

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krx, y, zs. Then we have two different minimal primary decompositions

a “ px, yq X px, zq X px, y, zq2 “ px, yq X px, zq X px2 , y, zq.
Note by the way that a “ px, yq ¨ px, zq. △
If q is a p-primary ideal, then we have
‚ pq : xq “ A if x P q
‚ pq : xq “ q if x R p
‚ pq : xq is p-primary if x P q.
Furthermore, there is an element x P A such that p “ pq : xq. Using this observation, one
shows that the set of prime ideals pi appearing in the decomposition of a is exactly the prime
ideals of the form pq : xq where x P A (in particular, they are uniquely determined by q).
These prime ideals are called the associated primes of a, and the set of these is sometimes
denoted by Asspaq.
The minimal prime ideals pi associated to a (minimal with respect to inclusion) play an
important role: V ppi q give the irreducible components of the closed set V ppq. The non-
minimal associated prime ideals are sometimes called the embedded components of a.
a the associated prime ideals are p1 “ px, yq and
For instance, in the example above,
p2 “ px, zq. The prime ideal p3 “ px2 , y, zq “ px, y, zq is an embedded prime.
We will need two results about primary decompositions in localizations.

Proposition A.25. Let A be a Noetherian ring and let S Ă A be a multiplicative subset.

Let a Ă A be an ideal, and a “ q1 X ¨ ¨ ¨ X qr be a mimimal primary decomposition
?
of a with associated prime ideals pi “ qi . Assume that the qi are ordered such that
pi X S “ H for every i “ 1, . . . , s and pi X S ‰ H for i “ s ` 1, . . . , r. Then
i “ 1, . . . , s, the ideal S ´1 qi is S ´1 qi -primary
(i) For each Ş
´1 s
(ii) S a “ i“1Ş S ´1 qi
s
(iii) S ´1 a X A “ i“1 S ´1 qi .
Moreover, the decompositions in (ii) and (iii) are minimal.

Here S ´1 a X A denotes the preimage of S ´1 a via the localization map A Ñ S ´1 A.

Proof This is Proposition 4.9 in Atiyah (2018).

Proposition
Şr A.26. Let a Ă A be an ideal with a minimal primary decomposition
a “ i“1 qi , where each qi is pi -primary. Then, for each index i such that pi is a
minimal prime ideal, we have
qi “ aApi X A.
In this case, we call qi the pi -primary component of a.

In particular, the primary components qi corresponding to minimal primes pi are unique

and depend only on a.
Given a Noetherian ring R, we can use the uniqueness result mentioned above to define the
n-th symbolic power of any prime ideal p Ă R for n ě 1. Consider a primary decomposition

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of pn , and note that p is the only minimal prime ideal of pn . The p-primary component of pn
is unique and given by
ppnq :“ pn Rp X R.

This is called the n-th symbolic power of p. Clearly, pn Ă ppnq and equality pn “ ppnq holds
if and only if pn is primary.

A.13 Dimension theory

The Krull dimension of a ring A, denoted by dimpAq, is the supremum of the lengths n of
chains
p0 Ĺ p1 Ĺ ¨ ¨ ¨ Ĺ pn

of distinct prime ideals pi in A. The height of a prime ideal is the supremum of the lengths
of chains of prime ideals contained in p. By the correspondence between primes in Ap and
primes contained in p, we have htp “ dim Ap .

Example A.27. The Krull dimension of a field is zero. The Krull dimension of Z equals 1. △

Example A.28. If p is a prime ideal in a finitely generated k -algebra A, then

dimpAq “ dimpA{pq ` htppq.

△

Theorem A.29 (Krull’s Principal Ideal Theorem). Let A be a Noetherian ring and
I “ pf1 , . . . , fr q a proper ideal of A. Then each prime ideal that is minimal among those
that contain I has height at most r.

Corollary A.30. Let A be a Noetherian local ring with maximal ideal m and let f P m.
Then
dimpA{pf qq ě dim A ´ 1
with equality if f is not contained in any minimal prime ideal of A.

Proposition A.31. If A is a Noetherian ring, then the Krull dimension of Arxs equals
dim A ` 1.

In particular, dim krx1 , . . . , xn s “ n and dim Zrx1 , . . . , xn s “ n ` 1.

Proof Here is a proof in the case A “ k is a field. The proof goes by induction on n. The
case n “ 1 is clear.
Consider an irreducible nonconstant polynomial f in A “ krt1 , . . . , tn s. As in Lemma A.35,
let ui “ ti ´ ts1 Then

f pt1 , . . . , tn q “ f pt1 , u2 ` ts1 , . . . , un ` ts1 q

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is a monic polynomial in t1 with coefficients in B “ kru2 , . . . , un s Ă A, and the ui ’s are

algebraically independent. By induction dim B “ n ´ 1.
Consider now the algebra A{pf qA. The algebra B maps injectively into A{pf qA: a
polynomial in the kernel depends only on the ui ’s, but it also is a multiple of f (which
depends on t1 ), hence it must vanish. The extension

B Ă A{pf qA

is integral because A{pf qA is generated over B by the class of t1 , which is integral since f
is monic. The Going-Up Theorem then shows that dim A{pf qA “ n ´ 1.

Example A.32. If K{k and L{k are field extensions, then

dimpK bk Lq “ minptrdegk pKq, trdegk pLqq.

A.14 Noether’s Normalization Lemma

We now turn to one of the key results in the theory of varieties, Noether’s Normalization
Lemma, due to Emmy Noether.

Theorem A.33 (Noether’s Normalization Lemma). Let k be a field and let B be an

integral domain which is finitely generated as a k -algebra and let n be the transcendence
degree of the fraction field KpBq over k . Then there are algebraically independent
elements x1 , . . . , xn P B such that B is finite as a module over krx1 , . . . , xn s.

Noether’s Normalization Lemma is a powerful tool, which allows us to reduce many ques-
tions about k -algebras to statements about the polynomial ring krx1 , . . . , xn s. It has many
fundamental results as consequences; in particular, we will obtain Hilbert’s Nullstellensatz as
a corollary.
With only minor modifications, the standard proof of the classical version of the Normal-
ization Lemma yields a more general result, which will be useful for our purposes.

Theorem A.34. Let A Ă B be two integral domains with B of finite type over A, and
let n be the transcendence degree of the quotient field KpBq over KpAq. Then there are
elements x1 , . . . , xn in B which are algebraically independent over A and an element
f P A such that Bf is a finite module over Af rx1 , . . . , xn s.

When A is a field, the localization has no effect, and Theorem A.33 follows directly from
Theorem A.34.
We prove the theorem by induction on the minimal number of generators of A. The key to
the inductive step is the following lemma:

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Lemma A.35. Let A be a ring and let P P Art1 , . . . , tn s. Then if s is a sufficiently large
integer, we have
n´1
P pt1 ` tsn , . . . , tn´1 ` tsn , tn q “ g ¨ tdn ` pterms of order ă d in tn q
where g P A is nonzero.
i
Proof If we replace ti with ti ` tsn for i “ 1, . . . , n ´ 1 in a monomial te11 . . . tenn , we
get a polynomial in tn of the form tN n `(lower degree terms), where N “ en ` e1 s `
2 n´1
e2 s ` ¨ ¨ ¨ ` en´1 s . The key observation is that for s sufficiently large, the expressions
en ` e1 s ` e2 s2 ` ¨ ¨ ¨ ` en´1 sn´1 will all be different. Indeed, for two n-tuples pe1 , . . . , en q,
pe11 , . . . , e1n q, an equality
e1 ` e2 s ` ¨ ¨ ¨ ` em sm´1 “ e11 ` e12 s ` ¨ ¨ ¨ ` e1m sm´1
holds only for finitely many integers s because non-zero polynomials have only finitely many
zeros. Since f has only finitely monomials, we can simply choose s so that no such equalities
take place.
2 n´1
This means that if we expand P pt1 ` tsn , t2 ` tsn , . . . , tn´1 ` tsn q into powers of tn ,
there cannot be any cancellation between the leading terms, and so the leading term is of the
desired form gtN n , where g is a non-zero element of A.

Proof of Theorem A.34 Choose generators w1 , . . . , wm for B as an algebra over A. Then

KpBq is generated as a field over KpAq by the wi ’s as well. It follows that m ě n “
tr.degKpAq pKpBqq. In the case m “ n, the w1 , . . . , wm are algebraically independent over
A, and B “ Arw1 , . . . , wm s is a polynomial ring, so the theorem holds in this case.
If m ą n, the elements are not algebraically independent, so there is a non-zero polynomial
P pt1 , . . . , tm q P Art1 , . . . , tm s such that P pw1 , . . . , wm q “ 0. Pick an s P N as in Lemma
A.35. Then the polynomial
2 m´1
Qpt1 , t2 , . . . , tm q “ P pt1 ` tsm , t2 ` tsm , . . . , tm´1 ` tsn , tm q
i´1
s
has a leading coefficient g P A as a polynomial in tm . If we define zi “ wi ´ wm for
´1
i “ 1, . . . , m ´ 1, then Qpz1 , z2 , . . . , zm´1 , wm q “ 0. The polynomial g Q has a leading
term equal to 1 in tm , so the localization Bg is an integral extension of the subalgebra
B 1 “ Ag rz1 , . . . , zm´1 s.
Now, KpBq “ KpBg q is an algebraic field extension of KpB 1 q, so the two fields have
the same transcendence degree over KpAq “ KpAg q. Moreover, by construction, B 1 is gen-
erated by fewer than m elements over Ag . By induction, there is an h P Ag and algebraically
independent elements x1 , . . . , xn P B 1 such that Bh1 is finite over Agh rx1 , . . . , xn s. Now, g
is invertible in B 1 , so Bh1 “ Bgh , and taking f “ gh, we are done.
Example A.36. The algebra krx, x´1 s is not finite over krxs, as it requires all the negative
powers x´n as generators. However, for any elements a, b of k with ab ‰ 0, the ring
krx, x´1 s is finite over krax`bx´1 s. Indeed, krx, x´1 s is generated by x over krax`bx´1 s,
and x satisfies the monic equation
T 2 ´ a´1 pax ` bx´1 qT ` ba´1 “ 0.

