TODAY, I LEARNED...

SAAD SLAOUI

Contents
1. November 2020 2
1.1. 11/30/2020 (homotopical algebra, formal moduli problems) 2
2. December 2020 3
2.1. 12/01/2020 (complex geometry, sheaf cohomology) 3
2.2. 12/02/2020 (homological algebra, formal geometry) 3
2.3. 12/03/2020 (scheme theory, Chern-Weil theory) 4
2.4. 12/04/2020 (scheme theory, representation theory) 5
2.5. 12/05/2020 (motives) 6
2.6. 12/06/2020 (representation theory) 7
2.7. 12/07/2020 (homotopical algebra) 7
2.8. 12/08/2020 (geometric Langlands) 8
2.9. 12/09/2020 (geometric Langlands) 9
2.10. 12/10/2020 (homotopical algebra) 10
2.11. 12/14/2020 (algebraic groups) 10
2.12. 12/15/2020 (formal groups and Lie algebras) 10
2.13. 12/16/2020 (scheme theory) 12
2.14. 12/17/2020 (scheme theory) 12
2.15. 12/18/2020 (representation theory) 13
2.16. 12/30/2020 (homotopical algebra) 13
3. January 2021 14
3.1. 01/12/2021 (Goodwillie calculus) 14
3.2. 01/19/2021 (classical Langlands, arithmetic topology) 15
3.3. 01/25/2021 (HRR via DAG) 15
3.4. 01/29/2021 (infty categories and chromatic homotopy theory) 16
3.5. 01/31/2021 (formal group laws) 17
4. February 2021 17
4.1. 02/01/2021 (étale cohomology) 17
4.2. 02/03/2021 ((co)bar resolution) 18
4.3. 02/08/2021 (underlying interests) 18
4.4. 02/10/2021 (homotopical algebra) 19
4.5. 02/16/2021 (Grothendieck fibration and algebraic de Rham complex) 20
4.6. 02/18/2021 (formal moduli problems, HKR theorems) 21
4.7. 02/22/2021 (Grothendieck duality) 21
4.8. 02/24/2021 (∞-category of spectra) 22
4.9. 02/24/2021 (Serre duality, proper morphisms) 22
4.10. 02/26/2021 (∞-category of chain complexes) 23
4.11. 02/27/2021 (derived Satake) 24
4.12. 02/28/2021 (étale fundamental group) 25
1
2 SAAD SLAOUI

5. March 2021 26
5.1. 03/01/2021 (de Rham stack and D-modules) 26
5.2. 03/11/2021 (Hochschild homology) 26
5.3. 03/12/2021 (Chern classes) 27
6. April 2021 27
6.1. 04/02/2021 (factorization homology) 27
6.2. 04/02/2021 (complete Segal spaces) 28
6.3. 04/06/2021 (Chern classes) 29
6.4. 04/07/2021 (genus, lifting criteria in AG) 29
6.5. 04/11/2021 (categorical group actions) 30
6.6. 04/16/2021 (Milnor numbers, vanishing cycles, matrix factorization) 31
6.7. 04/21/2021 (topos theory) 32
6.8. 04/22/2021 (TQFT, B-models) 33
6.9. 04/23/2021 (categorified HKR, Koszul duality) 33
6.10. 04/29/2021 (∞-categorical Yoneda embedding, homology theories) 34
7. May 2021 35
7.1. 05/02/2021 (Stable ∞-categories, AG) 35
7.2. 05/11/2021 (Gelfand and Koszul) 36
7.3. 05/13/2021 (deformation theory over a DVR) 38
7.4. 05/17/2021 (Balmer spectrum and Gelfand’s categorification) 38
7.5. 05/21/2021 (Euler and Riemann) 39
References 40

1. November 2020
1.1. 11/30/2020 (homotopical algebra, formal moduli problems).
• Whitehead’s theorem can be stated and proven in any model category as the
fact that if A, X are both fibrant and cofibrant, then a map f : A → X is a
weak equivalence iff it admits a homotopy inverse (in the model categorical
sense of homotopy between maps, using either cylinder or path objects).
In this framework, assuming we know what weak/homotopy equivalences
should be, the bulk of the work in generalizing Whitehead’s theorem to
other settings is shifted to exhibiting a model category structure on the
category at hand such that the objects one is interested in (e.g. smooth
projective varieties) are both fibrant and cofibrant.
• With respect to an appropriate notion of covering, one can think of a simpli-
cial resolution of a commutative ring A as the same thing as a hypercovering
of A, meaning roughly the data of a cover of A, together with a covering
of each pairwise intersection of elements of the cover, and so on... In this
language, a projective resolution is the same thing as a cofinal hypercover,
i.e. one such that any other hypercover factors through it.
• I started getting a better feel for what the formal moduli problem X as-
sociated to a given elliptic curve E should assign to the ring k[]/(2 ): an
element of X(k[]/(2 )) consists of E together with (scheme theoretic) in-
finitesimal data of order 1 indicating the “germ of a deformation of E”. For
instance, if we start with a family of elliptic curves over A1 with fiber E
at 0, we can restrict to the tangent space at 0 to get such an element. We
 TODAY, I LEARNED... 3

can’t expect to get all elements of X(k[]/(2 )) in this way, roughly because
some deformations correspond to “power series”.

2. December 2020
2.1. 12/01/2020 (complex geometry, sheaf cohomology).
• Identifying sl2 -representations appearing “in nature” can be a fruitful way
to better understand algebraic objects with an underlying vector space
structure. For instance, one can recover the hard Lefschetz theorem on
the cohomology ring of a compact Kähler manifold by realizing that wedg-
ing with the Kähler form can be interpreted as the “e” action of an sl2 -
representation. Another example arises in studying the general structure
of semisimple complex Lie algebras: several steps of the structure theorem
involve looking at the various sl2 -module structures obtained by having the
subalgebra sα = gα ⊕ g−α ⊕ hα ' sl2 associated to some root α ∈ R act on
g via the adjoint action. This tells us for instance that for k ∈ Z within a
certain range, we get isomorphisms:
'
ad(eα ) : gβ+kα −
→ gβ+(k+1)α .
• Taking cohomology is a construction that makes sense in the category
ShvAb (C) of abelian sheaves on any Grothendieck site C: this is still an
abelian category and one can run the usual right derived functor on global
sections construction. For a scheme X, one usually works with C the (étale
or otherwise) site over X (think: étale maps to X), and the global sections
functor Γ : ShvAb (C) → Ab is given by evaluation at X. This language
carries over to commutative rings: if B is an A-algebra, we can take C
to be A-algebras over B, coverings to be set-theoretically surjective maps,
and then, given a B-module M (equivalent to an abelian group object
B ⊕ M in C), we can take the sheaf cohomology of the representable sheaf
DerA (−, M ) = HomB (−, B ⊕ M ) in that Grothendieck site:
Dq (B, M ) := HTq (B, DerA (−, M )).
2.2. 12/02/2020 (homological algebra, formal geometry).
• There are at least three ways to associate a chain complex to a given sim-
plicial abelian group A• , all equivalent up to chain homotopy equivalence.
The most “direct” construction is called the Moore complex P A∗ , for which
n
one takes An in degree n and sets the differential to be ∂ := i=0 (−1)i di ,
where di : An → An−1 , 0 ≤ i ≤ n are the face maps in degree n. Al-
ternatively, one can restrict to the subcomplex DA• with DAn generated
by degenerate n-simplices (those in the image of a degeneracy map). Per-
haps most useful
Tn−1is the normalized chain complex N A• , given in degree n
by N An := i=0 ker(di ) (note that we omitted dn ), and with differential
given by (−1)n dn . The assignment A• 7→ N A• gives one direction of the
Dold-Kan correspondence, which is an equivalence of categories:
'
N : sAb −
→ Ch≥0 (Ab).
• In attempting to generalize the Lie group-Lie algebra correspondence to
the settings of group schemes over some field k, one is led to looking for a
way to formalize what it means to consider an “infinitesimal neighborhood
4 SAAD SLAOUI

of the identity 1 ∈ G” - the so called “formal group” associated to G.

Under the yoga that algebraic geometry studies functors out of categories
of commutative rings, formal geometry focuses on functors of the form
X : Artaug
k → Set,
where Artaug
k denotes the category of local Artinian k-algebras (k → A → k
composing to the identity on k), or left Kan extensions thereof to functors
CAlg → Set. The intuition is that if X is a scheme, x ∈ X is a given
closed point, and A is a local Artinian k-algebra, then an A-valued point
f : Spec(A) → X taking the maximal ideal of A to x is the right way to
describe “an arbitrary infinitesimal neighborhood of X in x”. As such, the
formal group associated to G is defined to be the functor
aug
G∧
1 : Artk → Grp
given by G∧ 1 (R) := G(R) ×G(k) {1}. One may more generally associate
to any closed embedding Y ,→ X a formal scheme XY∧ called the formal
completion of Y inside X. In the language of DAG, this object may be
succinctly constructed via the following pullback square:
XY∧ /X

 
YdR / XdR .
• Just came across my new favorite proof of the fundamental theorem of
algebra: given a polynomial p ∈ C[z] of degree n ≥ 1, view it as the
characteristic polynomial of a matrix A (the so-called companion matrix),
which can be assumed to be invertible (o/w done). Existence of a root for
p is equivalent to existence of an A-eigenvector, which is equivalent to the
CP n−1 endomorphism induced by A having a fixed point. Since GLn (C)
is path connected, A is homotopic to I as CP n−1 endomorphisms, so they
have the same Lefschetz number, and L(I) = χ(CP n−1 ) = n > 0, hence A
has a fixed point by the Lefschetz fixed point theorem!
2.3. 12/03/2020 (scheme theory, Chern-Weil theory).
• Guiding principle: “solutions to Diophantine equations may be thought of
as sections of a fiber bundle over the arithmetic curve Spec Z”. Suppose we
are interested in finding solutions to a given polynomial equation f ∈ Z[x]
modulo various primes p. We attempt to interpret the problem geometri-
cally as follows. Let A1Z = Spec Z[x] denote the affine line, which may be
viewed together with a flat morphism down to Spec Z. The polynomial f
defines a closed subscheme Spec Z[x]/(f ) ⊆ A1Z , and we may consider its
pullback to any fiber corresponding to a prime p ∈ Z:
Spec Fp [x]/(f ) / Spec Z[x]/(f )

 
Spec Fp / Spec Z

Solutions to f mod p correspond to maps Spec Fp → Spec Fp [x]/(f ), which

necessarily compose to the identity on Spec Fp in the above square: these
 TODAY, I LEARNED... 5

look like sections of the RHS vertical map interpreted as a “fiber bundle”
over the various closed points of Spec Z. More to follow.
• Chern-Weil theory provides a differential-geometric construction of char-
acteristic classes for vector bundles (and more generally for principal G-
bundles) through the machinery of connections. Start with a complex
vector bundle with connection (E, ∇) of rank r over a smooth manifold
M . View the characteristic polynomial (evaluated at −1) as a conjugation-
invariant function on Mr (C), and express it as a sum of homogeneous sym-
metric polynomials Pk for 0 ≤ k ≤ r, which may be alternatively viewed
as elements of Symk (Mr (C))GLr (C) . One may formally enable the Pk ’s to
take in and return differential forms. Letting F∇ ∈ A2 (End(E)) denote the
curvature form of ∇, we may define
2k
ck (E) := [Pk (F∇ )] ∈ HdR (M ; C).

