Journal of Algebra 278 (2004) 32–42

www.elsevier.com/locate/jalgebra

Serre finiteness and Serre vanishing

for non-commutative P1 -bundles
Adam Nyman
Department of Mathematical Sciences, Mathematics Building, University of Montana,
Missoula, MT 59812-0864, USA
Received 21 October 2002
Available online 25 May 2004
Communicated by J.T. Stafford

Abstract
Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free OX -
bimodule of rank 2, A is the non-commutative symmetric algebra generated by E and Proj A is
the corresponding non-commutative P1 -bundle. We use the properties of the internal Hom functor
Hom Gr A (−, −) to prove versions of Serre finiteness and Serre vanishing for Proj A. As a corollary
to Serre finiteness, we prove that Proj A is Ext-finite. This fact is used in [I. Mori, J. Pure Appl.
Algebra, in press] to prove that if X is a smooth curve over Spec K, Proj A has a Riemann–Roch
theorem and an adjunction formula.
 2004 Elsevier Inc. All rights reserved.

Keywords: Non-commutative geometry; Serre finiteness; Non-commutative projective bundle

1. Introduction

One of the major problems in non-commutative algebraic geometry is to classify non-

commutative surfaces [1,8]. Since intersection theory on commutative surfaces facilitates
the classification of commutative surfaces, one expects intersection theory to be an
important tool in the classification of non-commutative surfaces.
Non-commutative P1 -bundles over curves play a prominent role in the theory of non-
commutative surfaces. For example, certain non-commutative quadrics are isomorphic
to non-commutative P1 -bundles over curves [10]. In addition, every non-commutative

E-mail address: nymana@mso.umt.edu.

0021-8693/$ – see front matter  2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jalgebra.2004.04.003
 A. Nyman / Journal of Algebra 278 (2004) 32–42 33

deformation of a Hirzebruch surface is given by a non-commutative P1 -bundle over P1

[9, Theorem 7.4.1, p. 29]. For these reasons, it is important to develop an intersection
theory for non-commutative P1 -bundles over curves.
In the intersection theory of curves on a (commutative) surface, the Riemann–Roch the-
orem and the adjunction formula are particularly useful for carrying out computations. The
proofs of these results depend on the cohomology of the underlying surface. The purpose of
this paper is to study properties of the cohomology of non-commutative P1 -bundles over
curves. In order to state our main results, we recall the definition of a non-commutative
P1 -bundle over a commutative smooth scheme (see Section 2 for more details).

Definition 1.1 [9]. Suppose X is a smooth scheme of finite type over a field K, E is a
locally free OX -bimodule of rank 2 and A is the non-commutative symmetric algebra
generated by E. Let Gr A denote the category of graded right A-modules, let Tors A denote
the full subcategory of Gr A consisting of direct limits of right-bounded modules, and let
Proj A denote the quotient of Gr A by Tors A. The category Proj A is a non-commutative
P1 -bundle over X.

This notion generalizes that of a commutative P1 -bundle over X as follows. Let E be

an OX -bimodule on which OX acts centrally. Then E can be identified with the direct
image pri∗ E for i = 1, 2. If, furthermore, E is locally free of rank 2 and A is the non-
commutative symmetric algebra generated by E, van den Bergh proves [9, Lemma 4.2.1]
that the category Proj A is equivalent to the category Mod PX (pri∗ E), where PX (−) is the
usual (commutative) projectivization.
Suppose X is a smooth scheme of finite type over K, E is a locally free OX -bimodule
of rank 2, A
is the non-commutative symmetric algebra generated by E, ek A is the right
A-module l
k Akl and π : Gr A → Tors A is the quotient functor. Our main result is the
following theorem (Theorem 3.5):

Theorem. For any noetherian object N in Gr A, and any finite set I ⊂ Z × Z:

(1) (Serre finiteness) ExtiProj A ( (j,k)∈I π(OX (j ) ⊗ ek A), πN ) is finite-dimensional over
K for all i
0, and

(2) (Serre vanishing) for i > 0, ExtiProj A ( (j,k)∈I π(OX (j ) ⊗ ek A), πN ) = 0 whenever
j  0 and k  0.

As a corollary to Serre finiteness, we prove that non-commutative P1 -bundles are Ext-

finite; that is, we prove the following

Corollary. If M and N are noetherian objects in Gr A, ExtiProj A (πM, πN ) is finite-

dimensional for i
0.