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Corollary A.37. Let A be a finitely generated algebra over a field k and let m be a
maximal ideal in A. Then A{m is a finite field extension of k .

Proof The field K “ A{m is finitely generated as a k -algebra because A is. By Noether’s
Normalization Lemma, K is an integral extension over some polynomial ring kry1 , . . . , yr s.
If r ě 1, then K has dimension ě 1, which is not possible. Therefore, r “ 0, and K is a
finite field extension of k .

A.15 Prime ideals in k-algebras

Lemma A.38. Let A be an integral domain which is finitely generated as a k -algebra. If

p Ă A is a prime ideal of height one, then A{p has transcendence degree n ´ 1 over k .

Proof Assume first that A “ krx1 , . . . , xn s. Then p is a principal ideal, say p “ pf q, for
some polynomial f P krx1 , . . . , xn s. After a change of coordinates, we may assume that f
is of the form
f “ ad px1 , . . . , xn´1 qxdn ` ¨ ¨ ¨ ` a0 px1 , . . . , xn´1 q
Consider the polynomial ring B “ krx1 , . . . , xn´1 s Ă A. We have B X pf q “ p0q. There-
fore, the ring map B Ñ A{p is injective, and the classes of x1 , . . . , xn´1 are algebraically
independent in A{p. On the other hand, by the relation f , the element x1 is algebraic over
the fraction field KpBq “ kpx1 , . . . , xn´1 q. Hence tr.degk pA{pq “ n ´ 1.
For the general case, let B “ krx1 , . . . , xn s Ă A be a Noether normalization of A.
Applying the Going-Down theorem, we see that the prime q “ p X B has height 1 in B . By
the previous paragraph, B{q has transcendence degree n ´ 1, and so

A.16 Graded rings

In this book, a graded ring will refer to a ring R which is graded by the non-negative integers,
i.e. a ring which admits a decomposition
à
R“ Rn “ R0 ‘ R1 ‘ ¨ ¨ ¨
ně0

as an abelian group, such that Rm ¨ Rn Ă Rm`n for each m, n ě 0. Occasionally, we will

also discuss Z-graded rings, where we allow negative degrees as well.
A ring map ϕ : R Ñ S between two graded rings R and S is said to be a map of graded
rings if it respects the grading, that is, if ϕpRn q Ă Sn for all n.
Note that R0 is a subring of R and that R is an algebra over R0 . Moreover, each Rn is
an R0 -module. The elements in Rn are said to be homogeneous of degree n, and one writes
deg x “ n when x P Rn . (By convention, 0 is considered homogeneous of any ř degree.)
Every non-zero element x P R can be expressed uniquely as a finite sum x “ n xn with
xn P Rn , and the non-zero terms in the sum are called the homogeneous components of x.

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Example A.39. The simplest examples of graded rings are the polynomial rings R “
Art0 , . . . , tr s, with the standard grading where each variable ti has degree 1 and the elements
from A have degree 0. Each graded piece Rn is a free module over R0 “ A with the
monomials of degree n serving as a basis. △
À
An R-module M is graded if it has a similar decomposition M “ nPZ Mn as an
abelian group and Rm Mn Ă Mm`n for all n and m. Note that we allow also elements of
negative degrees. A map of graded R-modules is an R-linear map ϕ : M Ñ N satisfying
ϕpMn q Ă Nn for all n P Z. With this notion of morphisms, the graded R-modules form a
category, denoted GrModR .
As for graded rings, a non-zero element x P M is homogeneous of degree nřif it lies in
Mn . Any element x P M may be expressed in a unique way as a finite sum x “ n xn with
each xn in Mn , and the non-zero terms are called the homogeneous components of x.
Most of the familiar definitions for modules carryÀ over to the graded setting. For instance,
the direct sum of a collection ofÀ graded modules i Mi is graded in a natural way such
that canoncal inclusions Mj ãÑ i Mi preserve the grading. Likewise, the kernel and the
cokernel of a map of graded modules are also graded in a natural way.
A sequence of graded modules

0 M1 M M2 0,

is exact if it is exact as sequence of ordinary modules. As maps of graded modules preserve

the grading, this is equivalent to saying that each of the sequences

0 Mn1 Mn Mn2 0.

are exact (as a sequence of R0 -modules).

An ideal a Ă R is called homogeneous if the homogeneous
À components of each element
in a belongs to a. In other words, we may write a “ n an with an “ a X Rn . An ideal a
is homogeneous if and only if it is generated by homogeneous elements (see Exercise 5.4.4).
It is readily verified that radicals, intersections, sums and products of homogeneous ideals
À If a is an homogeneous ideal, the quotient R{a inherits a grading from R
are homogeneous.
and R{a “ n Rn {an .
A homogenous ideal p is prime if and only if for each pair of homogeneous elements
f, g P R, f g P p implies f P p or g P p. À
We will write R` for the direct sum ną0 Rn . This is naturally a homogeneous ideal of
R, which we call the irrelevant ideal.
Example A.40. The irrelevant ideal of a polynomial ring R “ Art0 , . . . , tr s is equal to
R` “ pt0 , . . . , tr q. △
Example A.41 (Veronese subrings). À For a graded ring R and d P N, we define Rpdq to
be the subring of R defined by ně0 Rnd . As every element of Rpdq has degree equal to a
multiple of d, it is often useful to redefine the grading on Rpdq by dividing all degrees by d.
For instance, when R “ krx0 , x1 s, the Veronese subring Rp2q is given by krx20 , x0 x1 , x21 s.
We may adjust the grading by defining deg x0 “ deg x1 “ 12 , the elements of the ring Rp2q
will have integer degrees. A different viewpoint is to note that Rp2q is isomorphic to the

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graded ring kru0 , u1 , u2 s{pu21 ´ u0 u2 q, where in the latter ring, the variables have degree 1.
△
If S Ă R is a multiplicative system consisting of homogeneous elements, and M is a
graded module, the localization S ´1 M is naturally a graded R-module with degree n part
equal to
␣ (
pS ´1 M qn “ m{s P S ´1 M | m P M homogeneous, s P S and deg m ´ deg s “ n .
In particular, if f is a homogeneous element of positive degree, the localization Rf is
a Z-graded ring. As we will see, the degree 0 part pRf q0 will play a crucial role in the
Proj-construction.
Example A.42. In the polynomial ring R “ Art0 , . . . , tn s, with the standard grading, the
elements of degree zero in the localization Rtj are polynomials in the ratios t0 {tj , . . . , tn {tj ,
so the piece of degree zero pRtj q0 is the polynomial ring
„ ȷ
t0 tn
pRtj q0 “ A ,..., .
tj tj
△

A.17 Regular local rings

If A is a local Noetherian ring with maximal ideal m, the cotangent space m{m2 is naturally
a vector space over the residue field A{m. This is finite dimensional, because m is finitely
generated. If x1 , . . . , xr are elements from A that generate m{m2 as a A{m-vector space,
then Nakayama’s lemma implies that the x1 , . . . , xr generate m. But then Krull’s Principal
ideal theorem implies dim A “ ht m ď r “ dimA{m m{m2 . This gives:

Proposition A.43. Let A be a Noetherian local ring with maximal ideal m. Then
dim A ď dimA{m m{m2 ă 8.

We say that the ring A is a regular if equality holds, that is, if m can be generated by n
elements, where n “ dim A is the Krull dimension. A general Noetherian ring A is said to
be regular if every localization Ap is a regular local ring.

A.18 Discrete valuation rings

Let K be a field. A discrete valuation is a surjective function v : K ´ 0 Ñ Z Y t8u
satisfying, for each f, g P K :
(i) νpf gq “ νpf q ` νpgq.
(ii) νpf ` gq ě mintνpf q, νpgqu, with equality in the latter when νpf q ‰ νpgq.
(iii) νpf q “ 0 if and only if f “ 0.
An integral domain A with fraction field K is called a discrete valuation ring if there exists a
valuation ν : K Ñ Z Y t8u such that
A “ t x P K | νpxq ě 0 u

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Given a valuation ν , the set t x P K | νpxq ě 0 u is a subring of K . Therefore the discrete

valuation ring determines and is determined by ν .