This definition turns out to be independent of ∇, and to coincide with the

standard definition of ck (E) ∈ H 2k (M ; Z). This procedure produces char-
acteristic classes from other choices of conjugation-invariant homogeneous
polynomials, and more generally in the settings of principal G-bundles one
may define the so-called Chern-Weil homomorphism to be the following
map of graded C-algebras:

Sym∗ (g)G → HdR

∗
(M ; C).

The above map may be interpreted as a “way to produce differential form-

valued invariants of principal G-bundles”. In [3], Freed-Hopkins attempt to
formalize the mathematical context of these invariants, and show that the
Chern-Weil homomorphism is in some precise way “the only natural differ-
ential form-valued invariant of principal G-bundles on smooth manifolds”.

2.4. 12/04/2020 (scheme theory, representation theory).

• Affine schemes are fully determined by their ring of functions: the ring R
“knows all there is to know” about Spec R. For instance, if R is a k-algebra
over some field, then points of Spec R may be recovered by looking at ring
homomorphisms R → k 0 for k 0 /k a field extension. This is not true of
non-affine schemes - for instance, O(Pnk ) = k, yet Spec k only has a single
point. To formulate an analogous reconstruction result for schemes, one
needs to categorify functions to sheaves. It is then true that a scheme is
fully determined by the associated symmetric monoidal category of quasi-
coherent sheaves (QCoh(X), ⊗). The fact that the ring of functions sufficed
in the affine case is reflected in the identification QCoh(SpecR) ' ModR .
More generally, stacks which may be recovered from their category of quasi-
coherent sheaves are called “1-affine”.
• (Reminiscence) Serre’s GAGA principle allows us to import methods of
complex geometry to bear on projective complex algebraic geometry. It
states that for any smooth projective complex algebraic variety, the analyti-
fication functor induces an equivalence of categories QCoh(X) ' QCoh(X an ),
as well as a natural isomorphism on sheaf cohomology groups:

H q (X, F) ' H q (X an , F an ).
6 SAAD SLAOUI

In particular, the fact that O(PnC ) = C follows from this isomorphism and
the fact that holomorphic functions on compact complex manifolds are
constant, together with the identification O(PnC ) = H 0 (PnC , OPnC ).
• (To be fleshed out at a later date) Given a monoidal category (C, ⊗), one
may consider its Bernstein center, which is the ring Z(C) = End(IdC ) of
endo-natural transformations of the identity functor. This ring naturally
acts on C, and one may expect some form of “spectral decomposition of
C” accordingly. In AG terms, form the space Spec Z(C). One may then
interpret C as forming in some sense the data of a “sheaf of categories over
Spec Z(C)”:
C

Spec Z(C).
In the context of representation theory, this viewpoint should lead to an in-
tuitive understanding of the decomposition of BGG category O into blocks:
M
O' Oλ .
λ∈h∗ /(W,•)

Compare also to the statement that for a finite group A, we may identify
Rep(A) with the category Vect(Â) of vector bundles over the character
group  = Hom(A,
L U (1)), via the fact that any representation V splits as a
direct sum V ' α∈Â Hom(Vα , V ), where Vα denotes the irreducible A-rep
associated to α ∈ Â:
Rep(A)

Â.
For much more on categorical centers and their applications to geometric
rep theory, see [1].
2.5. 12/05/2020 (motives).
• Whatever the category of (pure) motives (over a ground field k) M(k)
may be, it should have the structure of a Tannakian category, which makes
it susceptible to a Tannakian reconstruction theorem exhibiting it as the
category of representations of an affine group scheme G, called the motivic
Galois group. The group G should be given by the ⊗-automorphism group
scheme of a fiber functor, G := Aut⊗ (ω), where ω : M(k) → VectfQd
should be thought of as a “cohomology theory” valued in Q-vector spaces.
Furthermore, to any smooth projective algebraic variety X, there should be
i
associated a sequence of objects Hmot (X), which should induce the various
classical cohomology groups (de Rham, Betti, crystalline,...) under the
application of various realization functors out of M(k). See [2] for more.
Also fascinating: there is an analogy
finite G-sets : finite étale covs / S :: G-reps : Tannakian cat / k
i.e. one may think of the Tannakian formalism as a linearization of the
theory of étale coverings and étale fundamental groups.
 TODAY, I LEARNED... 7

2.6. 12/06/2020 (representation theory).

• DBZ made an interesting comment today, adding to the “sheaves of cate-
gories over the Bernstein center” intrigue (see 12/04/2021): there should
be an analogy in the settings of linear operators on (f.d.) vector spaces
viewed as “sheaves over C”:
eigenspace : generalized eigenspace (Jordan block) ::
support on a point : support on the formal nbhd of a point
i.e., the nilpotency of Jordan blocks is geometrically incarnated by sheaf-
theoretic infinitesimal behavior. The idea should go as follows: A : V → V
turns V into a C[t]-module, and one may view

C[t] − mod

Spec C[t].

This picture should also apply to category O = U g − modf.g.,n−locf.,h−s.s. ,

wherein generalized central characters should coincide with sheaves scheme
theoretically supported on the formal neighborhood of a point in the char-
acter group of the torus (so that one should expect the Bernstein center of
O to be Hom(T, Gm )?). This is now getting closer to the Rep(A) over Â
picture in the finite group settings from before - and indeed one can show
that C-points of Spec C[A] correspond to characters of A (!).

2.7. 12/07/2020 (homotopical algebra).

• The construction of the cotangent complex LDB/A of an A-algebra B con-
sists in viewing the structure map A → B as a map cA → cB in sCAlg
between the corresponding constant simplicial rings, choosing a projective
factorization
> P•
'

. !!
cA / cB,

and then defining

LDB/A := ΩP• /A ⊗P• B,
where Ω−/A ⊗− B is the abelianization functor on the category CAlgA/,\B ,
applied component-wise to the simplicial object P• . Viewing this object
as a chain complex by Dold-Kan, one may define the cohomology of the
A-algebra B with coefficients in a B-module M to be:
Di (B/A, M ) := Hi (LDB/A ⊗B M )

i.e. “take ith homology of the derived functor applied to M ”. This is the
same general procedure that one first encounters when defining the Tor
groups for modules over a ring, and somewhat more generally for defining
the left derived functors associated to a right exact functor between abelian
8 SAAD SLAOUI

categories F : A → B, provided A has enough projectives:

A / Ch(A)
fibrant replacement F / Ch(B) Hi
/B

A / P•  / F (P• )  / Hi (F (P• )) =: Li F (A).

One should be able to define a model structure on Ch(A), resp. Ch(B) in

such a way that this process is an instance of the derived functor construc-
tion in the settings of model categories.

2.8. 12/08/2020 (geometric Langlands).

• ((lots of) missing adjectives) Start with a reductive algebraic group G and
a smooth algebraic curve X. The geometric Langlands conjecture roughly
states that there should be an equivalence of derived categories

QCoh(LocSysG∨ )
' / D(BunG )

between quasi-coherent sheaves on the moduli stack of G∨ -local systems

on X and D-modules on the moduli stack of G-bundles on X. The RHS
is called the automorphic side, and categorifies the notion of “automorphic
forms”, while the LHS is called the spectral side and categorifies the notions
of “Galois representations” in the classical settings.
The correspondence should send the skyscraper sheaf associated to an
irreducible local system E to a D-module M which is a Hecke eigensheaf for
E on BunG . This roughly means that M transforms “like an eigenvector”
under the action (“integral transform”) of the various Hecke functors
Hλ : D(BunG ) → D(X × BunG )
for λ ∈ P+ a dominant weight of G∨ .
Goal 2.1. Make some sense of what a Hecke eigensheaf should be.
These Hecke functors mentioned above are “integral transforms” induced
via a pull-push process by a correspondence at the level of stacks:
H
h1 h2

| %
BunG X × BunG
where H is the Hecke stack parametrizing data (P, P 0 , x, β) for P, P 0 a pair
of principal G-bundles P, P 0 over X trivialized away from x ∈ X via β.
The maps above are respectively given by
P o  (P, P 0 , x, β)  / (P 0 , x).

Let’s run the story over C, where we may work with perverse sheaves
on the automorphic side (stacky
F Riemann-Hilbert?). Here, the Hecke stack
admits a stratification H = λ∈P+ Hλ induced by the Bruhat-type stratifi-
cation of the affine Grassmannian (which appears when studying the fiber
 TODAY, I LEARNED... 9

of h2 over various (x, P 0 )’s). Via this stratification, one may define ana-
logues of IC sheaves ICλ on H for each λ ∈ P+ . The functor Hλ is then
given by
Hλ : PG(O) (BunG ) → PG(O) (X × BunG )
Hλ (M) := h2,∗ (h∗1 (M ⊗ ICλ )).
Now, start with a G∨ -local system E on X and a G∨ -irrepn Vλ associated
to λ ∈ P+ . Form the balanced product VλE := E ×G∨ Vλ ∈ LocSysG∨ . We
have an exterior product operation:

LocSysG∨ × PG(O) (BunG )

 / PG(O) (X × BunG ).

Definition 2.2. Call M ∈ PG(O) (BunG ) a Hecke eigensheaf for the G∨

local system E if it comes with the data of isomorphisms for each λ ∈ P+ :

ηλ : Hλ (M)
' / V E  M,
λ

which are compatible with the monoidal structure on (Rep(G∨ ), ⊗).

For more on this story, see section 6.1 of [4].

2.9. 12/09/2020 (geometric Langlands).

• Abelian class field theory studies the absolute abelian Galois group Gal(F ab /F )
associated to some number field F/Q, and posits the existence of an iso-
morphism:
Gal(F ab /F ) ' F × \A×


F c.c.

where the subscript c.c. denotes connected components, and AF is the ring
of adèles of F , obtained by putting together the various completions of F .
On the one hand, we have the identification of Gal(F ab /F ) with the
abelianization Gal(F̄/F )ab of the absolute Galois group, whose structure is
completely determined by the set Hom(Gal(F̄/F ), GL1 )
of its 1-dimensional
linear representations. Likewise knowledge of (F × \A× F c.c. may be reduced
to understanding certain of its 1-dimensional representations, or equiva-
lently certain reps of A× × ×
F occurring in Fun(F \AF ). We are thus led to the
equivalent isomorphism:
1-dim reps of Gal(F̄/F ) ' Reps of GL1 (AF ) in functions on GL1 (F × )\GL1 (A×
 
F)

modulo an appropriate adjective on which reps appear on the RHS.

By Tannakian reconstruction, understanding Gal(F̄/F ) is roughly equiv-
alent to understanding Repf d (Gal(F̄/F )) as a Tannakian category. The
classical Langlands correspondence for GLn gives us some traction on this
category of representations by positing the existence of a correspondence
(generalizing the above):
n-dim reps of Gal(F̄/F ) ' Reps of GLn (AF ) in functions on GLn (F × )\GLn (A×
 
F)

which furthermore carries meaningful structure from either side to one an-
other (namely, Frobenius conjugacy classes are matched with Hecke eigen-
values). This is all part of section 1 of [4].
10 SAAD SLAOUI

2.10. 12/10/2020 (homotopical algebra).