Let X be a smooth curve over K. In [7], we prove that a non-commutative P1 -bundle

over X satisfies Serre duality and has cohomological dimension two (see Section 4 for
a precise statement of these results). As in the commutative case, these facts, along with
the fact that non-commutative P1 -bundles over X are Ext-finite, are used to prove that
34 A. Nyman / Journal of Algebra 278 (2004) 32–42

non-commutative P1 -bundles over X have a Riemann–Roch theorem and an adjunction

formula [4]. We conclude the paper by stating the versions of the Riemann–Roch theorem
and the adjunction formula which hold for non-commutative P1 -bundles.
The results of this paper have also been used to compute particular intersections on
non-commutative P1 -bundles. In [5, Section 6], Mori and Smith study non-commutative
P1 -bundles Proj A such that A is generated by a bimodule E with the property that E ⊗ E
contains a non-degenerate invertible bimodule. Using cohomological properties of Proj A
(including the fact that Proj A satisfies Serre duality, has cohomological dimension two
and is Ext-finite), they develop an intersection theory on Proj A. Without the use of the
Riemann–Roch theorem or the adjunction formula, they compute various intersections on
Proj A. In particular, they prove that distinct fibers on Proj A do not meet, and that a fiber
and a section on Proj A meet exactly once.
In what follows, K is a field, X is a smooth, projective scheme of finite type over
Spec K, Mod X denotes the category of quasi-coherent OX -modules, and we abuse
notation by calling objects in this category OX -modules.

2. Preliminaries

Before we prove Serre finiteness and Serre vanishing, we review the definitions of
bimodule, non-commutative symmetric algebra and the definition and basic properties of
the internal Hom functor Hom Gr A (−, −) on Gr A.
Let S be a scheme of finite type over K and let X be an S-scheme. For i = 1, 2, let
pri : X ×S X → X denote the standard projections, let δ : X → X ×S X denote the diagonal
morphism, and let ∆ denote the image of δ.

Definition 2.1. A coherent OX -bimodule, E, is a coherent OX×S X -module such that

pri| Supp E is finite for i = 1, 2. A coherent OX -bimodule E is locally free of rank n if
pri∗ E is locally free of rank n for i = 1, 2.

Now assume X is smooth. If E is a locally free OX -bimodule, then let E ∗ denote the
dual of E [9, p. 6], and let E j ∗ denote the dual of E j −1∗ . Finally, let η : O∆ → E ⊗OX E ∗
denote the counit from O∆ to the bimodule tensor product of E and E ∗ [9, p. 7].

Definition 2.2. Let E be a locally free OX -bimodule.

The non-commutative symmetric
algebra generated by E is the sheaf-Z-algebra A = i,j ∈Z Aij with components

• Aii = O∆ ,
• Ai,i+1 = E i∗ ,
• Aij = Ai,i+1 ⊗ · · · ⊗ Aj −1,j /Rij for j > i + 1, where Rij ⊂ Ai,i+1 ⊗ · · · ⊗ Aj −1,j
is the OX -bimodule
j −2

Ai,i+1 ⊗ · · · ⊗ Ak−1,k ⊗ Qk ⊗ Ak+2,k+3 ⊗ · · · ⊗ Aj −1,j ,
k=i

and Qi is the image of the unit map O∆ → Ai,i+1 ⊗ Ai+1,i+2 , and

 A. Nyman / Journal of Algebra 278 (2004) 32–42 35

• Aij = 0 if i > j ,

and with multiplication, µ, defined as follows: for i < j < k,

Ai,i+1 ⊗ · · · ⊗ Aj −1,j Aj,j +1 ⊗ · · · ⊗ Ak−1,k

Aij ⊗ Aj k = ⊗
Rij Rj k

∼ Ai,i+1 ⊗ · · · ⊗ Ak−1,k
=
Rij ⊗ Aj,j +1 ⊗ · · · ⊗ Ak−1,k + Ai,i+1 ⊗ · · · ⊗ Aj −1,j ⊗ Rj k

by [6, Corollary 3.18]. On the other hand,

Rik ∼
= Rij ⊗ Aj,j +1 ⊗ · · · ⊗ Ak−1,k + Ai,i+1 ⊗ · · · ⊗ Aj −1,j ⊗ Rj k

+ Ai,i+1 ⊗ · · · ⊗ Aj −2,j −1 ⊗ Qj −1 ⊗ Aj +1,j +2 ⊗ · · · ⊗ Ak−1,k .