Example A.44. Let K “ kpxq be the field of rational functions in one variable. Then
any element f P K can be written as f “ xn gpxq{hpxq where n P Z; and gpxq, hpxq
are polynomials such that f p0q, gp0q ‰ 0. The ‘order of vanishing at 0’ gives a valuation
va : K ˆ Ñ Z by setting vpf q “ n. In this case, the elements of non-negative valuation are
exactly the ones of the form gpxq{hpxq, where hp0q ‰ 0. Therefore, the valuation ring is the
localization of krxs at pxq:
A “ krxspxq .
△
Example A.45. Let K “ kpxq be the field of rational functions in one variable. Define the
valuation v8 : K ˆ Ñ Z by setting
ˆ ˙
f
v8 “ deg g ´ deg f
g
One can check that this defines a valuation on kpxq. The valuation v8 is supposed to measure
the order of a pole ‘at infinity’. The corresponding valuation ring is

A “ tf {g P kpxq| deg f ď deg gu.

with maximal ideal m “ tf {g P kpxq| deg f ă deg gu. △
Example A.46. Let K “ Q be the field of rational numbers, and let p be a prime number.
Any y P Q can be expressed as y “ pn a{b where n P Z and a, b are coprime to p. We can
define the p-adic valuation vp : Qˆ Ñ Z by setting vpyq “ n. In this case, the valuation
ring is the localization of Z at ppq:
!m )
A “ Zppq “ P Q | gcdpp, nq “ 1
n
△

Proposition A.47. Let A be a discrete valuation ring with fraction field K . Then
(i) A is a local ring with maximal ideal
m “ t f P K | νpf q ě 1 u
and group of units
Aˆ “ t f P K | νpf q “ 0 u.
(ii) If t P A is an element with νptq “ 1, then m “ ptq.
(iii) If t is any element with νptq “ 1, then every non-zero ideal of A is of the
form ptn q for some n ě 1.
(iv) A is Noetherian and of Krull dimension 1.
(v) A is normal.

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Proof (i): Let f P A be a nonzero elemement. Then f is a unit if and only if the inverse
1{f P K belongs to A, which is if and only if νp1{f q “ ´νpf q ě 0. But as f P A,
νpf q ě 0, and so νpf q “ 0. Therefore Aˆ “ t f P K | νpf q “ 0 u and the non-units is
given by m “ t f P K | νpf q ě 1 u.
(ii) In general, if f, g P A and vpgq ě vpf q, then vpf {gq ě 0, so f {g P A, which means
that g P pf q. In particular, m “ ptq for any element t P A with vptq “ 1. (The discrete
valuation ν is assumed to be surjective, so there exists at least one such element t.)
(iii) Let a be an ideal, and let f P a be such that vpf q is minimal. Then for any g P a, we
have vpgq ě vpf q, so as above we find g P pf q. Consequently, a Ă pf q and hence a “ pf q.
(iv) By the previous two points, we have that the ideals of A are p1q, pxq, px2 q, . . . , and
p0q. It is easy to see that pxq and p0q are the only prime ideals of A, so A is Noetherian of
Krull dimension 1.
(v) Let x P K , and assume that x is integral over A. We must show that x P A. Since x is
integral over A, we can find a relation

xn ` an´1 xn´1 ` ¨ ¨ ¨ ` a0 “ 0, ai P A
so
xn “ ´an´1 xn´1 ´ ¨ ¨ ¨ ´ a0 .
If vpxq “ d, we then get

vpxn q “ nvpxq “ nd “ vp´an´1 xn´1 ´ ¨ ¨ ¨ ´ a0 q ě minpvp´ai xi qq.

Hence there exists an i ď n ´ 1 such that

nd “ vpxd q ě vp´ai xi q “ vp´ai q ` vpxi q ě id.

This gives pn ´ iqd ě 0, and so d ě 0. Hence, x P A.

An element t P A with νptq “ 1 is called a uniformizing parameter for A. By the above

proof, any element t P m ´ m2 is a uniformizing parameter. In fact, we can recover the
valuation from the maximal ideal m: any non-zero element in the fraction field K may be
written as utn with u a unit in A and n an integer. Indeed, if f P A and f ‰ 0, we let νpf q
be the unique non-negative integer n such that f P mn ´ mn`1 , then f “ utvpf q for a unit u.
For a general non-zero element f {g in K , one has f {g “ utνpf q´νpgq with u a unit.

Proposition A.48. Let A be a Noetherian local domain of dimension 1 with maximal

ideal m. Then the following are equivalent:
(i) A is a discrete valuation ring.
(ii) The maximal ideal m is principal.
(iii) Every non-zero ideal is a power of m.
(iv) A is integrally closed.
(v) A is regular, i.e., dimk m{m2 “ 1.

Proof (i) ñ (ii), (iii) and (iv) by Proposition A.47.

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(ii) ñ (i). We will need the fact that

8
č
mn “ p0q (A.10)
n“1

Ş8
Let t be a generator for m. Suppose a P n“1 ptn q. This means that for every n P N, there is
an an P A such that a “ an tn . As A is an integral domain, we have an`1 “ an t for every
n ě 1 and hence there is a chain of ideals pa1 q Ą pa2 q Ą pa3 q Ą ¨ ¨ ¨ . As A is Noetherian,
this chain stabilizes, meaning that there is an N such that paN q “ paN `1 q “ ¨ ¨ ¨ , or in other
words, an`1 “ un an for n ě N and un is a unit. Then a “ an`1 tn`1 “ an tn , implies
pun t ´ 1qan tn “ 0 for n ě N . But un t ´ 1 is clearly a unit, as t P m. Therefore an “ 0
for n ě N , and hence a “ 0.
From this it follows that any f P A can be written as f “ utn , where n ě 0 and u P Aˆ
is a unit. Explicitly, n is the unique integer n such that f P mn ´ mn`1 . (Note that mn ´ mn`1
nonempty for every n by (A.10).) If we extend this via νpf {gq “ νpf q´νpgq for f {g P K ˆ
and νp0q “ 8, then A is exactly the discrete valuation associated to ν .
(ii) ñ (iii). Let t be a generator for the maximal ideal m and let a Ă A be a non-zero
ideal. Let n be the largest integer such that a Ă mn . (Such an n exists beccause a cannot
be contained in all powers of m by (A.10).) Since a Ę mn`1 , there is an a P a such that
a “ btn with b R m; that is, b is a unit since the ring is local. It follows that ptn q Ă a, and
hence a “ mn .
(iv) ñ (ii). Assume that A is integrally closed and let x P m be any non-zero element.
Since A is Noetherian and of dimension 1, m is the only non-zero prime ideal in A, and so m
is associated to pxq.
There is an integer n such that mn Ă pxq. If we choose n to be minimal among these, so
that mn´1 Ć pxq, and mn ´ pxq is non-empty. Pick y P mn´1 ´ pxq. Set z “ x{y P K .
Then we claim that z P A and m “ ptq.

z ´1 m Ă x´1 m Ă x´1 mn Ă x´1 pxq Ă A

Therefore z ´1 A is an ideal of A. There are two cases:

Case 1: z ´1 P m. Then consider the endomorphism ϕ : A Ñ A given by ϕpaq “ z ´1 a,
takes A into m. By the Cayley–Hamilton theorem, this would mean that z ´1 is integral over
A. By assumption, A is normal, so z ´1 P A. Hence z ´1 “ yx´1 P A and hence y P A. But
this contradicts the definition of y .
Case 2: z ´1 R m. Then z ´1 A “ A is the unit ideal. But then m “ pzq, and m is principal.
(ii) ô (v). If m “ pxq, then m{m2 is generated by the class of x modulo m2 . We also have
m ‰ m2 (since A has dimension 1), so dimk m{m2 “ 1. The converse implication follows
by Nakayama’s lemma.

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A.19 Normal domains

Theorem A.49 (Serre). Let A be Noetherian local ring. Then A is normal if and only if
(i) Ap is regular for every prime ideal of height at most 1.
(ii) A is reduced, and for every nonzerodivisor f P A, the associated primes of
A{pf q are of height 1 in A.

Theorem A.50 (Krull). Let A be an integral domain, then A is normal if and only if
(i) Ap isŞa discrete valuation ring for every non-zero minimal prime ideal p
(ii) A “ p Ap , where the intersection is taken over all minimal prime ideals
of A.

Let us recall that an ideal quotient pb : aq “ t x | xa P pbq u equals the annihilator of the
class of a in A{pbqA. A basic result from the theory of primary decomposition in Noetherian
rings asserts that each proper annihilator is contained in a maximal annihilator. In this case,
these are precisely the prime ideals associated with pbq.
Ş
Lemma A.51. A Noetherian domain A equals the intersection p Ap , where p runs
through the prime ideals associated with principal ideals.

Proof ŞAssume for contradiction that there is an element ab´1 in the fraction field of A that
lies in p Ap but not in A. The ideal quotient pb : aq “ t x P A | xa P pbq u is a proper
ideal because ab´1 R A, and so pb : aq is contained in a maximal ideal, that is, a prime p
associated with pbq. Since ab´1 P Ap , we can write ab´1 “ cd´1 with c, d P A and d R p.
Hence ad “ bc, implying that d P pb : aq Ă p, which is a contradiction.

Theorem A.52. Let A be a Noetherian domain. Then A is normal if and only if the
following conditions are met:
(i) The local ring Ap at each height 1 prime ideal p is a DVR.
(ii) Each principal ideal has no embedded components, that is, all associated
primes have height 1.