• Let X be a complex projective manifold, and notice that any codimension
k complex submanifold Y ⊂ X induces via Poincaré duality an integral
form of type (k, k), called an algebraic class. The Hodge conjecture asks
whether every cohomology class in H k,k (X) ∩ H 2k (X; Q) can be obtained
as a Q-linear combination of algebraic classes. The only known case of this
conjecture for general Kähler manifolds is the Lefschetz theorem on (1, 1)
classes, which states that taking the Chern class of a line bundle gives
a surjection c1 : Pic(X)  H 1,1 (X) ∩ H 2 (X; C)Z , where H 2 (X; C)Z :=
im(H 2 (X; Z) → H 2 (X, C)).
• With respect to the model category structure on the category of non-
negatively graded cdga’s over C where weak equivalences are quasi-isomorphisms
and fibrations are degree-wise surjective chain maps, a minimal model for
A∗ is a valid cofibrant replacement of A∗ , i.e. a cofibrant cdga M∗ to-
gether with an acyclic fibration M∗  A∗ factorizing the map 0 → A∗ .
One defines a minimal cdga M∗ to be one such that there exist elements
(xi ∈LMi )i∈I ,
1 ≤ d1 ≤ d2 ≤ ..., generating M∗ as a cdga, i.e. M∗ =
V ∗
i∈I Cxi , and such that dxi ∈< x1 , ..., xi−1 >>0 for each i (where
the subscript “> 0” indicates that we are excluding constants). Work of
Sullivan shows that minimal models are valid cofibrant replacements with
respect to the model category structure described above, and that any
simply connected cdga over C admits a minimal model (which is then nec-
essarily unique up to quasi-isomorphism).
2.11. 12/14/2020 (algebraic groups).
• The Lie algebra g associated to an algebraic group G defined over k is
defined to be the Zariski tangent space at the identity T1 G, with Lie bracket
given by the adjoint representation of g induced by the action of G on itself
by conjugation:
[x, y] := ad(x)(y).
From the functor of points perspective, for a given k-algebra R, we can
define g(R) explicitly as follows: write R[] := R[x]/(x2 ). Then R[] has
the structure of an augmented R-algebra via

R
r7→r+0
/ R[] a+b7→a
/ R.

This produces a corresponding sequence of group homomorphisms:

G(R)
i / G(R[]) π / G(R),

so that we may set

π

g(R) := ker G(R[]) −
→ G(R)
to consist of “tangent vectors of G(R) starting at 1 ∈ G(R).” For more on
this, see [10].
2.12. 12/15/2020 (formal groups and Lie algebras).
• (Variation on a theme) In a variety of contexts, one can fruitfully discuss a
correspondence between “spaces” and “groups” according to the following
blueprint:
 TODAY, I LEARNED... 11

differentiation /
Spaces o Groups.
exponentiation

The prototypical example comes from the classical Lie group-Lie algebra
correspondence, which is an actual equivalence of categories as long as we
restrict our attention to simply connected Lie groups, in which case the
above blueprint becomes:

LieGrp o
Lie / LieAlg.
exp

Notice that the Baker-Campbell-Hausdorff formula gives us an explicit way

to generate a group law from a Lie algebra, in that if exp(x) exp(y) =
exp(z), then we may write z explicitly as:
1 1

z = x + y + [x, y] + [x, [x, y]] + higher order brackets in x, y .
2 12
In AG over a field k of characteristic zero, one may play a similar game,
only replacing Lie groups by formal groups, according to the philosophy
that small neighborhoods have to be replaced by infinitesimal neighbor-
hoods. We get an analogous equivalence of categories, carrying over the
BCH formula “formally”:

FormalGrp o
Lie / LieAlg .
exp k

In the land of rational homotopy theory, Quillen tells us that there is an

equivalence of categories:

SQ≥2
' / dgLieAlg

between simply connected spaces up to rational homotopy equivalence and

dg Lie algebras (essentially Lie algebra objects in chain complexes). In the
land of classical homotopy theory, the loop space construction produces an
equivalence of categories on simply connected pointed spaces:

S∗≥1 o
Ωx
/ Grp (S)
E1
B

where the inverse equivalence is given by passing to the classifying space.

(To be further elucidated) This admits an analogue in formal DAG,

which provides a similar group-like characterization of formal moduli prob-
lems (thought of as infinitesimal moduli spaces in derived algebraic geom-
etry):
FMPk
Ω / GrpE (FMPk )
1

Lurie-Pridham take this a step further and identify the RHS with derived
Lie algebras by passing through the world of formal groups:

FMPk
' / GrpE (FMPk ) ' FormalGrpk ' / LieAlgk
Ω 1 Lie=T−,1
12 SAAD SLAOUI

2.13. 12/16/2020 (scheme theory).

• The forgetful functor
obliv
CAlgR −−−→ ModR
admits a left-adjoint given by the symmetric algebra V 7→ Sym∗R (V ). One
may then think of the assignment
M 7→ Spec(Sym∗R (M ))
as a “canonical” way of turning an R-module M into a space living over
Spec R. By Serre-Swan, for reasonable R, this procedure should produce
algebraic vector bundles out of finitely generated projective modules. Over
a more general base scheme S, one can make sense of a relative Spec con-
struction associating to any sheaf of algebras A ∈ CAlg(QCoh(S)) a scheme

SpecS (Sym∗OS (A))


S.
Under this assignment, locally free sheaves of finite rank E correspond to
algebraic vector bundles over S. Affine locally on some U = Spec R ⊆ S,
one retrieves the data of an R-module A such that Γ(U, A) = A, which we
require to be isomorphic to a polynomial algebra over R in the case where
A|U is free (corresponding to the usual local trivializability statement after
taking relative Spec).

2.14. 12/17/2020 (scheme theory).

• Oftentimes, adjectives appended to scheme morphisms (e.g. smooth, pro-
jective, affine) are ways of carrying over the property we understand for
a given variety (defined over a field) to the relative context over a general
base scheme, i.e. of making sense of the concept “in families”. For instance,
a smooth morphism may be defined as a (flat, locally of finite type) scheme
morphism such that each geometric fiber is smooth as a scheme over the
algebraic closure of the residue field at that point. Exhibit 2, following up
on the relative Spec construction above: an affine morphism X → S may
be characterized by the fact that one may find a sheaf of OS -algebras A
such that X ' SpecS (A) is given by a relative Spec (hence affine locally by
a classical Spec).
This philosophy fruitfully applies to arithmetic geometry, for instance in
the following context: start with a scheme X locally of finite type over
Spec Z (read: a family of polynomial equations in finitely many variables
defined over Z), and suppose that its base change to C (hence also to the
generic fiber Q) is smooth (in the usual Jacobian sense). Say that X has
bad reduction at a prime p ∈ Z if Xp is not smooth over Fp . Then one
may prove that X can only have bad reduction at finitely many primes
using a theorem of Grothendieck stating that the set of points of Spec Z at
which X → Spec Z is smooth is Zariski open, non-empty in our case since
it contains the generic point, and thus its complement is Zariski closed in
Spec Z hence finite.
 TODAY, I LEARNED... 13

2.15. 12/18/2020 (representation theory).

• (Following chapter VIII of [14]) Given a complex connected Lie group G,
there is a contravariant relationship between the size of π1 (G) and the size
of Z(G). This relates to the fact that a covering map between Lie groups

G̃
p

G

exhibits G as a quotient of G̃ by the normal discrete subgroup ker p ⊆ Z(G̃),

while the map p induces an inclusion p∗ : π1 (G̃) ,→ π1 (G) (hence G has a
smaller center and a larger fundamental group than G̃, and vice versa).
Now, suppose we start with a complex semisimple Lie algebra (g, h) with
coroot, resp. coweight lattice ΛR∨ ⊆ Λ∨ . If G is any complex connected
Lie group with maximal torus T such that Lie(G) = g and Lie(T ) = h,
then the kernel Γ(T ) of the exponential map exp : h → T always satisfies
ΛR∨ ⊆ Γ(T ) ⊆ Λ∨ . In fact, there is a 1-1 correspondence between com-
plex connected Lie groups G with Lie(G) = g (up to isomorphism) and
intermediary subgroups
ΛR∨ ⊆ Γ ⊆ Λ∨ .
The group corresponding to Γ = ΛR∨ is simply connected, and the group
corresponding to Γ = Λ∨ (called the adjoint group) has trivial center.
Dually (e.g the weight lattice is dual to the coroot lattice), one may also
work with intermediary subgroups
ΛR ⊆ X ∗ ⊆ Λ,
where ΛR ⊆ Λ denote the root resp. weight lattice of (g, h), and various
G’s produce various character groups X ∗ (T ) = Hom(T, C× ). For a given
G, the g-representations which induce group representations of G are then
those whose weights lie inside X ∗ (T ). This is always the case for simply
connected G, but may fail for certain representations in general.

2.16. 12/30/2020 (homotopical algebra).

• The notion of chain homotopy between chain maps can be related to the
traditional notion of homotopy between continuous maps as follows: define
I• to be the following chain complex:
(+−)
... → 0 → Z −−−→ Z ⊕ Z → 0 → ... ,
where the term Z ⊕ Z lives in degree 0. (This is equivalently the nor-
malized chain complex associated to the simplicial chains on the standard
1-simplex.)
Given a chain complex C• , one may consider the total chain complex
C• ⊗I• . The data of a chain homotopy between two chain maps f, g : C• →
D• may then be expressed as the data of a chain map
s : C• ⊗I• → D•
14 SAAD SLAOUI

with appropriate “boundary conditions” regarding f and g. More explic-

itly” in degree n, we have
(C• ⊗I• )n = Cn ⊕ Cn+1 ⊕ Cn ,
 
d 1 0
and the boundary map is given by the matrix 0 −d 0. In particular,
0 1 d
we have natural inclusion maps C• → C• ⊗I• . For s to be a chain homo-
topy between f and g, we then simply require that the following diagram
commute:
C•
f

 #
C• ⊗I• / D•
O ;
g

C•
The boundary map then gives rise to the traditional boundary condition
sd + ds = f − g.
This whole discussion should make sense in the context of model cate-
gories: C• ⊗I• should be a cylinder object associated to the chain complex
C• , with obvious structure maps
C• ⊕ C• → C• ⊗I• → C• ,
so that the commutativity condition becomes

C• ⊕ C•
f +g
/ D• .
:
s

C• ⊗I•

3. January 2021
3.1. 01/12/2021 (Goodwillie calculus).
• (Rêve éveillé) Goodwillie calculus starts off with the idea that categories
can be fruitfully thought of as “manifolds”, among which stable categories
correspond to linear manifolds, i.e. vector spaces - with the category ChR
of chain complexes of R-modules over a ring being the canonical example
thereof.
Following this analogy, one should be able to take the “tangent category”
to a category C at a given object X, encoding “infinitesimal directions” in
C away from X. Taking S to be the category of spaces and pt to be the
one point space, it turns out that
Tpt S = ChZ
is the category of chain complexes of abelian groups. Furthermore, given
any space X, the terminal map X → pt can be thought of as a “path” in
S , whose “derivative at time zero” outputs as “tangent vector” precisely
the chain complex H∗ (X) encoding the singular homology groups of X!
 TODAY, I LEARNED... 15

3.2. 01/19/2021 (classical Langlands, arithmetic topology).

• Weil’s Rosetta stone hints at a similarity between the features of three
classes of mathematical objects: number fields (alt. the space Spec OF
associated to the ring of integers of that number field), function fields (alt.
smooth projective curves over a finite field), and Riemann surfaces (alt.
their field of meromorphic functions).
When looking at things from the étale viewpoint, one is led to realize
that Fp feels more like a circle, in that it has étale fundamental group
π1ét (SpecFp ) = Gal(Fp /Fp ) = Ẑ
(generated by the Frobenius element). As such, function fields should really
correspond to Riemann surface bundles over S 1 , which are a special class
of 3-manifolds - the idea being that the Frobenius endomorphism should
provide gluing data with respect to the usual clutching construction for
bundles over S 1 . Similar thinking can lead to 3-manifolds associated to
number fields.
This marks the start of arithmetic topology, with ties to the classical
Langlands program. For instance, DBZ suggests that an instructive way
to think of the passage in the Langlands conjecture from a number field F
to the associated space V of “automorphic forms” is by considering the 3-
manifold M associated to F , then obtaining the vector space V by feeding
M into some 4 dimensional TQFT. Many features of automorphic forms
(such as conductors and singularities/ramification behavior along them)
can thereby be encoded fruitfully in the language of physics - in the sense
of TQFTs.