Thus there is an epi µij k : Aij ⊗ Aj k → Aik .

If i = j , let µij k : Aii ⊗ Aik → Aik be the scalar multiplication map O µ : O∆ ⊗ Aik →
Aik . Similarly, if j = k, let µij k : Aij ⊗ Ajj → Aij be the scalar multiplication map µO .
Using the fact that the tensor product of bimodules is associative, one can check that
multiplication is associative.

Definition 2.3. Let Bimod A − A denote the category of A − A-bimodules. Specifically:

• an object of Bimod A − A is a triple

 
C = {Cij }i,j ∈Z , {µij k }i,j,k∈Z , {ψij k }i,j,k∈Z

where Cij is an OX -bimodule and µij k : Cij ⊗ Aj k → Cik and ψij k : Aij ⊗ Cj k → Cik
are morphisms of OX2 -modules making C an A − A bimodule.
• A morphism φ : C → D between objects in Bimod A − A is a collection φ = {φij }i,j ∈Z
such that φij : Cij → Dij is a morphism of OX2 -modules, and such that φ respects the
A − A-bimodule structure on C and D.

Let B denote the full subcategory of Bimod A − A whose objects C = {Cij }i,j ∈Z have
the property that Cij is coherent and locally free for all i, j ∈ Z.
Let Gr A denote the full subcategory of B consisting of objects C such that for some
n ∈ Z, Cij = 0 for i = n (we say C is left-concentrated in degree n).
36 A. Nyman / Journal of Algebra 278 (2004) 32–42

Definition 2.4 [7, Definition 3.7]. Let C be an object in B and let M be a graded right A-
module. We define Hom Gr A (C, M) to be the Z-graded OX -module whose kth component
is the equalizer of the diagram

 ∗ α  ∗
i Mi ⊗ Cki j Mj ⊗ Ckj

β γ (1)
   ∗
 ∗
   ∗

j i Mj ⊗ Aij ⊗ Cki j i Mj ⊗ (Cki ⊗ Aij )
δ

where α is the identity map, β is induced by the composition

η µ
Mi → Mi ⊗ Aij ⊗ A∗ij → Mj ⊗ A∗ij ,

γ is induced by the dual of

µ
Cki ⊗ Aij → Ckj ,

and δ is induced by the composition

   
Mj ⊗ A∗ij ⊗ Cij∗ → Mj ⊗ A∗ij ⊗ Cki
∗
→ Mj ⊗ (Cki ⊗ Aij )∗

whose left arrow is the associativity isomorphism and whose right arrow is induced by the
canonical map [7, Section 2.1]. If C is an object of GrA left-concentrated in degree k, we
define HomGr A (C, M) to be the equalizer of (1).

Let τ : Gr A → Tors A denote the torsion functor, let π : Gr A → Proj A denote the
quotient functor, and let ω : Proj A → Gr
A denote the right adjoint to π . For any k ∈
Z,

let e k A denote the right-A-module

l
k Akl . We define ek A
k+n to be the sum
i
0 ek Ak+n+i and we let A
n = k ek A
k+n . From now on, we take S = Spec K.

Theorem 2.5. If M is an object in Gr A and C is an object in B, Hom Gr A (C, M)

inherits a graded right A-module structure from the left A-module structure of C, making
Hom Gr A (−, −) : Bop × Gr A → Gr A a bifunctor. Furthermore,

(1) τ (−) ∼
= limn→∞ Hom Gr A (A/A
n , −).
(2) If F is a coherent, locally free OX -bimodule,

HomGr A (F ⊗ ek A, −) ∼
= (−)k ⊗ F ∗ .