Proof Assume first that the two conditions are fulfilled. By Lemma A.51, A is the inter-
section of the local rings Ap where p runs over the prime ideals associated to a principal
ideal. Each such p has height 1 by Krull’s Principal Ideal Theorem, and by assumption (i) the
localization Ap is a DVR, hence normal. Intersections of normal rings are normal, so A is
normal.
Conversely, assume that A is normal. Then the localizations Ap at height 1 primes are
also normal, and being 1-dimensional, they are discrete valuation rings XXX. For item (ii):
let p be a prime in A associated with a principal ideal pbq. In the local ring Ap , its maximal
ideal m “ pAp remains associated with the principal ideal pbqAp . By Proposition A.48, m is
principal. Hence, Ap is a DVR, so p is of height 1 and cannot be an embedded prime.
When the first condition of the criterion is met, the p-primary ideals of height 1 are well
understood. They are all symbolic powers ppνq “ A X pν Ap . Indeed, if q is p-primary, then

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q “ A X qAp , and since Ap is a DVR, every ideal in Ap , including qAp is a power of the
maximal ideal pAp .
If A is normal and f P A is a nonzero element, then the primary decomposition of pf q
has the form
pν q
pf q “ p1 1 X ¨ ¨ ¨ X ppν
r
rq
.
where the pi are height 1 primes. The primes pi and the exponents νi completely determine
pf q, that is, they determine f up to an invertible factor.

Theorem A.53 (Algebraic Hartogs’ theorem). Let A be a Noetherian normal integral

domain. Then
č
A“ Ap . (A.11)
htp“1

Proof If g P A is a nonzero element, then each prime ideal p associated to pgq is of height
1. This is because pAp is associated to pgqAp , and Ap is a discrete valuation ring, hence
htppq “ dim Ap “ 1.
Now assume for a contradiction that there is an element f {g P K which lies in every Ap
for p of height 1, but not in A. In other words, f P pgqAp for every height 1 prime p.
?
Let pgq “ q1 X ¨ ¨ ¨ X qr be a primary decomposition of pgq, and let pi “ qi be the
assocated prime ideals. By the above paragraph, the pi have height 1, we get f P pi for every
i. By Proposition A.26, we get
r
č r
č
fP ppgqApi X Aq “ qi “ pgq
i“1 i“1

In other words, there is an element a P A such that f “ a ¨ g , or in other words f {g “ a P A.

Theorem A.54 (Generic freeness). Let A be a Noetherian integral domain and let B be
a finitely generated A-algebra.

Proof See (?, p.185) or (?, Tag 051S).

A.20 Unique factorization domains

In a Noetherian ring A, any nonzero nonunit x P A admits a factorization

x “ x1 ¨ ¨ ¨ xn (A.12)

where each xi is irreducible (that is, a non-zero element that is not a unit and is not the
product of two non-units). To see this, suppose for a contradiction that no such factorization
exists. Set a1 “ x. We must be able to write x as x “ yz where neither y nor z is irreducible,
nor a unit. Without loss of generality, we may assume that y cannot be written as a product
of irreducibles. Set a2 “ y . Writing y in the same way, and so on, we arrive at a sequence
a1 , a2 , a3 , . . . of elements of A such that ai`1 divides ai for each i ě 1 and ai`1 {ai is not

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a unit. This means that the chain pa1 q Ă pa2 q Ă pa3 q Ă . . . is an infinite strictly increasing
sequence of ideals, contradicting the fact that A is Noetherian.
We say that A is a unique factorization domain (UFD) if the factorization in (A.12) is
unique up to order and multiplication by units.

Example A.55. Examples of UFDs include Z, Zris, fields, and principal ideal domains. If A
is a UFD, then so is Arxs. △

Proposition A.56 (Gauss’ Lemma). Let A be a UFD with fraction field K . If f P Arxs
is a polynomial which factors into a product of non-constant polynomials in Krxs, then
it does so also in Arxs.

Proposition A.57. Let A be an integral domain and let f P A be a nonzero element

which is not a unit. Then if pf q is prime, then f is irredcucible. If A is a UFD, then the
converse also holds.

Proof If pf q is prime and f “ gh is a factorization, then g P pf q or h P pf q. If e.g.,

g “ a ¨ f , then f “ paf qh, implying that h is a unit. Hence f is irreducible.
Suppose that f is irreducible and that xy P pf q. Then we can write xy “ af for some
a P A. Write x “ x1 ¨ xs , y “ y1 ¨ ¨ ¨ ys where the xi and yj are irreducible. As A is a UFD
and f is irreducible, we see that f has to be one of the xi or the xj , that is, f divides either x
or y .

Proposition A.58. Let A be a Noetherian integral domain. Then A is a UFD if and only
if every height 1 prime ideal is principal.

Proof Suppose first that A is a UFD and let p be a height 1 prime ideal. Take x P p non-zero
and let x “ x1 ¨ ¨ ¨ xn be a factorization into irreducible elements. Since p is prime, we must
have, say, x1 P p. Then p Ą px1 q Ą p0q. A is UFD, px1 q is prime, and hence p “ px1 q
because p has height 1.
Conversely, suppose that every height 1 prime is principal. By (A.12), any element can be
written as a product of irreducibles, so we need to show that the factorization is unique up to
order and multiplication. For this, it suffices to prove that an irreducible element is prime.
But this follows by Proposition A.57.

A.21 Modules over a PID

A very useful theorem for modules over a principal ideal domain R is the fact that any
matrix with entries in R can be put into so-called Smith normal form (see below). This is a
generalization of the diagonalization process from Linear Algebra; finding the normal form
involves only basic elementary row and column operations, and multiplying by units.

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Theorem A.59 (Smith Normal form). Let R be a PID and let A be an m ˆ n matrix
with entries from R. Then there exist invertible matrices P and Q, with dimensions
m ˆ m and n ˆ n respectively, such that
P AQ “ diagps1 , . . . , sv , 0, . . . , 0q
is a diagonal matrix where the non-zero entries satisfy si |si`1 for i “ 1, . . . , v ´ 1. The
elements si are unique up to multiplication by a unit of R and are called the invariant
factors of A.

If A is a matrix A and si P R are the elements in the Smith Normal Form of A, then we
have
Coker A » R{ps1 q ‘ ¨ ¨ ¨ ‘ R{psv q ‘ Rm´v .
Over a PID, any finitely generated module M has a presentation 0 Ñ E Ñ F Ñ M Ñ 0,
where E and F are free modules of finite rank.

Corollary A.60 (Modules over a PID). Let R be a principal ideal domain and let M be
a finitely generated R-module. Then there are elements d1 , . . . , dm P R with di |di`1 for
each i, such that
M » Rr ‘ R{pd1 q ‘ ¨ ¨ ¨ ‘ R{pdm q.

In particular, if M is a free R-module of rank m (so M » Rm ), then any submodule

N Ă M is free of rank at most m.

A.22 Symmetric powers and Exterior powers

For an A-module M , the tensor algebra T pM q is defined as
8
à
T pM q “ M bn
n“0
bn
where M is the n-fold tensor product M bA ¨ ¨ ¨ bA M . T pM q has a multiplication given
by the tensor product, but this is not commutative except in a few special cases. However, we
can define SympM q as the quotient of T pM q by the two-sided ideal generated by all elements
of type x b y ´ y b x for x, y P M . Then SympM q is a graded, commutative, A-algebra. For
n P N, we define Symn pM q to be the n-graded piece of SympM q. As the multiplication is
commutative, we write x1 x2 ¨ ¨ ¨ xn for the image of x1 b¨ ¨ ¨bxn in SympM q. So, explicitly,
Symn pM q is generated as an A-module by all symbols x1 x2 ¨ ¨ ¨ xn where x1 , . . . , xn P M ,
modulo the relations x1 x2 ¨ ¨ ¨ xn “ xσp1q xσp2q ¨ ¨ ¨ xσpnq where σ P Sn is a permutation.
Example A.61. If M “ Ar , then SympM q “ Arx1 , . . . , xr s and Symn pAr q is the degree-
n piece. In particular,
`n`r´1 ˘ if M is a free A-module of rank r, then Symn pM q is a free module of
rank r´1 , generated by elements of the form xe11 ¨ ¨ ¨ xerr , where e1 ` ¨ ¨ ¨ ` er “ n. △
Ź
Next, we define the exterior algebra M as the quotient of T pM q by the two-sided ideal
generated by products x b x, where x P M . We let x1 ^ ¨ ¨ ¨ ^ xn denote the image of

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A.23 The Koszul complex

For n P N, consider An with the standard basis
Let A be a ring and let x1 , . . . , xn P A. Ź
p p
e1 , . . . , en . For each p P N, define K “ An and
dp : K p ÝÝÑ K p`1
by the formula
dp pwq “ px1 e1 ` ¨ ¨ ¨ ` xn en q ^ w
Then we get a complex of A-modules
d0 d1 dn´2 dn´1
K‚ : K 0 ÝÑ K 1 ÝÑ . . . ÝÝÝÑ K n´1 ÝÝÝÑ K n Ñ 0. (A.13)
More generally, for any A-module M , we get a complex
K ‚ px1 , . . . , xn ; M q “ K ‚ bR M.
Example A.64. For n “ 1, the Koszul complex is given by
x
0ÑAÑ
Ý A Ñ 0.
This is exact if and only if x is not a zerodivisor. △
Example A.65. For n “ 2 and n “ 3, the Koszul complexes take the form
¨ ˛
´ ¯
x
˝ 1‚
x1 , x2 x2
0 Ñ A ÝÝÝÝÝÑ A2 ÝÝÝÑ A Ñ 0
and
¨ ˛ ¨ ˛
´x2 ´x3 0 ‹
˚
x
˚ 1‹
´ ¯
x 1 0 ´x ˚
3
˚
˝
‹
‹
‚
˚x2 ‹
˚
˝
‹
‚
x1 , x2 , x3 3
0 x2 x3 3
x3
0 Ñ A ÝÝÝÝÝÝÝÝÑ A ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ A ÝÝÝÑ A Ñ 0.
△

Theorem A.66. If x1 , . . . , xn forms a regular sequence, then K ‚ gives a resolution of

A{px1 , . . . , xn q.