3.3. 01/25/2021 (HRR via DAG).

• Start life with a smooth projective variety X defined over a field of charac-
teristic zero k. One may look at the K-theory spectrum K(X) associated
to X, globalizing the algebraic K-theory of a ring and encompassing the
Grothendieck group K0 (X) of algebraic vector bundles on X, or alterna-
tively at K(C) where C = QCoh(X) (where the category C is compactly
generated by perfect complexes, i.e. bounded complexes with locally free
cohomology sheaves) on the one hand, and the Hochschild homology com-
plex HH(X) on the other hand, which globalizes the construction A⊗LA⊗A A
(a sort of derived self intersection). There is then a trace map relating the
two:
tr : K(C c ) → HH∗ (C),
where we interpret the k-vector space on the RHS as a spectrum, and where
C c denotes the subcategory of compact objects in C, so that K(C c ) = K(X)
by the above discussion.
Next, because our variety is smooth, the HKR theorem states that
M
HH∗ (X) ' ΩiX [i],
i

which in degree zero specializes to the statement

M
HH0 (X) ' H i (X, ΩiX ).
i
16 SAAD SLAOUI

It turns out that the composite of the trace map followed by the HKR
isomorphism recovers precisely the Chern character map on vector bun-
dles (interpreting Chern classes as Dolbeault cohomology classes of equal
bidegree).
Now, the trace map is functorial with respect to functors between com-
pactly generated categories which preserve compact objects. The particular
instance of such a functor which we care about is the pushforward func-
p
tor associated to the structure map X − → pt = Spec k, which preserves
compact objects because X is proper; we therefore end up with a commu-
tative square, which we can piece up with the HKR isomorphism to get the
following diagram (focusing on degree 0):

K0 (X)
tr / HH0 (X) ' / L H i (X, Ωi )
i X

χ p∗
  v pair with Td

Z = K0 (Vect)
incl / HH0 (pt) = k

Where the diagonal map is the top degree L trace map on cohomology
applied to the pairing of a given element of i H i (X, ΩiX ) with the Todd
(X, Ωn−i
L n−i
class of X, viewed as an element of iH X ). Commutativity
of this diagram together with an explicit interpretation of the inverse of
the HKR isomorphism therefore recovers the algebro-geometric Hirzebruch-
Riemann-Roch theorem: given an algebraic vector bundle E over X, one
has the relationship:
Z
χ(E) = ch(E)td(X).
X

3.4. 01/29/2021 (infty categories and chromatic homotopy theory).

• To say that an object obtained via a UP in classical category theory is
unique up to unique isomorphism is to say that the category of possible
choices forms a contractible groupoid, meaning a category such that there
exists a unique isomorphism between any two objects. This is the discrete
analogue of a contractible ∞-groupoid, which is the contractible choice of
spaces one obtains when formulating UPs in the ∞-categorical settings.
• (Based on lecture 1 of [7]) To a complex orientable cohomology theory E
with a choice of generator t ∈ E 2 (pt), so that E ∗ (CP ∞ ) = E ∗ (pt)[[t]], one
may associate a formal group law f E ∈ E ∗ (pt)[[u, v]] = E ∗ (CP ∞ × CP ∞ )
encoding the product rule satisfied by the corresponding theory of Chern
classes on line bundles:
0 0
cE E E
1 (L ⊗ L ) = f (c1 (L), c1 (L )).

This power series in two variables is subject to conditions reflecting the

monoidal structure of Pic (X). This structure in turn defines a group oper-
1
ation on the formal affine line A\ , which turns out to be independent
∗ E (pt)
of the choice of t up to “coordinate change”. It turns out that the as-
signment R 7→ FGL(R) ⊆ R[[x, y]] is corepresentable by a ring L called
the Lazard ring: formal group laws on R are in 1-1 correspondence with
 TODAY, I LEARNED... 17

ring maps L → R, or scheme-theoretically with affine scheme morphisms

Spec R → Spec L.
As such, we obtain an assignment:
 
cplx oriented cohlgy thies → formal groups
going from some kind of “spaces” (spectra) to some kind of “algebra”.
Chromatic homotopy teaches us that there is much to be learned from
this assignment, which may be more precisely formulated as mapping into
(algebro-geometric objects over) the moduli stack of formal groups MF G =
(Spec L)/G, where G takes into account the aforementioned coordinate
changes. For instance, the Landweber exact functor theorem states that any
flat morphism Spec R → Spec L (i.e. any affine scheme flat over Spec L) is
guaranteed to correspond to a formal group law coming from some complex
oriented cohomology theory - thereby providing us with a rich source of
possibly new cohomology theories. For instance, complex K-theory can
be obtained as the cohomology theory associated to the formal group law
f (u, v) = u + v + uv ∈ Z[[u, v]]. It turns out that not-necessarily-complex
orientable can also be made to fit into the picture as different types of
“algebraic geometry over (Spec L)/G, namely as quasi-coherent sheaves
over it. Whence the following:
Slogan: The structure of the stable homotopy category is controlled by
the geometry of the stack (Spec L)/G.

3.5. 01/31/2021 (formal group laws).

• Lazard’s theorem states that the Lazard ring is a polynomial ring in count-
ably many variables, L ∼ = Z[b1 , b2 , ...], so that is geometrically looks like an
“infinite dimensional affine space” Spec L ∼ = A∞ Z . By the UP of L, this iso-
morphism can be obtained by exhibiting an appropriate formal group law
F on the ring Z[b1 , b2 , ...], corresponding to a ring map L → Z[b1 , b2 , ...].
The actual proof involves working one degree at a time with the module of
indecomposables QL := I/I 2 , where I = L>0 , and showing that QL2n ∼ =Z
for every n. This leads to a nice reinterpretation of the problem in terms
of computing certain Ext groups of comodules - more on this later.

4. February 2021
4.1. 02/01/2021 (étale cohomology).
• Here is one way in which étale cohomology is a “better” algebraic invariant
for algebraic varieties than sheaf cohomology with respect to the Zariski
topology: for X an irreducible algebraic variety, one can directly show using
that Zariski opens in X are connected that H k (X, Λ) = 0 for any k ≥ 1
and any constant sheaf Λ (essentially because Λ itself is then flasque). On
the other hand, for a smooth variety X over C, one can show that we have
isomorphisms for every k ≥ 1 and every finite abelian group Λ:
k
Hét (X, Λ) ∼ k
= Hsing (X, Λ).
This follows from the fact that étale covers can be refined by complex
π
analytic coverings, in the sense that for any étale cover (U, u) −
→ (X, x)
18 SAAD SLAOUI

one can find a complex analytic neighborhood x ∈ V ⊂ X together with a

factorization of the corresponding inclusion map:

(U, u)
;
π

$
(V, x) / (X, x).
incl

This essentially corresponds to the idea that covering spaces are trivializable
in the analytic topology (so that we may map V into a “single sheet” of
the covering U over X).

4.2. 02/03/2021 ((co)bar resolution).

• Start life with a commutative k-algebra R and an R-module M . There is
a canonical way to construct a projective resolution of M , called the bar
construction, and defined as follows:

Bar• (M ) = (... → R⊗3 ⊗ M → R⊗2 ⊗ M → R ⊗ M )

(all tensor products are implicitly taken over k). The differential is given
in eg degree 2 by

d[r1 |r2 |r3 |m] := [r1 r2 |r3 |m] − [r1 |r2 r3 |m] + [r1 |r2 |r3 m].

There is an dual construction, called the cobar construction, exhibiting a

“universal” resolution of a comodule over a coalgebra by projective comod-
ules. The latter can be used to resolve a crucial lemma (the symmetric
2-cocycle lemma) along the way to proving Lazard’s theorem (see note
01/31/2021).
Notice the analogy with the bar-type construction one encounters when
trying to form the groupoid associated to a non-free group action of G on
X: this involves taking a directed colimit over the following sequence of
spaces (with simplicial structure maps mimicking the differential described
above):

→ →
(...→ →
→G × G × X →G × X).
→G × G × G × X →
→
For instance, the rightmost top map is given by [g1 |g2 |x] 7→ [g1 g2 |x], while
the rightmost bottom map is given by [g1 |g2 |x] 7→ [g1 |g2 x]. Note that for
free actions, the last two terms suffice to define the quotient space X/G.

4.3. 02/08/2021 (underlying interests).

• (flow of consciousness) Parametrized geometry, deformation theory, ho-
motopy theory: studying mathematical objects in families; studying the
structure underlying the dynamics of a deformation problem; studying the
essential shape underlying the various realizations/manifestations of an ob-
ject - the homotopy type/quasi-isomorphism class/motivic homotopy type.
Studying the theories that govern such phenomena, in a way that maintains
the original motivation visible.
 TODAY, I LEARNED... 19

4.4. 02/10/2021 (homotopical algebra).

• Recall that one may define the cotangent complex associated to a com-
mutative k-algebra A by first choosing a simplicial resolution by smooth
≈
algebras A• − → A and then defining LA to be the left derived functor of the
algebraic differential forms functor (A 7→ Ω1A/k ) applied to A• - explicitly,
letting Tot denote the “Dold-Kan complex” associated to a simplicial ring,
we set
LA := Tot([n] 7→ Ω1An ).
(This is an application of the blueprint of abstract homotopy theory, which
for a general model category dictates that we apply the functor at hand to
a cofibrant replacement.)
Let us begin to make precise the slogan that “the cotangent complex of
A encodes the deformation theory of A”. Let’s say that à is an infinitesimal
deformation of A if we have a cartesian square of the form:

k[ε]/(ε2 )
ε7→0 /k

 
à /A

(essentially because the cotangent complex classifies square-zero extensions

by its defining UP), corresponding to the algebro-geometric situation

X /X

 
Spec k / Spec k[ε]/(ε2 ).

It turns out that we have a 1-1 correspondence:

infinitesimal deformations of A ∼= Ext1A (LA , A),

with the higher Ext groups encoding whether it is possible to extend a given
deformation to a larger Artinian ring of the form k[ε]/(εn ).

This story admits a global analogue following Grothendieck and Illusie’s

formulation of a global cotangent complex (essentially by resolving the
structure sheaf by a sheaf of simplicial algebras A∗ → OX which locally
resolves OX by polynomial algebras), with a corresponding statement re-
lating the Ext groups Ext∗OX (LX , OX ) to deformations of X, in a way
which recovers the usual Kodaira-Spencer theorem as a special case - stat-
ing that for X smooth complex projective, deformations of X are encoded
by H 1 (X, TX ). In this case, LX is simply Ω1X concentrated in degree 0,
with Ω1X a vector bundle (locally free sheaf) of rank equal to the dimen-
sion of X, and the reduction to the Kodaira-Spencer cohomology group is
essentially given as follows:

Ext1OX (LX , OX ) ∼
= Ext1OX (OX , (Ω1X )∨ ) ∼
= H 1 (X, TX ),
20 SAAD SLAOUI

where we have used the general fact that for F, G vector bundles over X,
we have an identification:
ExtiOX (F, G) ' ExtiOX (OX , F ∗ ⊗ G).
(Induced from the usual hom-tensor adjunction and F ∨ ⊗ G ∼ = hom(F, G)
for i = 0.)
For more on this and a host of other exciting ideas, see Toën’s DAG
survey [18].