(3) If L is an OX -module and M is an object of Gr A,

 
HomOX L, HomGr A (ek A, M) ∼= HomGr A (L ⊗ ek A, M).
 A. Nyman / Journal of Algebra 278 (2004) 32–42 37

Proof. The first statement is [7, Proposition 3.11], (1) is [7, Proposition 3.19], (2) is
[7, Theorem 3.16(4)] and (3) is a consequence of [7, Proposition 3.10]. 2

By Theorem 2.5(2), HomGr A (−, M) is F ⊗ ek A-acyclic when F is a coherent, locally

free OX -bimodule. Thus, one may use the resolution [9, Theorem 7.1.2] to compute the
derived functors of HomGr A (A/A
1 , −). By Theorem 2.5(1), we may thus compute the
derived functors of τ :

Theorem 2.6. The cohomological dimension of τ is 2. For i < 2 and L a coherent, locally
free OX -module,

Ri τ (L ⊗ ek A) = 0

and
 2  
L ⊗ Q∗l−2 ⊗ A∗l−2−i,l−2 if i
0,
R τ (L ⊗ el A) l−2−i ∼
=
0 otherwise.

Proof. The first result is [7, Corollary 4.10], while the remainder is [7, Lemma 4.9]. 2

3. Serre finiteness and Serre vanishing

In this section let I denote a finite subset of Z × Z. The proof of the following lemma
is straightforward, so we omit it.

Lemma 3.1. If M is a noetherian object in Gr A, πM is a noetherian object in Proj A

and M is locally coherent.

Lemma 3.2. If M is a noetherian object in Gr A, Ri τ M is locally coherent for all i
0.

Proof. The
module OX (j ) ⊗ ek A is noetherian by [7, Lemma 2.17] and the lemma holds
with M = (j,k)∈I OX (j ) ⊗ ek A by Theorem 2.6.
To prove the result for arbitrary noetherian M, we use descending induction on i.
For i > 2, Ri τ M = 0 by Theorem 2.6, so the result is trivial in this case. Since M is
noetherian, there is a finite subset I ⊂ Z × Z and a short exact sequence

0→R→ OX (j ) ⊗ ek A → M → 0
(j,k)∈I

by [7, Lemma 2.17]. This induces an exact sequence of A-modules

  
   
··· → R τ
i
OX (j ) ⊗ ek A → Ri τ M l → Ri+1 τ R l → · · · .
(j,k)∈I l
38 A. Nyman / Journal of Algebra 278 (2004) 32–42

The left module is coherent by the first part of the proof, while the right module is
coherent by the induction hypothesis. Hence the middle module is coherent since X is
noetherian. 2

Corollary 3.3. If M is a noetherian object in Gr A, Ri (ω(−)k )(πM) is coherent for all

i
0 and all k ∈ Z.

Proof. Since (−)k : Gr A → Mod X is an exact functor, Ri (ω(−)k )(πM) ∼= Ri ω(πM)k .

Now, to prove ω(πM)k is coherent, we note that there is an exact sequence in Mod X
 
0 → τ Mk → Mk → ω(πM)k → R1 τ M k → 0

by [7, Theorem 4.11]. Since Mk and (R1 τ M)k are coherent by Lemma 3.1 and Lem-
ma 3.2 respectively, ω(πM)k is coherent since X is noetherian.
The fact that Ri ω(πM)k is coherent for i > 0 follows from Lemma 3.2 since, in this
case,
   
Ri ω(πM) ∼
= Ri+1 τ M k (2)
k

by [7, Theorem 4.11]. 2

Lemma 3.4. For N noetherian in Gr A, R1 ω(πN )k = 0 for k  0.

Proof. When N = (l,m)∈I (OX (l) ⊗ em A), the result follows from (2) and Theorem 2.6.
More generally, there is a short exact sequence
 
0→R→π OX (l) ⊗ em A → πN → 0
(l,m)∈I

which induces an exact sequence

  
··· → R ω π
1
OX (l) ⊗ em A → R1 ω(πN ) → R2 ω(R) = 0.
(l,m)∈I

where the right equality is due to (2) and Theorem 2.6. Since the left module is 0 in high
degree, so is R1 ω(πN ). 2

Theorem 3.5. For any noetherian object N in Gr A,

(1) ExtiProj A ( π(OX (j ) ⊗ ek A), πN ) is finite-dimensional over K for all i
0,
(j,k)∈I


(2) for i > 0, ExtProj A ( (j,k)∈I π(OX (j ) ⊗ ek A), πN ) = 0 whenever j  0 and k  0.
i

Proof. Let d denote the cohomological dimension of X. Since ExtiProj A (−, πN ) com-
mutes with finite direct sums, it suffices to prove the theorem when I has only one element.
 A. Nyman / Journal of Algebra 278 (2004) 32–42 39