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Further constructions and examples

B.1 Grassmannians
Grassmannian varieties are important examples in algebraic geometry. Just like projective
space parameterize lines through the origin of a vector space, Grassmannians are designed to
parameterize linear subspaces of a fixed vector space. In fact, for natural numbers d and n,
there is a scheme Grpd, nq of finite type over Z such that for any field k , the k -points are
given by the set
" *
Grpd, nqpkq “ rLs | L Ă k n is a linear space of dimension d . (B.1)

Projective space Pn´1

Z is the special case when r “ 1.

In the present discussion, we will find it notationally simpler to work with linear spaces of
codimension r. The basic idea is to parameterize the r “ n ´ d linear equations defining L.
a1,1 x1 ` a1,2 x2 ` ¨ ¨ ¨ ` a1,n xn “ 0
a2,1 x1 ` a2,2 x2 ` ¨ ¨ ¨ ` a2,n xn “ 0
.. (B.2)
.
ar,1 x1 ` aq,2 x2 ` ¨ ¨ ¨ ` ar,n xn “ 0
or equivalently, we write
¨ ˛
a1,1 ¨ ¨ ¨ a1,n
L “ KerpAq Ă k n where A “ ˝ ... .. .. ‹
˚
. . ‚
ar,1 ¨ ¨ ¨ ar,n
Of course, the matrix A is not unique in determining L. For instance, the two matrices
ˆ ˙ ˆ ˙
1 1 0 1 1 0 1 0
and (B.3)
1 0 1 0 0 1 ´1 1
give rise to the same 2-dimensional subspace in k 4 , as the second matrix is the row reduction
of the first. In fact, two matrices A and A1 give rise to the same subspace L precisely they
are related by an element of GLr pkq, that is, A1 “ gA for some g P GLr pkq. Moreover, the
matrix A must have rank r for L to have dimension n ´ r.
Let Arn pkq denote the k -points of the affine space of dimension rn with affine cordinates
xij , i “ 1, . . . , r, j “ 1, . . . , n. We will think of the points of Arn pkq as r ˆ n-matrices
498

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with entries in k . Let Z Ă Arn pkq denote the closed subset defined by the vanishing set of
all r ˆ r-minors of the generic matrix pxij q. Then Z corresponds to the matrices of rank at
most r ´ 1.
Based on the above paragraph, we want to consider the quotient space
pArn pkq ´ Zq { GLr pkq (B.4)
This is of course possible in the category of topological spaces, as we can simply give the set
on the right-hand side the quotient topology. What is not yet clear, is that this works on the
level of schemes. While there is a vast theory on constructing quotients of schemes, it turns
out in the present setting to be simpler to construct the scheme Grpr, kq by gluing together
affine spaces, like we did for projective space.
To explain how the gluing works we consider again the situation above. If a subspace L Ă
n
k is represented by A, we can by Gaussian elimination find a more canonical representative,
by putting the matrix A in reduced echelon form. In other words, we may represent each L
by a matrix A with some r ˆ r identity matrix as a submatrix. Conversely, any such matrix
A determines a subspace L of k n and now two matrices A, B give the same L if and only if
A “ B . Note that matrices A with a fixed r ˆ r identity submatrix are parameterized by an
affine space of dimension nr ´ r2 “ rpn ´ rq. It therefore makes sense to try to construct
the variety Grpr, nq by gluing together these affine spaces.
Write
Arn “ Spec Zrxij : 1 ď i ď r, 1 ď j ď ns
We think of Arn as an affine space parameterizing r ˆ n-matrices.
We write M “ pxij q for the r ˆ n-matrix of indeterminates. If I Ă t1, . . . , nu is a subset
of size r and A is an r ˆ n-matrix, we let AI denote the r ˆ r submatrix of A given by the
columns in I .
For each subset I Ă t1, . . . , nu of size r, consider the closed subscheme of Arn defined
by
UI “ V pxij ´ δij : i, j P Iq » Arpn´rq
` ˘ ` ˘
Note that there are nr choices for the r columns, so nr affine spaces in total.
If J is another r-subset, let UI,J Ă UI denote the distinguished open set Dpdet MJ q in
UI .
Example B.1. For n “ 4, r “ 2, there are 6 affine spaces, each isomorphic to A4 . In
suggestive notation:
„ ȷ „ ȷ
1 0 x13 x14 1 x12 0 x14
U12 “ Spec Z U13 “ Spec Z
„0 1 x23 x24 ȷ „0 x22 1 x24 ȷ
1 x12 x13 0 x 1 0 x14
U14 “ Spec Z U23 “ Spec Z 11
„ 0 x 22 x 23 1 ȷ „ 21 0 1 x24 ȷ
x
x 1 x13 0 x x12 1 0
U24 “ Spec Z 11 U34 “ Spec Z 11 .
x21 0 x23 1 x21 x22 0 1
△
If T is a scheme, the T -valued points UI pT q is the set of r ˆ n-matrices with entries in

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OT pT q and the identity matrix in the columns indexed by I . Furthermore, UI,J pT q is the set
of r ˆ n-matrices A P UI pT q such that the determinant of MJ is invertible, i.e., belongs to
OT pT qˆ .
Note that if A is an rˆn matrix and det AI is invertible, then A´1
I A has the identity matrix
in columns I . Hence, A ÞÑ pAI q´1 A defines a bijection of sets UJ,I pT q Ñ UI,J pT q. Note
that if f : S Ñ T is a morphism of schemes, the pullback induces a map UI pT q Ñ UI pSq.
In this way, we obtain natural transformations betwen the functors of points hUI,J and hUJ,I .
By Yoneda’s Lemma, we obtain isomorphisms of affine schemes
τJI : UI,J ÝÝÑ UJ,I
These isomorphisms moreover satisfy the gluing condition
τKI “ τKJ ˝ τJI . (B.5)
Indeed, both sides of the equation, when applied to a matrix A with the identity matrix in
the I -columns, row-reduce A to a matrix with the identity matrix in the K -columns. The
left-hand side does this directly, while the right-hand side does so in two steps: first to the
identity in columns J , then to the identity in columns K .
It follows that the affine spaces UI glue to a scheme, which we call the Grassmannian
Grpd, nq. The next proposition tells us that the k -points of Grpd, nq are in bijection with the
set of d-dimensional linear subspaces of k n , as we alluded to in (B.1).

Proposition B.2. Let k be a field. Then the k -points of Grpr, nq are in bijection with the
set of equivalence classes of r ˆ n-matrices of rank r modulo the action of GLr pkq.

Proof If A is an r ˆ n-matrix with entries in k , and the rank of A is equal to r, then some
submatrix AI must be invertible. The bijection is defined by sending A to A´1 I A, which
defines a k -point in UI , and hence in Grpd, nq. If A1 is another matrix equivalent to A, then
´1
also A1I is invertible, and A1 I A “ A´1 I A. It follows that the assignment is injective. It is
also surjective, as any k -point Grpd, nq lies in some UI , and hence comes from a matrix A
with identity matrix in the I -columns.
The Grassmannians resemble projective spaces in several ways:
Generalized homogeneous coordinates. In light of Proposition B.2, we may also talk
about generalized homogeneous coordinates on Grpr, nq: for a field k , the k -points are
equivalence classes rM s of r ˆ n-matrices, where we say rM s “ rM 1 s if M 1 “ gM for
some g P GLr pkq. For instance, for the two matrices in (B.3) we have
„ˆ ˙ȷ „ˆ ˙ȷ
1 1 0 1 1 0 1 0
“ .
1 0 1 0 0 1 ´1 1
Local coordinates. Just as the ratios x0 {xi , . . . , xn {xi form affine coordinates on D` pxi q Ă
Pn , the affine coordinates of UI are given by the entries of the matrix MI´1 M . More precisely,
if ZrMI´1 M s denotes the k -algebra generated by the entries in the matrix MI´1 M , then
Grpr, nq is obtained by gluing together the affine spaces
UI “ Spec ZrMI´1 M s » Arn (B.6)

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using the isomorphisms given by the following equality of subrings of Qpx11 , . . . , xnn q:

ZrMI´1 M sdetpMI´1 MJ q “ ZrMJ´1 M sdetpMJ´1 MI q . (B.7)

Quotient morphism. If Z Ă Arn denotes the closed subset defined by all the r ˆ r-minors
of the matrix of variables M , then there is a a quotient morphism

π : Arn ´ Z ÝÝÑ Grpr, nq. (B.8)

This morphism is obtained by gluing together morphisms between affine schemes

πI : Dpdet MI q ÝÝÑ UI (B.9)