4.5. 02/16/2021 (Grothendieck fibration and algebraic de Rham com-

plex).
• Start with a category C. Grothendieck fibrations are a certain class of
F
functors E → C which are classified by functors C −
→ Cat, in the sense that
each fiber Fib(c), c ∈ C coincides with the designated category F c ∈ Cat.
Pictorially:
E


C
F / Cat
(Compare with the universality of maps X → BG in classifying principal
G-bundles over X.)
In ∞-category land, Lurie’s formalism allows us to establish an equivalence
of categories
Fun(C, Cat∞ ) ' coCart/C
between functors C → Cat∞ and so-called co-cartesian fibrations over C.
• Algebraic differential forms are actually quite friendly. Starting from the
observation that polynomials can be formally differentiated over any field,
one constructs (for a smooth affine variety V = Spec k[x1 , ..., xn ]/(f1 , ..., fm ))
the O(V )-module of algebraic 1-forms as follows:
Ω1V /k := O(V ) < dx1 , ..., dxn > /(df1 , ..., dfm ),
where we follow the classical settings of differential topology and set:
n
X ∂f
df := dxi .
j=1
∂xi

It turns out that the rank of Ω1V /k as an O(V ) module will always equal the
dimension of V in the smooth settings. Higher degree algebraic differential
forms can be constructed in the naive way, by setting ΩkV := Λk Ω1V . Pro-
ceeding naively further, we obtain an algebraic de Rham complex (Ω•V , d)
and corresponding de Rham cohomology groups:
i
HdR (V ) := H i (Ω•V , d).
In the non-affine case, one constructs a sheaf which locally looks like the
above - and one may most concisely obtain Ω1X as the pullback ∆∗ I/I 2
∆
under the diagonal morphism X −→ X × X. Everything still goes through,
 TODAY, I LEARNED... 21

with the exception that we need to take the hypercohomology of the result-
ing de Rham complex of sheaves (really, just pass to an injective resolution
in the appropriate derived category):
i
HdR (X) := Hi (Ω•X ).
One of the amazing results one can establish is that we have a period
isomorphism recovering singular cohomology groups for smooth projective
varieties over C:
H i (X) ∼
dR = H i (X(C), C).
sing

(And this even though the LHS involved purely algebraic data, while the
RHS involves the analytic structure of X: a GAGA-type surprise.)
Another lovely result is that one may recover de Rham cohomology as
defined above for not-necessarily affine smooth varieties as the abutement of
the so-called Hodge-to-de-Rham spectral sequence whose E1 page consists
of Dolbeault cohomology groups:
E1pq := H p (X, ΩqX ) =⇒ HdR
p+q
(X).
An excellent treatment of this story is given in [5].

4.6. 02/18/2021 (formal moduli problems, HKR theorems).

• (Variation on 12/15/20) The philosophy of formal moduli problems dictates
that we should always be able to find some dg Lie algebra encoding the
infinitesimal behavior of a given FMP. Begin with a scheme X, a point
x ∈ X, and consider the formal completion Xx∧ as a formal moduli problem.
Then the corresponding dg Lie algebra turns out to be the shifted tangent
complex Tx [−1] associated to the group (in an appropriate homotopical
sense) Ωx X. The Lie algebra structure on Tx [−1] may be obtained by
thinking about Atiyah classes, a formalization of the attempt to import an
analogue of Chern-Weil theory to the land of AG.
The underlying philosophy is the following: every space in AG can be
fruitfully thought of as X = BG for some “group” G, in such a way that
every sheaf on X can be interpreted as carrying an action of G. From
this perspective, Grothendieck-Riemann-Roch really provides a character
formula for G-representations in this setting, in a way which interpolates
between classical characters associated to representations of finite groups
(think: V → BG) and Chern characters associated to vector bundles (think:
E → X). More to come...

4.7. 02/22/2021 (Grothendieck duality).

f
• Start life with a reasonable morphism of schemes X − → Y (i.e. of finite type
between Noetherian schemes). Then at the level of the respective derived
categories of complexes of quasi-coherent cohomology sheaves, aside from
the usual adjunction Lf ∗ a Rf∗ , Grothendieck teaches us that we also
obtain an extraordinary pullback functor fitting in an adjunction Rf∗ a f × .
That is, we have the data of a natural isomorphism:
hom(Rf∗ A, B) ∼
= hom(A, f × B)
for any A ∈ Dq.c. (X), B ∈ Dq.c. (Y ).
22 SAAD SLAOUI

f
Let’s specialize to the case where X −
→ Spec k is a smooth and proper
scheme over a field k. Then one has that f × OSpec k = ωX = ΩnX [n] is
given by volume forms on X, concentrated in degree n = dim X. The
above adjunction applied to a quasi-coherent sheaf F over X and to OSpec k
translates to:
hom(Rf∗ F, OSpec k ) ∼
= hom(F, ΩnX [n]), a.k.a.
(Rf∗ F)∨ ∼
= hom(F, ΩnX [n]).
Taking (i − n)th cohomology, we recover the familiar statement of Serre du-
ality (remembering that shifting complexes results in shifts in cohomology
degree):
H n−i (X, F)∨ ∼
= Exti (F, ΩnX ).
In the more general case where X is no longer assumed to be smooth, but
only Cohen-Macaulay, we still recover the dualizing sheaf ωX concentrated
in degree n as f × OSpec k , and therefore obtain the more general form of
Serre duality:
H n−i (X, F)∨ ∼= Exti (F, ωX ).
As such, Grothendieck duality can be seen as a “maximal generalization”
of Serre duality in relative form and in not necessarily smooth or proper
context - in the general absolute case, f × OSpec k outputs the dualizing
complex ωX , which may no longer be concentrated in a single degree. For
an excellent treatment of this story, see [11].

4.8. 02/24/2021 (∞-category of spectra).

• (Ressenti) Elements of the ∞-category of spectra, analogously to derived
R-modules, admit a “minimal” characterization in terms of their homotopy
groups and k-invariants, where the latter encode the data of the maps of
spectra necessary to “reconstruct” a given spectrum “one homotopy group
at a time”, by means of iterated pushout diagrams of the form:
X≤(n+1) /0

 
X≤n
kn
/ Σn+2 πn+1 (X).

The trick is, while these kn maps are relatively straightforward to con-
struct in the case of chain complexes, maps of spectra can be much more
subtle (e.g. something as elementary as π∗ homSp (F2 , F2 ) already outputs
the Steenrod algebra A), and so while this construction is formally analo-
gous to the situation in DR , it may be of lesser use in actually pinpointing
the weak equivalence class of a given spectrum.

4.9. 02/24/2021 (Serre duality, proper morphisms).

• In its most elementary form, Serre duality posits the existence of an iso-
morphism, for any coherent sheaf F over a (smooth?) projective scheme X
over a field k,
hom(F, ωX ) ∼
= H n (X, F)∨ ,
 TODAY, I LEARNED... 23

where ωX denotes the dualizing sheaf of X. This isomorphism is obtained

from the data of a non-degenerate pairing given by the composite:
tr
hom(F, ωX ) ⊗ H n (X, F) → H n (X, ωX ) −
→k
where the first map is the “Yoneda pairing” applying the map given by the
first term to the given cohomology class of the second term, while the second
map is a trace map, to be thought of as the AG analogue of integrating a
top degree form over a compact manifold (with properness here playing the
role of compactness).
For dualizable F (e.g. F an algebraic vector bundle over X), we may
shuffle things around to formulate the above pairing in the form:
H 0 (X, F ∨ ⊗ ωX ) ⊗ H n (X, F) → k,
whose sibling pairings for different cohomological degrees recover the “usual”
form of Serre duality.
• A brisk walk through useful facts on proper morphisms: in terms of “cat-
egorical invariants”, proper morphisms are useful because proper pushfor-
ward is guaranteed to preserve coherent sheaves. Compare this to the non-
proper morphism A1k → Spec k, under whose pushforward the structure
sheaf is sent to the infinite dimensional vector space k[t] over k. In terms
of scheme adjectives, one can show that a map is proper and affine iff it
is a finite map, meaning that affine locally it consists of finite ring maps
- thus proper morphisms between affine schemes are rather uninteresting.
On the other hand, we get lots of interesting examples of proper morphisms
by first realizing that projective space is proper over its field of definition,
and that as a result projective schemes are guaranteed to be proper.
One quickly recovers a fundamental theorem in valuation theory in this
f
framework: given a morphism Pn − → Pm between projective spaces, cut out
by homogeneous polynomials f1 , ..., fm in terms of a choice of coordinates
on the domain, we are guaranteed that the map is proper because the
domain is proper and the target is separated, and as such it has closed
image. Thus the image of f may be described as the vanishing locus of
homogeneous polynomials g1 , ..., gs with coordinates in the target - as such,
we can systematically “eliminate the parameters” involved in defining f .
4.10. 02/26/2021 (∞-category of chain complexes).
• Here’s a “geodesic” way to form the ∞-category DR of derived R-modules,
starting from the category ChR of chain complexes of R-modules: view
ChR as enriched over chain complexes of abelian groups, by equipping each
homChR (M, N ) with a chain complex structure, essentially with R-linear
maps M → Σn N living in degree n. Now, apply the truncation functor
τ≥0 hom-wise, and follow this up with a hom-wise Dold-Kan construction,
resulting in an enrichment of ChR over simplicial abelian groups, hence
simplicial sets after applying a forgetful functor. This category may now
be ∞-localized at weak equivalences to obtain the derived ∞-category of
R-modules.
Notice that we seem to have “killed off” half of the information we started
with by applying a truncation functor. It turns out that this process is not
really destructive, a realization that crucially relies on the fact that DR is
24 SAAD SLAOUI

a stable ∞-category, and that as such the suspension/shift functor is an

autoequivalence, allowing us to “lift back up” negative information from
the dead - compare with the notion that spectra are “infinite loop-spaces”,
in that a given spectrum X may be presented as X = Ωn Σn X for any
n ≥ 0. (Story to be made more precise at a later time.)
More generally, for any pointed ∞-category C, one may form a category
Sp(C) of “spectrum objects” of C consisting of infinite loop objects in an
analogous sense to the above (with the classical case corresponding to C = S
and Sp(S) = Sp). The resulting category Sp(C) is a stable ∞-category (the
“stabilization of C”), and generalized Brown representability theorem char-
acterizes cohomology theories on C as being precisely classified by elements
of Sp(C). As such, the stabilization procedure feels like a way to pass from
a “category of spaces” to a category of “coefficients” with respect to which
to measure these spaces. This is part of section 1.4 in [6].

4.11. 02/27/2021 (derived Satake).