     
HomProj A π OX (j ) ⊗ ek A , πN ∼= HomGr A OX (j ) ⊗ ek A, ωπN
 
∼
= HomOX OX (j ), HomGr A (ek A, ωπN )
 
∼
= HomOX OX (j ), ω(πN )k
 
∼
= Γ OX (−j ) ⊗ ω(−)k (πN )

where the second isomorphism is from Theorem 2.5(3), while the third isomorphism is
from Theorem 2.5(2). Thus,
     
ExtiProj A π OX (j ) ⊗ ek A , πN ∼= Ri Γ ◦ (OX (−j ) ⊗ ω(−)k ) πN .

If i = 0, (1) follows from Corollary 3.3 and [2, III, Theorem 5.2a, p. 228].
If 0 < i < d + 1, the Grothendieck spectral sequence gives us an exact sequence
   
· · · → Ri Γ OX (−j ) ⊗ ω(πN )k → Ri Γ ◦ OX (−j ) ⊗ ω(−)k πN
 
→ Ri−1 Γ R1 OX (−j ) ⊗ ω(−)k πN → · · · . (3)

Since ω(πN )k and R1 (OX (−j ) ⊗ ω(−)k )πN ∼ = OX (−j ) ⊗ R1 (ω(−)k )πN are coherent
by Corollary 3.3, the first and last terms of (3) are finite-dimensional by [2, III,
Theorem 5.2a, p. 228]. Thus, the middle term of (3) is finite-dimensional as well, which
proves (1) in this case. To prove (2) in this case, we note that, since ω(πN )k is coherent,
the first module of (3) is 0 for j  0 by [2, III, Theorem 5.2b, p. 228]. If i > 1, the last
module of (3) is 0 for j  0 for the same reason. Finally, if i = 1, the last module of (3) is
0 since R1 ω(πN )k = 0 for k  0 by Lemma 3.4.
If i = d + 1, the Grothendieck spectral sequence gives an isomorphism
    
Rd+1 Γ ◦ OX (−j ) ⊗ ω(−)k πN ∼ = Rd Γ R1 OX (−j ) ⊗ ω(−)k πN .

In this case, (1) again follows from Corollary 3.3 and [2, III, Theorem 5.2a, p. 228], while
(2) follows from Lemma 3.4. 2

Corollary 3.6. If M and N are noetherian objects in Gr A, ExtiProj A (πM, πN ) is finite-

dimensional for i
0.

Proof. Since M is noetherian, there is an exact sequence

 
0→R→π OX (j ) ⊗ ek A → πM → 0.
(j,k)∈I

Since the central term is noetherian by Lemma 3.1, so is the R. Since HomProj A (−, πN )
is left exact, there are exact sequences
  
0 → HomProj A (πM, πN ) → HomProj A π OX (j ) ⊗ ek A , πN → (4)
(j,k)∈I
40 A. Nyman / Journal of Algebra 278 (2004) 32–42

and, for i
1,

→ Exti−1
Proj A
(R, πN ) → ExtiProj A (πM, πN )
  
→ ExtiProj A π OX (j ) ⊗ ek A , πN → . (5)
(j,k)∈I

Since π commutes with direct sums, the right-hand terms of (4) and (5) are finite-
dimensional by Theorem 3.5(1), while the left hand term of (5) is finite-dimensional by
the induction hypothesis. 2

4. Riemann–Roch and adjunction

Let X be a smooth projective curve over K, let A be the non-commutative symmetric

algebra generated by a locally free OX -bimodule E of rank 2, and let Y = Proj A. In this
section, we state the Riemann–Roch theorem and adjunction formula for Y . In order to state
these results, we need to define an intersection multiplicity on Y . This definition depends
on the fact that Y has well behaved cohomology, so we begin this section by reviewing
relevant facts regarding the cohomology of Y .
Let OY = π pr2∗ e0 A. By [7, Theorem 5.20], Y satisfies Serre duality, i.e., there exists
an object ωY in Proj A, called the canonical sheaf on Y , such that

∼
Ext2−i
Y (OY , −) = ExtY (−, ωY )
i
(6)

for all 0  i  2. Mori has proven that ωY is isomorphic to the noetherian module
π((ωX ⊗ Q0 ) ⊗ e2 A), where ωX is the canonical sheaf on X [3].
By [7, Theorem 4.16], Y has cohomological dimension two, i.e.,


2 = sup i | ExtiY (OY , M) = 0 for some noetherian object M in Proj A . (7)

We write D : Y → Y for an autoequivalence, −D : Y → Y for the inverse of D, and

M(D) := D(M) for M ∈ Y .