To define (B.9), we can use Yoneda’s Lemma. Note that a T -point of the open subscheme
Dpdet MI q Ă Arn is given by an r ˆ n-matrix A with entries in OT pT q so that det AI is
invertible in OT pT q. On the level of T -points, the morphism (B.9) then sends the matrix
A to A´1I A which is a well defined point in UI pT q. It is clear that this defines a natural
transformation between the functors of points, and that the morphism is surjective. Moreover,
two matrices A, B P Dpdet MI q map to the same point in UI if and only if A´1 ´1
I A “ BI B ,
which happens if and only if B “ gA where g “ AI BI´1 is an element in GLr pOT pT qq.
In fact, the multiplication map defines a bijection between GLr pOT pT qq ˆ UI pT q and
Dpdet MI qpT q, and consequently an isomorphism of schemes
»
GLr ˆUI ÝÝÑ Dpdet MI q (B.10)

It is straightforward to check that the morphisms (B.9) are compatible with the gluing of
Grpd, nq, so we obtain the morphism (B.8). In light of (B.10), π is a so-called ‘fiber bundle’
over UI with fiber GLr ’.
Universal sheaves. On projective space, Pn there is the invertible sheaf Op1q which is glued
x
together by copies of OUi using the transition functions gij “ xji over the distinguished opens
D` pxi q » An . Moreover, the homogeneous coordinates x0 , . . . , xn determine a surjection
OPn`1
n ÝÝÑ Op1q ÝÝÑ 0
Likewise, the Grassmannian carries a universal quotient sheaf Q, which satisfies Q|UI »
OUr I , and it is glued together using the transition functions
gIJ “ MI´1 MJ P GLr pOUI,J q. (B.11)

and there is a surjection

n
OG ÝÝÑ Q ÝÝÑ 0. (B.12)

Note that on Arn , there is such a map, namely, the map given by the r ˆ n matrix M
M
OAnrn ÝÝÑ OAr rn (B.13)
Ť
This is surjective on all stalks contained in the open set I Dpdet MI q. Moreover, restricting
this to the UI we obtain
M
OUnI ÝÝÑ OUr I (B.14)

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In terms of the affine charts in (B.6), the map (B.14) is induced by

M ´1 M
ZrMI´1 M sn ÝÝÝ
I
ÝÝÑ ZrMI´1 M sr
Passing to the localization, these maps glue to the sheaf map (B.12), because the following
diagram commutes.
MI´1 MJ
pZrMJ´1 M sdetpMJ´1 MI q qr pZrMI´1 M sdetpMI´1 MJ q qr

MJ´1 M pMI q´1 M

pZrMI´1 M sdetpMI´1 MJ q qn

The Plucker embedding Consider the morphism

n
p : Arn ÝÝÑ Ap r q “ Spec ZryI s
which is induced by all the r ˆ r determinants of M , that is, p7 pyI q “ det MI . This induces
a morphism
n
ρ : Arn ´ Z ÝÝÑ Pp r q´1
Ş
where Z “ I V pdet MI q is the closed subset defined by all the r ˆ r-minors of M .
We claim that ρ is a closed embedding. To do this, it suffices to show that ρ restricts to an
n
isomorphism over each UI , which maps into the distinguished open D` pyI q Ă Pp r q´1 . We
write ρI : UI Ñ D` pyI q for this morphism.

Lemma B.3. For each I , the morphism ρI is a closed embedding.

Proof After re-indexing, it suffices to consider the case I “ t1, 2, . . . , ru. Then, if we
expand the minors of the r ˆ n matrix (corresponding to a matrix UI )
¨ ˛
1 x1,r`1 . . . x1,n
˚ .. .. .. .. ‹
˝ . . . . ‚
1 xr,r`1 . . . xr,n
it is clear that the set of minors contains (up to sign) all the variables xij with j R I . Hence
n n
ρ : Arpn´rq Ñ Ap r q´1 is given by the graph of a morphism Arpn´rq Ñ Ap r q´1´rpn´rq ,
I
and hence is a closed immersion (Exercise 9.9.35).
n
This means that Grpr, nq embeds as a closed subscheme of Pp r q´1 . The image of ρ is
defined by all the polynomial relations between the r ˆ r-minors of the matrix M . These
relations are known to be generated by the so called Plucker quadrics. These quadrics take
the form
r`1
ÿ
p´1ql yi1 ,...,ir´1 ,jl yj1 ,...,ĵl ,...jr`1 “ 0. (B.15)
l“1

There is one relation for each pair of increasing sequences I “ pi1 ă ¨ ¨ ¨ ă ir´1 q,
J “ pj1 ă ¨ ¨ ¨ ă jr`1 q of length r ´ 1 and r ` 1 respectively, and j1 , . . . , ĵl . . . jr`1

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denotes the sequence j1 , . . . , . . . jr`1 with the term jl omitted. Moreover, in the sum (B.15),
the indexes in the variables may not be increasing: here we adopt the convention that yI is
equal to 0 if I contains repeated elements, and yi1 ,...,ir “ ´yi1 ,...,is´1 ,is`1 ,...,ir .
The proof that these quadrics generate the entire ideal of the Grassmannian is not partic-
ularly difficult, but it requires algebraic manipulations that make it a little bit lengthy (see
(?, Corollary 2.5)). Besides we do not need this fact in what follows. In fact, Grpk, nq is
probably the best example of a projective variety that is not best studied by its equations in
some projective space.
Example B.4. Let us continue the example of n “ 4, r “ 2. In this case, the image of ϕ is
the quadric
Q “ V py12 y34 ´ y13 y24 ` y14 y23 q Ă P5
This is a consequence of the following relation between the minors of a 2 ˆ 4 matrix
x11 x12 x13 x14 x x13 x12 x14 x x14 x12 x13
´ 11 ` 11 “0
x21 x22 x23 x24 x21 x23 x22 x24 x21 x24 x22 x23
and the fact that both Grp2, 4q and Q are integral and of the same dimension (over a field,
they are 4-dimensional). △
If R is a ring, we define GrR pd, nq » GrZ pd, nq ˆZ Spec R. The main case of interest is
when R “ k is a field.

Proposition B.5. For a field k , the Grassmannian Grk pr, nq is a non-singular projective
variety of dimension
dim Grk pr, nq “ rpn ´ rq.

B.1.0 The Grassmannian functor. Let r and n be non-negative integers and consider the
functor
" *
short exact sequences 0 Ñ U Ñ OTn Ñ Q Ñ 0
Gr,n pT q “ {„
where U, Q are locally free of ranks n ´ r and r
where two exact sequences are said to be equivalent if there are isomorphisms making the
following diagram commutative

0 U OTn Q 0
» »

0 U1 OTn Q1 0

Theorem B.6. The functor Gr,n is represented by the Grassmannian Grpr, nq.

Proof Suppose we have a scheme T and a short exact sequence of locally free sheaves
0 ÝÝÑ U 1 ÝÝÑ OTn ÝÝÑ Q1 Ñ 0. (B.16)
Choose an affine open covering of T given by Ti “ SpecpAi q so that Q1 |Ti » OTr i . The

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surjection OTni Ñ Q1 |Ti » OTr is given by some n ˆ r matrix Mi with entries in Ai , which
naturally determines a morphism gi : Spec Ai Ñ Arn .
As OTn Ñ Q is surjective, all of these morphisms factor through Arn ´ Z , so they
induce morphisms φ ˝ gi : SpecpAi q Ñ Grpr, nq. The morphisms hi “ φ ˝ gi glue
to give a morphism h : T Ñ Grpr, nq. Indeed, the hi are defined from the morphism of
locally free sheaves, which implies that they agree on overlaps. Moreover, it means that the
construction of h does not depend on the trivializations used, and also is independent of
choice of the sequence (B.16) within its equivalence class (as two equivalent sequences may
be simultaneously trivialized by the same data).
If ϕ : T Ñ Grpr, nq is any morphism, covering Grpr, nq by the open affine pieces UI
and then covering h´1 pUI q by open affine pieces SpecpAi q over which Q1 is trivial, we see
that ϕ pulls back the universal family to S if the following diagram commutes:
But this happens if and only if h|SpecpAi q “ gi . Therefore g is the unique morphism pulling
back the universal family to T , as required.
pn`rq´1
Remark B.7. The Plucker embedding P : Grpk, nq Ñ PZ r has an elegant interpretation
in terms of the functor of points. The morphism P comes from the natural transformation
which takes a T -valued point of Grpk, nq, that is, a quotient OTn Ñ Q Ñ 0, and sends it
pn`rq´1
to ^r On Ñ ^r Q Ñ 0, which since ^r Q has rank 1, defines a point in PZ r pT q. It is
possible to use this to produce a completely coordinate-free proof of the fact that P is an
embedding in this functorial language.
Example B.8. The functor F : Schop Ñ Sets given by
F pT q “ tlocally free sheaves of rank r on T u{isomorphism
is not representable. To see why, assume that there is a scheme X such that F » hX . For a
scheme T , let oT P F pT q denote the element corresponding to the trivial sheaf OTr .
Now, let T be any scheme, and let E be a locally free sheaf E or rank r on it. We may
choose a covering Ui of T such that E|Ui » OUr i . This means that the corresponding element
e P F pT q must map to oUi P F pUi q via the maps F pT q Ñ F pUi q. The same is of course
true for the element oT P F pT q. So by the above remark, we must have that e “ oT P F pT q.
However, this would mean that any locally free sheaf on T is trivial, which is false in general.
For instance, for T “ Pn and E » OP1 p1qr , then E is non-trivial. △
B.1.0 Enumerative problems. The Grassmannian appears in classical algebraic geometry
involving enumerative questions involving linear spaces in projective space. One example
is the following theorem, which is one of the most celebrated results in classical algebraic
geometry.