• Let’s try to reinterpret the bounded derived category of G(O)-equivariant
sheaves on the affine Grassmanian GrG in the special case where G = Gm ,
so that GrGm = C((t))/C[[t]] ∼= Z. As sheaves on an ind-scheme, they can
only be supported on a finite subset of points, hence
M
ShGm (GrGm ) ' Sh(pt/C∗ ).
i∈Z

Now, a theorem of Bernstein-Lunts gives us an equivalence component-wise:

M M
Sh(pt/C∗ ) ' ModfgC∗ (C∗ ) ,
i∈Z i∈Z

where C∗ (C ) = C[ε]/(ε ) denotes the cdga of cochains on C∗ . In DAG, the

∗ 2

latter category at each i may be reinterpreted as the category of coherent

sheaves on the affine derived scheme Spec Sym(C[1]). Remembering that
G∨ ∨
m = Gm so that g = g = C, we may further identify this category with
Coh(g [−1]) under the AG viewpoint on a vector space V as Spec Sym(V ∗ ).
∨

Thus
M
ShGm (GrGm ) ' Coh(g∨ [−1]) ' CohGm (g∨ [−1]),
i∈Z

under the usual correspondence between Z-grading and Gm -representation

structure. Finally, the HKR theorem enables us to identify, for a general
algebraic group G:
pt ×G pt = Ωpt G = LG ×G pt ' TG [−1] ×G pt = g[−1].
Therefore, we conclude with an identification:
ShGm (GrGm ) ' CohGm (pt ×Gm pt).
It turns out that this toy example leads to a sensible derived enhance-
ment of the classical geometric Satake equivalence, in the sense that by work
of Bezrukavnikov-Finkelberg, we have for any reductive group G an equiv-
alence of derived categories compatible with the geometric Satake equiva-
lence in the following sense:
 TODAY, I LEARNED... 25

ShG(O) (GrG )
' / CohG∨ (pt ×G∨ pt)
O O
π∗

PervG(O) (GrG ) / Rep(G∨ ) ' CohG∨ (pt).

'

Taking a stacky perspective on the RHS as

(pt ×G∨ pt)/G∨ ' LocG∨ (D tD∗ D),
i.e. as the moduli stack of local systems on the “ravioli” obtained from
gluing two formal disks D = Spec k[[t]] away from their closed point, and
likewise reinterpreting the space on the LHS as the moduli stack
G(O)\G(K)/G(O) = BunG (D tD∗ D)
of principal G-bundles on the ravioli, we see that this equivalence outputs a
sort of local geometric Langlands correspondence, in the form of an equiv-
alence of derived categories:
Sh(BunG (D tD∗ D) ' Coh(LocG∨ (D tD∗ D)).
L

4.12. 02/28/2021 (étale fundamental group).

• Start life with a smooth projective variety X over C. Then the Riemann
existence theorem leads to the data of an isomorphism:

π1ét (X) ∼ top

= π1 \ (X(C))
between the étale fundamental group of X and the profinite completion
of the topological fundamental group of its analytification. Together with
a base change theorem, this tells us that any smooth projective curve of
genus g over an algebraically closed field of characteristic zero admits 2g
topological generators (and something along similar lines can be said in
characteristic p can be said using a specialization argument).
This story, together with the existence of a SES for any variety X defined
over a subfield k ⊂ C:
1 / π1ét (XC ) / π1ét (Xk ) / Gal(k̄/k) / 1,

produces interesting arithmetic data of topological origin in the case where

X = P1Q − {0, 1, ∞} is the projective line over Q with three points removed:
in this case, the SES together with the base change theorem posit the
existence of a SES:
1 / (< γ0 , γ1 , γ∞ | γ0 γ1 γ∞ = 1 >)∧ / π1ét (P1 − {0, 1, ∞}) / Gal(Q̄/Q) / 1.
Q

Let H denote the profinite group on the LHS. Then the semi-direct product
structure on π1ét (P1Q − {0, 1, ∞}) corresponding to this group extension is
determined by the data of a map:
Gal(Q̄/Q) → Aut(H),
which upon linearization of the RHS produces a linear representation of the
absolute Galois group of Q.
26 SAAD SLAOUI

5. March 2021
5.1. 03/01/2021 (de Rham stack and D-modules).
• Given a scheme X, we may associate to it its de Rham stack XdR , given as
a functor of points by the formula
XdR := X(R/N (R))
for any R ∈ CAlg, where N (R) denotes the nilpotent radical of R. In spirit,
this is constructing XdR out of X by collapsing or canonically identifying
all infinitesimal neighborhoods in X (where infinitesimal is to be read in
the AG sense of nilpotency data). We attempt to begin making sense of
the analogy:
X : OX :: XdR : ΩX ,
where ΩX denotes the dg algebra of differential forms on X. The idea is
to view XdR as admitting as “underlying dg manifold” the shifted tangent
bundle TX [1], so that its structure sheaf (for smooth projective X?) should
be given by
OXdR = Sym(TX [1]∨ ) = Sym(Ω1X [−1]) = ΩX .
This leads to a nice categorification tower linking “linearizations of X” at
various categorical levels to “linearizations of X with added flatness condi-
tions” (with flatness here to be understood in the sense of flat connections):

Categorical level Linearization of X Flat linearization of X Linearization of XdR

1 OX ΩX OXdR
2 QCoh(X) DX -mod QCoh(XdR )
3 ShCat(X) CrysCat(X) ShCat(XdR )
Here, ShCat(X) denotes the 2-category of quasi-coherent sheaves of cate-
gories, which for reasonable X is equivalent to OX -mod-mod; resp. CrysCat(X)
is the “2-category of crystals of categories” (whatever this might mean in-
dependently of the identification with ShCat(XdR )). In particular, one may
think of DX -mod as “the sheaf theory one obtains from QCoh(X) by in-
sisting that the sheaves be equipped with a flat connection”, and such a
requirement is precisely encoded by the notion of a quasi-coherent sheaf
on the de Rham stack XdR ; furthermore, this behavior persists at different
categorical levels. This story is told in some more details in Safronov’s
shifted geometric quantization paper [13]
5.2. 03/11/2021 (Hochschild homology).
• Let A be an algebra over a field k. According to [16], Hochschild homology
should be thought of not as an invariant of A as a k-algebra, but rather
as an invariant of ModA as a k-linear dg-category - in those settings, one
may recover HH∗ (A) by taking the categorical trace of ModA viewed as a
dualizable object in dgcatk with dual ModAop . In this monoidal category,
the unit object is given by Vectk , endomorphisms of which are determined
by the image of the line k concentrated in degree zero. Furthermore, one
may show that the evaluation and coevaluation maps associated to ModA
compose to the unit endomorphism sending k to A⊗LA⊗Aop A. This is great,
because this complex precisely returns the Hochschild homology groups
 TODAY, I LEARNED... 27

HH∗ (A) upon passing to homology! As such, in analogy with the notion
of trace for dualizable elements of Vectk (namely finite-dimensional vector
spaces), we have realized one precise sense in which Hochschild homology
may be thought of as a categorified notion of dimension.

5.3. 03/12/2021 (Chern classes).

• There are at least five different approaches to defining Chern classes in al-
gebraic geometry, fitting into the paradigm that one is after an appropriate
collections of maps (for a given smooth projective variety X over C)
ci : K0 (X) → H∗ (X)
from the Grothendieck group of algebraic vector bundles on X to some
choice of algebro-geometric cohomology theory valued at X, or alternatively
to the de Rham or singular cohomology of X(C) viewed as a complex Kähler
manifold.
Start by fixing a rank n complex vector bundle E on X(C). In section 3.2
of [15], I describe the algebro-topological construction of the Chern classes
of E viewed as integral classes in singular cohomology, which consists of
first defining them for the universal bundle on BU (n), then pulling them
back via the classifying map corresponding to E in [X, BU (n)].
Next, Chern-Weil theory provides a second approach to defining Chern
classes as elements of the de Rham cohomology of X, as described in entry
2.3: one first equips E with some C ∞ connection : A0 (E) → A1 (E)

6. April 2021
6.1. 04/02/2021 (factorization homology).
• (First steps) Roughly speaking, factorization homology should be thought
of as a device enabling us to “integrate” any En -algebra A ∈ AlgB
En (V) over
a given n-manifold M , viewed as an element of the sym. mon. ∞-category
MfldB n , where B denotes a choice of “structure on the tangent bundle” (for
instance, B = pt gives framed manifolds). The output is an element
Z
A ∈ V.
M

One may define factorization homology concisely by means of a left Kan

extension. Namely, start with the full subcategory of disks DiskB
n , soRthat
A may be interpreted as a functor A : DiskBn → V, and simply define M
A
to be the image of M under the diagonal functor

DiskB
A /V
n =
R
 (−)
A

MfldB
n

which is the left Kan extension of A along the vertical inclusion, defined
explicitly pointwise (as for all left Kan extensions) as the colimit of the
28 SAAD SLAOUI

following composite of functors:

 
(DiskB
fgt
/ DiskB A /V .
R
M
A := colim n )/M n

This machine is wonderful. For the simple case M = S 1 , n = 1, and

V = Modk , so that A ∈ AlgE1 (Modk ) = Algk is a classical k-algebra, one
retrieves the Hochschild homology of A as a chain complex of k-modules:
Z
A = HH∗ (A).
S1
R
One may take S 1 A to be the definition of A for more general A. A beautiful
result further down the road states that for any perfect stack X, we have
an equivalence
Z
QCoh(X) ' O(Map(S 1 , X)),
M
1
where S denotes the simplicial circle viewed as a constant derived stack.
This statement therefore retrieves the DAG perspective on Hochschild ho-
mology as functions on the derived loop space LX = Map(S 1 , X) of X.
6.2. 04/02/2021 (complete Segal spaces).
• A (small) category may be encoded by the data of a simplicial set
X : ∆op → S et
satisfying the axiom that for every n, m ≥ 0, the diagram
Xn+m / Xn

 
Xm / X0
induced by the following maps in the simplex category

[m + n] o
k7→m+k
[n]
O O
k7→k 07→0

[m] o [0]
07→m

is a pullback square. This is encoding the fact that 1-categories admit

unique composition of composable sequences of morphisms. In these set-
tings, the data beyond n + m = 3 (associativity) is redundant. Notice
that the hom-spaces of X can be extracted by taking the pullback of the
following diagram:
homX (x, y) / pt

(x,y)
 
X1 / X0 × X0
In the context of simplicial spaces, i.e. functors of the form
X : ∆op → S
 TODAY, I LEARNED... 29

(where we may think of S as referring to sS et or Top up to localization),

one may state the Segal axiom as requiring that the above diagrams be
homotopy pullback squares. A model for ∞-categories is then obtained
by restricting one’s attention to complete Segal spaces, namely to simplicial
spaces satisfying the Segal axiom together with an additional condition
requiring that the (appropriately defined) map
pathX (x, y) → homX (x, y)inv
is an equivalence for every x, y ∈ X0 - this effectively formalizes the notion
that “invertible morphisms should correspond to paths between points” in
an ∞-category.

6.3. 04/06/2021 (Chern classes).

• Let X be a smooth projective algebraic variety over C, which may alter-
natively be viewed as a compact Kähler manifold X an by passing to its
analytification. Then there are several ways to define a theory of Chern
classes on X, beginning with an axiomatic characterization. Here is a di-
agram illustrating some of the possible approaches and how they relate to
one another:

2i
Hsing (X an ; Z) / H 2i (X an ; C) .
8 sing

'
 
K 0 (X)
ci
/ Ai (X) / H i,i (X)Z / H 2i (X an ; C)
O dR

'
&  
H i,i (X) /L H p,q (X)
p+q=2i

6.4. 04/07/2021 (genus, lifting criteria in AG).