Definition 4.1 [4, Definition 2.3]. A weak divisor on Y is an element OD ∈ K0 (Y ) of the

form OD = [OY ] − [OY (−D)] for some autoequivalence D of Y .

We now define an intersection multiplicity on Y following [4]. Let M be a noetherian

object in Proj A, and let [M] denote its class in K0 (Y ). We define, for OD a weak divisor
on Y , a map ξ(OD , −) : K0 (Y ) → Z by

∞
     
ξ OD , [M] = (−1)i dimK ExtiY (OY , M) − dimK ExtiY OY (−D), M .
i=0
 A. Nyman / Journal of Algebra 278 (2004) 32–42 41

This map is well defined by (7) and Corollary 3.6. We define the intersection multiplicity
of OD and M by
 
OD · M := −ξ OD , [M] .

Finally, we define a map χ(−) : K0 (Y ) → Z by

∞
  
χ [M] := (−1)i dimK ExtiY (OY , M).
i=0

Corollary 4.2. Let Y = Proj A, let ωY denote the canonical sheaf on Y , and suppose OD
is a weak divisor on Y . Then we have the following formulas:

(1) (Riemann–Roch)

  1
χ OY (D) = (OD · OD − OD · ωY + OD · OY ) + 1 + pa
2

where pa := χ([OY ]) − 1 is the arithmetic genus of Y .

(2) (Adjunction)

2g − 2 = OD · OD + OD · ωY − OD · OY

where g := 1 − χ(OD ) is the genus of OD .

Proof. The quasi-scheme Y is Ext-finite by Corollary 3.6, has cohomological dimension

2 by [7, Theorem 4.16], and satisfies Serre duality with ωY by [7, Theorem 5.20]. Thus, Y
is classical Cohen–Macaulay, and the result follows [4, Theorem 3.11]. 2

In stating the corollary, we defined the intersection multiplicity only for specific
elements of K0 (Y ) × K0 (Y ). In order to define an intersection multiplicity on the entire
set K0 (Y ) × K0 (Y ), one must first prove that Y has finite homological dimension. In [5,
Section 6], Mori and Smith study non-commutative P1 -bundles Y = Proj A such that A
is generated by a bimodule E with the property that E ⊗ E contains a non-degenerate
invertible bimodule. In this case, they use the structure of K0 (Y ) to prove that Y has finite
homological dimension. They then compute various intersections on Y without the use of
either the Riemann–Roch theorem or the adjunction formula.

Acknowledgment

We thank Izuru Mori for showing us his preprint [4] and for helping us understand the
material in Section 4.
42 A. Nyman / Journal of Algebra 278 (2004) 32–42

References

[1] M. Artin, Some problems on three-dimensional graded domains, in: Representation Theory and Algebraic
Geometry, in: London Math. Soc. Lecture Note Ser., vol. 238, Cambridge Univ. Press, Cambridge, 1995,
pp. 1–19.
[2] R. Hartshorne, Algebraic Geometry, Springer-Verlag, Berlin, 1977.
[3] I. Mori, private communication.
[4] I. Mori, Riemann–Roch like theorems for triangulated categories, J. Pure Appl. Algebra, in press.
[5] I. Mori, S.P. Smith, The Grothendieck group of a quantum projective space bundle, preprint.
[6] A. Nyman, The geometry of points on quantum projectivizations, J. Algebra 246 (2001) 761–792.
[7] A. Nyman, Serre duality for non-commutative P1 -bundles, Trans. Amer. Math. Soc., in press.
[8] J.T. Stafford, M. van den Bergh, Noncommutative curves and non-commutative surfaces, Bull. Amer. Math.
Soc. (N.S.) 38 (2) (2001) 171–216.
[9] M. van den Bergh, Non-commutative P1 -bundles over commutative schemes, submitted for publication.
[10] M. van den Bergh, Non-commutative quadrics, in preparation.