Theorem B.9 (Cayley–Salmon). Over an algebraically closed field, every smooth cubic
surface S Ă P3 contains exactly 27 lines.

Example B.10. For illustration, let us consider the lines on the Fermat cubic surface defined
by
F “ x30 ` ¨ ¨ ¨ ` x33 “ 0

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

We identify lines in P3k with planes in k 4 . Let us find equations for the set of points in
Grp2, 4q corresponding to lines in S “ V pF q. Let us consider lines L of the form
x0 ` a2 x2 ` b2 x3 “ 0,
x1 ` a3 x2 ` b3 x3 “ 0.
These correspond to points contained in the affine chart Ut1,2u Ă Grp2, 4q. Then L is
contained in S if and only if the cubic F restricts to the zero polynomial in L. In other words,
F p´a2 s ´ b2 t, ´a3 s ´ b3 t, s, tq is the zero polynomial in s and t. Expanding and collecting
coefficients, we obtain the following system of equations for a2 , a3 , b2 , b3 :
a32 ` a33 “1
a22 b2 ` a23 b3 “0
a2 b22 ` a3 b23 “0
b32 ` b33 “ 1.
Let a denote the ideal generated by these equations. Then a can be simplied to:
a “ pa2 a3 , a2 b2 , a3 b3 , b2 b3 , b32 ` b33 ´ 1, a33 ` b33 ´ 1, a32 ´ b33 q.
From this we see that either a2 “ 0 or a3 “ 0. If a2 “ 0, then the last equation implies that
b3 “ 0 as well, and the two remaining equations reduce to b32 “ 1 and a33 “ 1. Here a3 and
b3 can be any cubic root of unity, leading to 3 ˆ 3 “ 9 solutions. Doing the same for a3 “ 0
leads to a number of 9 ` 9 “ 18 lines in UI . By permuting the coordinates, we find the lines
contained in the other UI as well, leading to a total of 27 lines contained in V pF q:
x0 ` ω k x1 “ x2 ` ω j x3 “ 0, 0 ď j, k ď 2
x0 ` ω k x2 “ x1 ` ω j x3 “ 0, 0 ď j, k ď 2
x0 ` ω k x3 “ x1 ` ω j x2 “ 0, 0 ď j, k ď 2
where ω is a primitive cube root of 1. △

B.2 The Picard group of a Grassmannian

Proposition B.11. Let k be a field. Then the Picard group of the Grassmannian Grpr, nq
is isomorphic to Z. It is generated by the restriction of Op1q from the Plücker embedding.

Proof Consider Grpr, nq Ă PN embedded via the Plücker embedding. Fix an r-tuple
I Ă t1, . . . , nu. Then the divisor H “ divpxI q|X is a well-defined divisor on Grpr, nq.
The complement Grpr, nq is the open set UI Ă Grpr, nq (corresponding to n ˆ r-matrices
whose I -th minor is nonzero). As UI » Arpn´rq has trivial class group, the exact sequence
(17.15) shows that the class group of Grpr, nq is generated by H . Clearly, no multiple of H
is zero in ClpGrpr, nqq. That would mean that some OX pmq » OX . This is not possible, as
Op1q has N ` 1 linearly independent global sections.
Exercise B.2.1. Show that the set of lines on the quadric surface S “ V px0 x3 ´x1 x2 q Ă P3k
is a union of two conics in Grp2, 4q Ă P5k .

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Exercise B.2.2 (Gluing relative schemes). Let X be a scheme and suppose that we are given
the following data:
(i) For each affine open subscheme U Ă X , a scheme YU and a morphism
πU : YU Ñ U .
(ii) For any pair of affine open subschemes V Ă U Ă X a morphism ρU V : YV Ñ
YU such that the following diagram commutes:
ρU,V
YV YU
πV πU (B.17)

V U
and induces an isomorphism YV » πU´1 pV q.
(iii) For any affine open subschemes W Ă V Ă U Ă X , we have ρU,W “
ρV,U ˝ ρW,V .
Show that there exists a scheme Y together with a morphism π : Y Ñ X , and isomorphisms
ιU : π ´1 pU q Ñ YU , such that: for each V Ă U affine, the following diagram commutes
ιU πU
π ´1 pU q YU U
ρU,V

ιV πV
π ´1 pV q YV V

and the composition πU ˝ ιU “ π|U for each U . Show that as an X -scheme, Y is unique up
to isomorphism. H INT: See (?, Tag 01LG).

B.3 Relative Spec

Let X be a scheme and let A be a quasi-coherent sheaf of OX -algebras. This means that A
is a quasi-coherent sheaf and for each open set U Ă X , the group ApU q is an algebra over
the ring OX pU q.
Let us apply Proposition ?? to the case where
YU “ Spec ApU q
and πU : YU Ñ U is the morphism induced by the ring map OX pU q Ñ ApU q. Let us
check the that the second condition in the proposition is satisfied. If V Ă U is another
affine subset, we have there is a ring map ApU q Ñ ApV q which induces a morphism
YV Ñ YU making the diagram (B.17) commutative. The diagram is actually Cartesian,
because ApV q “ ApU q bOX pU q OX pV q, as A is quasi-coherent.
The third condition is also satisfied, because the ring map ApU q Ñ ApW q factors via
restriction to V .
It follows that the schemes YU glue together to a scheme, which we denote SpecpAq
which we call the ‘relative spectrum of A’. There is a morphism π : SpecpAq Ñ X which
satisfies
π˚ OSpecpAq “ A.

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

The scheme SpecpAq satisfies the following universal property: For each morphism h :
Z Ñ X with a map of OX -algebras A Ñ h˚ OZ , there should be a unique morphism
f : Z Ñ SpecpAq such that h “ π ˝ f .
Example B.12. For A “ OX rt1 , . . . , tn s, the relative Spec coincides with the relative affine
space AnX . △
Example B.13. Let X “ Ank “ Spec krx1 , . . . , xn s and let f P krx1 , . . . , xn s be a
polynomial. Then
A “ OX rts{ptm ´ f px1 , . . . , xn qq
is an OX -algebra. The relative spec Y “ Spec A is exactly the double cover of An ramified
along f . △
Example B.14. More generally, let X be a normal integral scheme, D Ă X an effective
divisor, and let L be an invertible sheaf on X such that Lbm » OX pDq. Let s P OX pDq be
the section that defines D; we will view it as a map s : OX Ñ Lbm . Define the OX -module
A “ OX ‘ L´1 ‘ ¨ ¨ ¨ ‘ L´m`1
This becomes an OX -algebra via the multiplication
idbs
Lá b L´b » Lá´b b OX ÝÝÝÑ Lá´b b Lm » Lá´b`m .
Let Y “ SpecA with the projection π : Y Ñ X . We call Y the ramified cyclic cover of s.
Over an open set U where L » OU , pick a local generator s. The image sm P ΓpU, Lm q.
On such an open, we have A|U » OUm , which is generated by 1 and f subject to the relation
zm “ f .
It is not hard to show that Z is nonsingular if and only if X and D are. △
Exercise B.3.1. Check that the scheme SpecpAq and the morphism π satisfies the above
universal property.
Example B.15 (Closed subschemes). An important special case is when A “ OX {I for
some quasi-coherent ideal I . In this case there is a morphism
i : Spec pOX {Iq ÝÝÑ Spec pOX q “ X
and Y “ Spec pOX {Iq is exactly the closed subscheme associated to I . △
Example B.16 (Vector bundles). Let E denote a locally free sheaf of rank r. The symmetric
algebra
Sym˚ pEq “ OX ‘ E ‘ S 2 pEq ‘ ¨ ¨ ¨
is naturally an algebra over OX . The corresponding relative Spec is denoted by V pEq. The
projection π : V pEq Ñ X is what’s known as a vector bundle; all the scheme-theoretic fibers
are affine spaces of dimension r. More precisely, if x P X , the fiber Epxq “ E bOX kpxq
is isomorphic to kpxqr , and so the scheme-theoretic fiber of π over x is isomorphic to the
spectrum of
Sym˚ pkpxqr q » kpxqrt1 , . . . , tr s
△

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Example B.17 (Normalization). TODO.

If U, V are two affines with V Ă U , we can consider the open subscheme W “
πU´1 pV q Ă YU . By assumption, this scheme is affine, because πU is an affine morphism, and
integral and normal, being an open set in YU . Furthermore,
č
OW pW q “ OYU ,p .
pPW

The intersection takes place inside K “ KpW q “ KpXq. As the local rings OYU ,p are
integrally closed, we see that OW pW q is normal. As W Ñ V is finite, OX pV q Ñ OW pW q
a finite ring extension, and we see that OW pW q is the integral closure of OX pV q in K .
In other words, we may canonically identify YV with πU´1 pV q, and consequently the fiber
product YU ˆU V . Finally, if W Ă V Ă U are three affines, the map YW Ñ YU clearly
factors via YV . Therefore the conditions of Proposition ?? are satisfied, and the schemes YV
glue to a scheme which we will denote by X .
Finally, we prove that the scheme X and πX : X Ñ X satisfy the universal property. So
let h : Z Ñ X be a dominant morphism from a normal integral scheme Z . Over each Ui ,
we have an induced dominant morphism h´1 pUi q Ñ Ui , which by the universal property
over the Ui must factor uniquely via Ui via a morphism hi : h´1 pUi q Ñ Ui . Again the
uniqueness in the universal property tells us that these maps must agree over the overlaps
h´1 pUij q. Since the h´1 pUi q form an open cover of Z , these maps glue to a map h : Z Ñ X
factoring h. △

B.4 Relative Proj

Let X be a scheme and let R be a quasi-coherent sheaf of graded OX -algebras. This means
that for each open set U Ă X , the group RpU q is a graded ring with degree 0 isomorphic to
OX pU q.
For an open affine U Ă X , set YU “ Proj RpU q, with projection π : YU Ñ U induced
by the natural map Proj RpU q Ñ Spec OX pU q “ U . If V Ă U is another affine, the map
RpU q Ñ RpV q is a map of graded rings, this induces a map YV Ñ YU . Checking that the
conditions of Proposition ?? are satisfied is similar to the Relative Spec-case. We call the
resulting scheme ProjpRq Ñ X the ‘relative Proj of R’.