• Let X be a smooth projective curve. Then Serre duality posits that there
is an isomorphism:
H 0 (X, Ω1X ) ' H 1 (X, OX )∨ .
This statement indicates that in these settings, arithmetic genus (which
is defined as the rank of H 1 (X, OX )) and geometric genus (defined as the
dimension of H 0 (X, Ω1X )) coincide. When X = V (f ) is given by a homo-
geneous polynomial of degree d, we further have that this genus is given by
the formula
(d − 1)(d − 2)
g(X) = .
2
• Let’s formalize the notion of “path lifting” in the context of algebraic geom-
etry. Here, paths should be interpreted as nilpotent thickenings, which in
the context of A-algebras for some commutative ring A can be expressed as
ring maps B → B/I where I ⊂ B is an ideal consisting of nilpotent elements
- to be thought of geometrically as the closed inclusion Spec B/I ,→ Spec B
of Spec B/I into an infinitesimal thickening of it.
30 SAAD SLAOUI

Now, let X be an A-scheme, thought of as a functor X : CAlgA → Set.

Then, given an infinitesimal thickening B → B/I in CAlgA , one may ask
for the corresponding map
X(B) → X(B/I)
to be:
– surjective, which is equivalent to requiring X to be a smooth A-
scheme;
– injective, which is equivalent to requiring the map X → Spec A to be
unramified;
– bijective, which is equivalent to requiring the map X → Spec A to be
étale.
Thus, étale A schemes are precisely those which admit unique path lifting in
the sense of algebraic geometry, pictorially corresponding to the existence
of a unique diagonal extension:

Spec B/I /
:X
cl

Spec B
This is the so-called nilpotent lifting condition.
Here is an interesting example from arithmetic geometry: suppose we
start with a scheme X smooth over Spec Z, thought of as an equation to
be solved over the integers. The existence of a solution mod p corresponds
to the existence of an Fp = Z/pZ point Spec Fp → X. Since X is smooth,
we are guaranteed that the solution mod p can be lifted to a solution mod
p2 , using the nilpotent lifting condition on the situation
/X
Spec Z/pZ :
cl

Spec Z/p2 Z

Iterating for increasing powers of p and passing to the limit, we actually

obtain the existence of a Zp -point Spec Zp → X. Thinking of Spec Zp as
a p-adic formal disk, this is telling us that for smooth schemes, pointwise
solutions can be lifted to solutions on “formal neighborhoods”. One may
then ask whether this “local-to-formal” extension can be “integrated out”
into a “local-to-global” extension, i.e. into an integer solution to the original
equation.

6.5. 04/11/2021 (categorical group actions).

• Given a group G, let BG denote the “delooping” category of G, i.e. the cate-
gory with one object (denoted by pt) and with morphism set endBG (pt) =
G. Then one may express the notion of an “object equipped with a G-
action” internally to any category C as the data of a functor

BG
X / C,
 TODAY, I LEARNED... 31

which is equivalent to the choice of an object X ∈ C together with a collec-

tion of endomorphisms g· ∈ endC (X) compatible with the group structure
on G.
In particular, one should be able to express the notion of “group action
on a category” as the data of a functor

BG
C / Cat.
The adequate notion may require a loosening of the notion of “equality”
between the composite of endofunctors (g·) ◦ (h·) and gh·, for instance as
the data of corresponding natural transformations.
6.6. 04/16/2021 (Milnor numbers, vanishing cycles, matrix factoriza-
tion).
• Start with a Henselian DVR S = Spec R with perfect residue field k and
fraction field K. (The running example will be Spec Zp , a Henselian DVR of
mixed characteristic. Think of an algebro-geometric formal neighborhood
of a point: this is one of the “smallest” settings in which to study families
of algebraic varieties, and we are particularly interested in “smooth degen-
eration to a singular fiber” over the closed point). Suppose we are given
a flat and proper morphism X → S, with X a regular scheme, such that
the generic fiber XK is smooth over K. Then Bloch’s conductor conjecture
states that we have an equality:
χ(Xk̄ ) − χ(XK̄ ) = [∆X , ∆X ]S + Sw(XK̄ ),
where Sw(Xeta ¯ ) is the Swan conductor, which carries arithmetic data about
XK̄ ; the Euler characteristics are taken with respect to `-adic cohomology;
and [∆X , ∆X ]S denotes a localized self-intersection number.

The above is to be thought of as a deformation theoretic/possibly sin-

gular generalization of the “`-adic Lefschetz fixed point formula”, which
states that for any smooth projective variety X over an algebraically closed
field, we have that
X
χ(X) = (−1)p+q hp,q (X) = [∆X , ∆X ],
p,q

where h p,q
(X) denotes the rank of the Dolbeault cohomology group H q (X, ΩpX ).

In their paper on the Bloch conductor formula[20], Toën-Vezzosi suggest

an approach via non-commutative geometry to a special class of cases (those
with “unipotent monodromy”) in which the Swan conductor vanishes. The
argument proceeds by building a dg category of matrix factorizations as-
sociated to a choice of uniformizer π for R, whose “Euler characteristic”
recovers the difference χ(Xk̄ ) − χ(XK̄ ). A general formalism of traces in
the nc-context then leads to the desired formula.

One source of motivation for considering categories of matrix factoriza-

tion as a reasonable approach to this problem comes from the notion of
Milnor number in differential topology. Given a function f : X → C with
an isolated critical point at 0, the Milnor number is a geometric invariant
32 SAAD SLAOUI

of the singular fiber f −1 (0). Near a point x ∈ f −1 (0), it may be computed

as the Euler characteristic of the cohomology ring H ∗ (Bδ (x) ∩ f −1 ()) for
small δ,  > 0. The latter cohomology groups admit an interpretation in
terms of nearby cycles of the constant sheaf Ψf C, and the Euler character-
istic may be alternatively obtained as the dimension of the Jacobian ring
of f , given by the quotient of the stalk of X at x by the ideal generated
by the partial derivatives of f at that point, or (appealingly) as the ring of
functions on the derived critical locus DCrit(f ) = Γ(df ) ∩ {0} ⊆ T ∗ X.

Now, start life with a complex manifold X together with a holomorphic

function (the “potential”) W : X → C. The Landau-Ginzburg model in
quantum field theory corresponds to a 2d TQFT LGB (via the “B-model”)
which assigns to a point the dg category of matrix factorizations MF(W ),
whose elements are given by complexes (P • , d) satisfying the relation d2 =
W ·. To start building some intuition for this category, notice that these
chain complexes are acyclic whenever W 0 6= 0, so that when interpreted as
“living over X” they are supported along the critical locus Crit(W ) ⊆ X.
As an extended TQFT, LGB takes the following values:
Categorical level Input Output
0 T2 Milnor number of (X, W )
1 S1 (Jacobian ring of W ) = ΦW C
2 pt MF(W )
Thus, M F (W ) appears as a “twice categorified” avatar of the Milnor num-
ber, which is built to capture invariants of singular fibers appearing as
degenerations of smooth fibers. Toën and Vezzosi’s main achievement was
to transport the intuition from the transcendental settings (over C) to the
mixed characteristic settings, developing the necessary framework at the
level of nc-schemes.

6.7. 04/21/2021 (topos theory).

• A meta-theorem in topos theory states that “any mathematical structure
which can be described by “geometric” axioms in a suitable language admits
a classifying topos” - for a more precise statement, see Chapter VIII of
[8]. For instance, there exists a classifying topos for ring objects in topoi.
Following the usual representability pattern, this consists of a topos R
together with a (universal) ring object R ∈ R such that, for any topos E,
f
geometric morphisms E − → R are in 1-1 correspondence with ring objects
in E via pullback f ∗ R ∈ E of the universal ring object:
homTopoi (E, R) ∼
= {ring objects OE ∈ E}.
This specific example is useful in formulating the analogue of “ringed spaces”
in these settings: a ringed topos is simply defined to be a topos X
equipped with a ring object OX ∈ X (thought of as the structure sheaf
of X ), or equivalently a topos X together with the data of a geometric
morphism X → R. When X = Sh(X) is the category of sheaves associated
to a Grothendieck site, OX being a ring object translates to the corre-
sponding sheaf being a sheaf of rings on X, as expected. This is part of the
general yoga that A-objects in a category of (pre)sheaves of sets correspond
 TODAY, I LEARNED... 33

to (pre)sheaves factoring through A, where A is an appropriate category

such as abelian groups or rings.

6.8. 04/22/2021 (TQFT, B-models).

• Suppose we are handed an n ≥ 2 dimensional TQFT Z, and let us evaluate
it on a codimension 2 sphere to obtain a category Z(S n−2 ). Then Z(S n−2 )
can always be equipped with a monoidal structure via the “little disks in
larger disks” picture, and because of the choice of co/dimension it turns out
that the resulting pair (Z(S n−2 ), ∗) is always symmetric monoidal. As such,
we may construct an algebro-geometric space out of it via the (categorified)
Spec construction:
M1Z := Spec (Z(S n−2 ), ∗)
This space allows us to formally associate a new TQFT in any dimension
m
m to Z, called the B-model associated to Z and denoted by BZ , according
to the blueprint
m
BZ (M m−k ) := kQCoh(Map(M, M1Z )).
Here, M is viewed as a homotopy type and interpreted as a constant derived
stack, placing it in the same world as M1Z ; Map implicitly refers to locally
constant maps; the terminology “kQCoh” refers to the k-category of sheaves
of (k − 1)-categories, a notion currently being formalized by Stefanich [17]
and possibly others. In low codimensions, we therefore obtain the following
outputs:
Categorical level Input Output
0 Mk O(Map(M, M1Z ))
1 N k−1 QCoh(Map(N, M1Z ))
2 P n−2 2QCoh(Map(P, M1Z ))
Of particular interest is the case m = n+1. In this case, it turns out that
the original TQFT Z may be viewed as a boundary condition for the
n+1
(n + 1)d TQFT BZ - this implies in particular that for any codimension
n−1
2 manifold N , the category Z(N n−2 ) sheafifies/spectrally decomposes
over the space Map(N, M1Z ).

6.9. 04/23/2021 (categorified HKR, Koszul duality).

• Start life with a perfect derived scheme X over a field of characteristic
zero. Then the derived HKR theorem states that there is an equivalence of
derived stacks:
LX ' TX[−1],
where the LHS denotes the derived free loop space of X, and the RHS
denotes the total space of the tangent complex of X, shifted by -1 - a space
which may alternatively presented as the relative Spec:
• ∗
Spec X (SymOX (TX [1])),

where T∗X [1] now denotes the cotangent complex shifted by 1, viewed as a
sheaf over X.
34 SAAD SLAOUI

In particular, for X a smooth scheme, we have that T∗X = Ω1X/k is the

sheaf of Kähler differentials, so that passing to functions yields the classical
version of the HKR isomorphism:
HH∗ (X) = Ω∗X/k
(using on the LHS that functions on the derived free loop space coincide
with Hochschild homology).

Now, it turns out that this story admits a categorification (in the sense
of a passage from functions to sheaves). Namely, it turns out that there
exists an equivalence of ∞-categories:
IndCoh(LX) ' Sym•OX (TX [−2])-mod.
This equivalence may be obtained as a special case of a more general result
which states that given any finite rank vector bundle E over a smooth
Noetherian scheme X, there exists an equivalence of ∞-categories:
IndCoh(E[−1]) ' Sym•OX (E[−2])-mod.
One retrieves categorified HKR at E = TX .
This result is to be thought of as a parametrized version/an incarnation
in families of the classical form of Koszul duality, which states that given a
finite dimensional vector space V over k, if we set SV := Sym•k (V [−2]) and
ΛV := Sym•k (V ∗ [1]), then there exists an equivalence of ∞-categories:
IndCoh(ΛV ) ' SV -mod.
This story, and much more, is beautifully explored in Rustam Antia’s thesis
[12]

6.10. 04/29/2021 (∞-categorical Yoneda embedding, homology theories).

• Let C be an ∞-category, and denote by Cˆ the ∞-category of presheaves on
C taking values in S = ∞-Gpd. Then there is an ∞-categorical version of
the Yoneda embedding, which posits the existence of a fully-faithful functor
h : C → Cˆ
exhibiting Cˆ as the completion of C with respect to all (small) colimits. This
means that, for any test ∞-category D, if we denote by hom! (C, ˆ D) the full
ˆ
∞-subcategory of hom(C, D) consisting of those functors which preserve
all (small) colimits, then restricting along the Yoneda embedding gives an
equivalence
ˆ D) − '
h∗ : hom (C,
! → hom(C, D).
This result leads as a special case to the Eilenberg-Steenrod characteri-
zation of (ordinary) homology theories as being determined by their value
at a point. Namely, we can take D = ∗ to be the one point ∞-category,
∗ ' S , and let D = DZ denote the derived ∞-category of (chain com-
with ˆ
plexes of) abelian groups. Now, one way to describe a homology theory is
as (the homotopy categorical truncation of) a colimit preserving functor
H : S → DZ .
 TODAY, I LEARNED... 35

from the ∞-category of spaces to DZ . Then, by the UP of the Yoneda

embedding, it follows that any such functor is determined up to equivalence
by its value at a point, H(∗) ∈ DZ .

7. May 2021
7.1. 05/02/2021 (Stable ∞-categories, AG).

• An ∞-category C is called stable if it has a zero object, admits finite

co/limits, and if any commutative square

X /Y

 
W /Z

is Cartesian iff it is coCartesian (i.e. it is a pullback square iff its is a

pushout square).

A key example is the stable infinity category of derived A-modules for

some commutative ring A, obtained as the Dwyer-Kan localization D(A) :=
L(ChA , q.iso). This notion expresses all of the properties of the 1-truncation
τ1 D(A) as a triangulated category “internally” to the theory, in that all of
the extra structures involved in a triangulated category appears as proper-
ties in the ∞-categorical settings. For instance, the distinguished triangles
are given precisely by the biCartesian squares, for any given morphism
f
X− → Y in D(A):

X /Y

 
0 / Y /X,

with the shift structure map given by further taking the suspension of X
on the RHS and invoking the UP of pushouts:

X /Y /0

  
0 / Y /X / ΣX,

Furthermore, the octahedral axiom is elucidated through the fact that given
a composite of morphisms X → Y → Z in a stable ∞-category, the central
36 SAAD SLAOUI

square in the following diagram:

X
f
/Y g
/Z /0

   
0 / Y /X / Z/X / ΣX

  
0 / Z/Y / ΣY

 
0 / ΣZ
is biCartesian.
• Start life with a commutative ring R (the story may also be told for R
a k-algebra, replacing Z by k throughout). Let’s make precise the idea
that “any element f ∈ R provides a function on Spec R”. By the UP of
polynomial rings, such an element f ∈ R is equivalent to the data of a ring
homomorphism Z[x] → R sending x to f . Passing to affine schemes, this is
the same data as a map
f
→ A1 .
Spec R −
Evaluation of f at a point of Spec R may also be interpreted geometri-
cally: any point p ∈ Spec R may be identified with the image of the map
Spec κ(p) → Spec R, where κ(p) = Frac(R/p) is the residue field of R at
p, and the map is opposite to the composite R  R/p → κ(p). Now, we
may consider the diagram
Spec κ(p) / Spec R

f
% 
A1 ,
which, upon passing to commutative rings, is equivalent data by the same
UP as above to an element f (p) ∈ κ(p): this is precisely the evaluation of
f at the point p, taking values in the appropriate residue field.

7.2. 05/11/2021 (Gelfand and Koszul).

• The slogan that “spaces may be studied via their ring of functions” may
be said to trace back to Gelfand’s representation theorem from the early
1940’s, stating the following: let X be a compact Hausdorff space, and
write C 0 (X) to denote the ring of continuous C-valued functions on X. If
mSpec(C 0 (X)) denotes the maximal spectrum of X (viewed as a topological
space by giving it the coarsest topology with respect to which the map
ϕf : mSpec(C 0 (X)) → C, m 7→ f¯ ∈ C 0 (X)/m ∼ = C is continuous for every
f ∈ C 0 (X)), then we have a homeomorphism:
∼
=
→ mSpec(C 0 (X))
X−
x 7→ mx := ker(evx ).
 TODAY, I LEARNED... 37

This assignment produces a fully faithful embedding

CpctHaus → BanachAlg

of the category of compact Hausdorff spaces into the category of Banach

C-algebras (by equipping each C 0 (X) with the sup norm), whose essential
image consists of commutative Banach algebras which admit an involu-
tion satisfying the axioms of a C ∗ -algebra. This suggests that it may be
reasonable to enlarge our notion of “space” by considering arbitrary, not
necessarily commutative Banach algebra, and exploring what kind of struc-
tures/theorems may be established in these broader settings. This turns
out to be a surprisingly fruitful philosophy referred to as “non-commutative
geometry”.
Suppose we set out to look for an analogue of Gelfand’s representation
theorem in the algebro-geometric settings, starting life with some k-scheme
X. From the get-go, the fact that e.g. O(Pnk ) ∼ = k indicates that trying to
study a variety via its ring of global functions will not do. This, together
with cohomological considerations, indicates that the appropriate lineariza-
tion procedure comes from considering the (bounded) derived category of
perfect complexes on X, denoted by D(X) in [19] - a k-linear dg category
which may be built out of derived categories of bounded complexes of f.g.
projective modules (denoted D(A) for some commutative k-algebra A) via
a (homotopy) limit operation taken in Catdg k with respect to an affine open
covering of X.
Now for the miracle: it can be proven that, as long as X is quasi-compact
and quasi-separated (read: AG-analogue of compact and Hausdorff), the
dg category D(X) admits a single compact generator EX ∈ D(X), i.e. a
compact object in D(X) with the property that the internal hom functor

hom(E, −) : D(X) → D(k)

is conservative. This is a situation in which Koszul duality may intervene.

Namely, let us consider the dg k-algebra BX := endD(X) (EX ). Then the
fact that EX is a compact generator guarantees that we obtain an equiva-
lence of dg categories:

D(X) ' D(BX ).

Thus, we may think of the dg algebra BX as an analogue of of C 0 (X) in

the AG context, leading to the following analogy:
Cpct Hausdorff X : C-algebra C 0 (X) :: qcqs X : dg k-algebra BX .
One should however keep in mind the caveat that the scheme X may not be
actually be recoverable from the data of the dg category D(BX ) - namely,
Morita equivalence is not an “injective” invariant of schemes. Nonetheless,
this perspective suggests that we enlarge the realm of algebraic geometry
by setting out to study general k-linear dg categories as “non-commutative
schemes” in their own right. This is the approach that Toën and Vezzosi
take towards a partial proof of the Bloch conductor conjecture, mentioned
in subsection 6.6.
38 SAAD SLAOUI

7.3. 05/13/2021 (deformation theory over a DVR).

• The situation considered by the Bloch conductor conjecture, as described
in subsection 6.6, appears naturally in various algebro-geometric contexts.
For instance, given a lft scheme X → Spec Z which is smooth over the
generic fiber Spec Q, we discussed in subsection 2.14 that X could only
possibly have bad reduction (i.e. fail to be smooth) at finitely many primes
p. Choose such a prime. “Zooming in” near p, by focusing on the formal
disk Spec Zp around Spec Fp ,→ Spec Z, we arrive to precisely a situation
where we have a Henselian DVR A = Zp of mixed characteristic over which
lives a family X → Spec Zp with smooth fiber Xt over the generic point
Spec Qp degenerating to a singular fiber X0 over the closed point Spec Fp .
Pictorially, by further base changing to the geometric fibers, we arrive to
the following diagram:

Xt /X o X0

  
Spec Q̄p / Spec Zp o Spec F̄p

7.4. 05/17/2021 (Balmer spectrum and Gelfand’s categorification).

• Let us state a more precise version of the algebro-geometric analogue of the
Gelfand representation theorem hinted at in subsection 7.2, issued from
Balmer’s survey on tensor triangulated (tt) categories in [9]. The spirit is
the following: it is possible to assign a so-called Balmer spectrum Spc T
to any tensor triangulated category T (i.e. T is a triangulated category
admitting a symmetric monoidal structure which is exact in each variable),
and in particular to the derived category Dperf (X) of perfect complexes as-
sociated to any scheme X: prime ideals in the classical settings are replaced
by prime thick-tensor (tt) ideals, and the basis for the closed subsets of the
Zariski topology is given by the subsets
supp(X) := {P ∈ Spc T | X ∈
/ P},
for X ranging over T (for instance, classical points feature as skyscraper
sheaves i∗ OSpec κ(x) ).
We then have the following theorem of Thomason, which provides a
concrete incarnation of the philosophy that categories of sheaves, as cat-
egorification of rings of functions, provide a more sensitive linearization
procedure for algebro-geometric objects:
Theorem 7.1. Let X be a qcqs scheme. Then there is a homeomorphism:
∼
=
→ Spc (Dperf (X))
|X| −
given by sending x ∈ X to the prime tt-ideal
P(x) = {Y ∈ Dperf (X) | Yx ∼
= 0}
given as the kernel of the residue functor Dperf (X) → Db (κ(x)).
(We should note however that this result only recovers X as a topological
space, and in particular it doesn’t account for the structure sheaf.)
 TODAY, I LEARNED... 39

7.5. 05/21/2021 (Euler and Riemann).

• Let X be a smooth projective curve defined over an algebraically closed
field k, and consider the category Coh(X) of coherent sheaves on X. This
category comes with two distinguished objects, the structure sheaf OX and
the dualizing sheaf ωX , and we may more generally consider the tensor
⊗n
powers ωX for n ∈ Z.
A canonical numerical invariant which can be associated to elements
F ∈ Coh(X) is their Euler characteristic, defined as:
X
χ(F) := (−1)i hi (X, F).

Using the Riemann-Roch theorem, we may compute that

⊗n ⊗n
χ(ωX ) = χ(OX ) + deg(ωX ) = (1 − g) + n(2g − 2).
In particular, the function
⊗−
χ(ωX ):Z→Z
is a linear polynomial in n, with coefficients depending only on the genus
of X. We may view it as the restriction to Z ⊆ C of the function of one
complex variable
f :C→C

s 7→ (2g − 2)s + 1 − g.
Now, observe that Serre duality gives rise to the following identity, for
each n ∈ Z:

⊗n ⊗n ⊗n
χ(ωX ) = h0 (ωX ) − h1 (ωX )
⊗−n+1 ⊗−n+1
= h1 (ωX ) − h0 (ωX )
⊗−n+1
= χ(ωX ).
This extends to a functional equation for f :
f (s) = −f (−s + 1).
Notice also that f has a unique zero at s = 1/2, and that the genus of g
may be recovered as the special value f (1) = g − 1.
Abstractly, working in the derived context and letting π : X → Spec k
denote the structure morphism, the dualizing sheaf of X could have been
obtained via exceptional pullback of the structure sheaf of the ground field,
ωX = π ! k, and the Euler characteristic could have been defined as the
following composite:
π∗
χ : Coh(X) −→ Coh(Spec k) ' Vectk → K0 (Vectk ) ∼
= Z.
Working with l-adic cohomology instead of coherent cohomology and
following the latter framework, this story can be carried out with respect to
a smooth algebraic variety X defined over F̄q so as to recover the Riemann
zeta function ζX associated to X, the central object of interest for the Weil
conjectures.
40 SAAD SLAOUI

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