Example B.18 (Projective bundles). △

Example B.19 (Hirzebruch surfaces). △
Example B.20 (Blow-ups). △

B.5 Pushouts of affine schemes

Gluing schemes along open subschemes have been a central theme in this book. In some
cases, we can also glue two schemes along a common closed subsheme. In this section, we
explain how this can be done for affine schemes.
Let A, B, C be rings and let f : A Ñ C , g : B Ñ C be surjections. From this data, we

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can form the pullback ring A ˆC B arising in the pullback diagram

A ˆC B B
g

f
A C
Explicitly, the ring A ˆC B is defined by
A ˆC B “ t pa, bq P A ˆ B | f paq “ gpbq u.
The diagram above induces a pushout diagram of schemes
Spec C Spec B

Spec A SpecpA ˆC Bq
This means that Spec A ˆC B satisfies a universal property dual to that of the fiber product:
it is universal among diagrams of the form (B.5) with SpecpA ˆC Bq replaced by some
other scheme.

Proposition B.21. As a topological space, SpecpA ˆC Bq is homeomorphic to

pSpec Aq YSpec C pSpec Bq (B.18)

Example B.22. The nodal cubic curve can be obtained from this construction; it is obtained
by identifying two points of A1k . [ADD MORE DETAILS.] △
Example B.23. Here is an example of a non-normal surface with an isolated singularity. We
let X be the scheme obtained by identifying two points in A2k ; X is the affine variety given
by the k -algebra
A “ tf P krx, ys | f p0, 0q “ f p0, 1qu.
Then the normalization X is the affine plane.

The algebra A is generated by the 4 polynomials

x, xy, y 2 ´ y, y 3 ´ y (B.19)
To see this, note that if f px, yq is any polynomial satisfying f p0, 0q “ f p0, 1q, we may
subtract products of the form bpxqpy 2 ´ yqk or bpxqpy 2 ´ yqk py 3 ´ yq until the y -degree of
f is at most 1; the remaining polynomials can be written as polynomials in x and xy .

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

The polynomials (B.19) define a closed embedding of the surface X in A4 . △

Theorem B.24. Let X and Y be schemes and let i : Z Ñ X and j : Z Ñ Y be closed

embeddings. Then the pushout X YZ Y exists as a scheme.
? ?
Example B.25 (Spec Z ´3). The spectrum of the ring R “ Zr ´3s is rather interesting.
It can be viewed as a sort of singular curve over Spec Z. As such it shares many properties
with the nodal cubic curve of Example 11.36.
Note first that R is not a unique factorization domain. For example,
? ?
2 ¨ 2 “ p1 ` ´3qp1 ´ ´3q.
? ?
It is also not normal, because the element ω “ p1 ` ´3q{2 P Qp ´3q satisfies the monic
equation x2 ` x ` 1 “ 0, but ω R R. In fact, the integral closure of R is given by the ring
the ring of Eisenstein integers Zrωs, and Zrωs is a unique factorization domain. In particular,
?
Zrωs is normal, and equals the integral closure of R inside the fraction field Qp ´3q. It
follows that the morphism
?
Spec Zrωs Ñ Spec Zr ´3s,
induced by the inclusion Zr2ωs Ñ Zrωs, is the normalization map.
Note that R is not a Dedekind domain: an integral domain is Dedekind if and only if each
? is a discrete valuation ring. However, the localization Rp at the prime ideal
of its localizations
p “ p2, 1 ` ´3q is not a discrete valuation ring; the maximal ideal requires two generators.
However the square of p is principal; it satisfies
?
p2 “ p2q Ă Zr ´3s.

There are two ring maps ϕ, ψ : Zrωs Ñ F2 rxs{px2 ` x ` 1q “ F4 , one sending ω to

x, and the other sending ω to x ` 1. The subring of Zrωs where these coincide is exactly
Zr2ωs “ R. We get a pushout diagram

F4 ψ
Zrωs
ϕ
?
Zrωs Zr ´3s
?
This induces a homeomorphism between Spec Zr ´3s and Spec Zrωs with two points
identified. Hence Spec R is obtained by identifying two points in the spectrum of the
Eisenstein integers. △

B.6 Multigraded rings

The Proj-construction has the following multigraded analogue.
Let R “ krx1 , . . . , xn s be a ring graded by the group Zn . This means that each variable
xi is assigned a degree di P Zn . Let f P R denote a homogeneous element with respect to
the Zn -grading. Let pRf q0 denote all the elements in the localization of degree 0 P Zn .

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Definition B.26. For w P Zn , we define the subring

à
Rpwq “ Rnw Ă R
ně0

and the w-irrelevant ideal as the graded ideal

à
Iw “ Rnw Ă R
ną0

Note that Rpwq and Iw are graded R0 -modules, in the usual sense. In fact, Rpwq is a
graded algebra over R0 .

Definition B.27. For a given w P Zn , we define the multigraded projective spectrum

w-ProjpRq as the set of homogeneous prime ideals p that do not contain the irrelevant
ideal Iw .

As in the usual Proj-construction, the set w-ProjpRq inherits a Zariski-topology, by

declaring that the closed sets are exactly the sets V paq of homogeneous prime ideals
p Ą a (not containing the irrelevant ideal Iw ). There is also the set of distinguished opens
D` pf q “ w-ProjpRq ´ V pf q, defined for Zn -homogeneous f . As before, these give a basis
for the topology on w-ProjpRq and

D` pf q » SpecpRf q0 .

Next, we define the structure sheaf on the topological space X “ w-ProjpRq. We define
it on the basis consisting on distinguished opens by

OX pD` pf qq “ pRf q0
and the restriction maps are as usual given by the localization maps Rf Ñ Rg for D` pf q Ą
D` pgq. This defines a sheaf of rings OX on X whose stalks at are the local rings pRp q0 .
Therefore, the locally ringed space pX, OX q is a scheme.

B.7 Examples
Example B.28. Let m, n ě 1 be integers and consider the polynomial ring R “ krx0 , . . . , xm , y0 , . . . , yn s
with the Z2 -grading given by deg xi “ p1, 0q and deg yj “ p0, 1q.
For the vector w “ p1, 1q, we find

Iw “ px0 y0 , . . . , xm yn q “ px0 , . . . , xm q X py0 , . . . , yn q.

Note that w-ProjpRq is covered by the affine schemes
„ ȷ
x0 xn y0 y0
Uij “ SpecpRxi yj q0 » Spec k ,..., , ..., » Akm`n .
xi x i yj yj
This is the usual affine cover of Xw “ Pm n
k ˆ Pk .
△

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Example B.29. Consider the polynomial ring R “ krx0 , x1 , y0 , y1 s with the grading given
by the columns of the matrix
ˆ ˙
1 1 0 ´1
M“
0 0 1 1
Let us choose the vector w “ p1, 1q as the degree vector. Then the irrelevant ideal is
generated by all?monomials of bidegree w, i.e., Iw “ px20 y1 , x0 x1 y1 , x0 y0 , x21 y1 , x1 y0 q.
This has radical Iw “ px0 y0 , x1 y0 , x0 y1 , x1 y1 q “ px0 , x1 q X py0 , y1 q. The localizations
are given by
” ı ” ı
pRpx0 y0 q q0 “ k xx01 , xy1 0y1 pRpx0 y1 q q0 “ k xx10 , xy0 1y0
” ı ” ı
pRpx1 y0 q q0 “ k xx01 , xy0 0y1 pRpx1 y1 q q0 “ k xx01 , xy0 1y0 .

This is the same affine cover as the blow-up of P2k at a point. △

Comments or corrections welcome: https://tinyurl.com/yc5y6dwp

Atiyah, Michael. 2018. Introduction to Commutative Algebra. CRC Press.

Eisenbud, David. 2013. Commutative Algebra: With a View Toward Algebraic Geometry. Vol. 150. Springer
Science & Business Media.
Godement, Roger. 1960. Topologie algébrique et théorie des faisceaux.
Griffiths, Phillip, and Harris, Joseph. 1979. Algebraic geometry and local differential geometry. Pages
355–452 of: Annales scientifiques de l’École Normale Supérieure, vol. 12.
Hartshorne, Robin. 2013. Algebraic Geometry. Vol. 52. Springer Science & Business Media.
Kempf, George. 1993. Algebraic varieties. Vol. 172. Cambridge University Press.
Serre, Jean-Pierre. 1955. Un théorème de dualité. Comment. Math. Helv, 29(9-26), 8.
Serre, Jean-Pierre. 2013. Local fields. Vol. 67. Springer Science & Business Media.
Stacks Project Authors. 2018. Stacks Project.

513

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