Prerequisites of homological algebra › ... › Touze.pdf · 2012-04-10 · 1.2 -functors...

Prerequisites of homological algebra

Antoine Touze

April 10, 2012

Contents

1 Derived functors 21.1 The philosophy of derived functors . . . . . . . . . . . . . . . 21.2 δ-functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Injectives and projectives . . . . . . . . . . . . . . . . . . . . 51.4 Derived functors. . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Examples of derived functors 102.1 Tor and Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 (Co)homology of discrete groups . . . . . . . . . . . . . . . . 122.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 The standard complex . . . . . . . . . . . . . . . . . . 122.2.3 Cohomology algebras . . . . . . . . . . . . . . . . . . . 14

2.3 Cohomology of linear algebraic groups . . . . . . . . . . . . . 152.3.1 Linear algebraic groups and representations . . . . . . 152.3.2 Cohomology of algebraic groups . . . . . . . . . . . . 172.3.3 Algebraic groups versus discrete groups . . . . . . . . 172.3.4 Hochschild complex and cohomology algebras . . . . . 18

2.4 (Co)homology of categories . . . . . . . . . . . . . . . . . . . 192.4.1 From groups to categories . . . . . . . . . . . . . . . . 192.4.2 The structure of functor categories . . . . . . . . . . . 192.4.3 Homological algebra in functor categories . . . . . . . 202.4.4 Homology and cohomology of categories . . . . . . . . 21

3 Spectral sequences 233.1 A quick overview . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Bigraded spectral sequences . . . . . . . . . . . . . . . . . . . 243.3 A practical example . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Problem 1: does the spectral sequence stop ? . . . . . 263.3.2 Problem 2: differentials are not explicit. . . . . . . . . 273.3.3 Problem 3: extension problems. . . . . . . . . . . . . . 27

1

Prerequisites - Antoine Touze

3.4 Additional structure . . . . . . . . . . . . . . . . . . . . . . . 283.4.1 Spectral sequences of algebras . . . . . . . . . . . . . . 283.4.2 Naturality of spectral sequences . . . . . . . . . . . . . 29

4 Examples of spectral sequences 304.1 Filtered complexes . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Bicomplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Grothendieck spectral sequences . . . . . . . . . . . . . . . . 344.4 Filtered differential graded algebras . . . . . . . . . . . . . . . 354.5 The LHS spectral sequence . . . . . . . . . . . . . . . . . . . 36

5 Exercises 375.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1 Derived functors

1.1 The philosophy of derived functors

Let A be an abelian category, for example A is the category of modulesover a ring R (or an algebra R over a commutative ring k). To analyze thestructure of an object A ∈ A, we often cut it into smaller pieces: we finda subobject A′ ⊂ A, and we study the properties of A′ and of the quotientA/A′. Then we try to recover some information on A from the informationwe have on A′ and A/A′. The notion of short exact sequences stems fromthis philosophy. A short exact sequence is a diagram in A of the followingform

0→ A′f−→ A

g−→ A′′ → 0

such that (i) f is a monomorphism, (ii) the kernel of g is equal to the imageof f and (iii) g is an epimorphism.

The information we want to get on an object of A is often encoded bya functor from A to an other abelian category B. For example, A = Ab thecategory of abelian groups and we want to determine the p-torsion part ofan abelian group. In this case, F is the functor

Ab → Z/pZ-modA 7→ pA := a ∈ A ; pa = 0 .

A functor F : A → B is exact if it sends short exact sequences in A toshort exact sequences in B. If F is exact and if we know two of the objectsF (A′), F (A) and F (A′′), then we have good chances to recover the thirdone. Unfortunately, many interesting functors are not exact. For example,the p-torsion functor is only left exact, that is it sends a short exact sequenceis to a sequence

0→ pA′ pf−→ pA

pg−→ pA′′

2

1.2 δ-functors Prerequisites - Antoine Touze

where (i) pf is injective, (ii) the image of pf equals the kernel of pg (butpg is not surjective in general as one easily sees by applying the p-torsion

functor to the short exact sequence 0→ Z ×p−−→ Z→ Z/pZ→ 0).When a functor F is left exact we can (under slight hypotheses on the

category A which will be made more precise below) define its right derivedfunctors RiF , i ≥ 1, which are designed so that the left exact sequence

0→ F (A′)F (f)−−−→ F (A)

F (g)−−−→ F (A′′)

fits into a long exact sequence (i.e. the kernel of each map equals the imageof the preceding map)

0→ F (A′)F (f)−−−→ F (A)

F (g)−−−→ F (A′′) δ−→ R1F (A′)R1F (f)−−−−−→ R1F (A)

R1F (g)−−−−→ . . .

. . .δ−→ RiF (A′)

RiF (f)−−−−→ RiF (A)RiF (g)−−−−→ RiF (A′′) δ−→ . . .

In particular, the cokernel of F (g) equals the kernel of R1F (f).The derived functors RiF encode many properties of F , and it is often

hard to understand F without looking at all its right derived functors.There is a similar situation for right exact functors, that is functors

G : A → B which send short exact sequences 0 → A′ → A → A′′ → 0 toright exact sequences G(A′)→ G(A)→ G(A′′)→ 0. Namely, we can defineleft derived functors LiG, i ≥ 1 fitting into a long exact sequences

. . . L1G(A)→ L1G(A′′)→ G(A′)→ G(A)→ G(A′′)→ 0 .

1.2 δ-functors

Definition 1.1. Let A and B be abelian categories. A homological δ-functor is a family of functors Gi : A → B, i ≥ 0, together with connect-ing morphisms δn : Gn(A′′) → Gn−1(A′) defined for each exact sequence0→ A′ → A→ A′′ → 0 such that:

(1) For each exact sequence 0 → A′ → A → A′′ → 0 the diagram in Abelow is a long exact sequence (by convention, Gi = 0 if i < 0):

. . .δi+1−−→ Gi(A′)→ Gi(A)→ Gi(A′′)

δi−→ Gi−1(A′)→ . . . .

(2) For each morphism of exact sequences, that is for each triple (f ′, f, f ′′)such that the following diagram commutes:

0 // A′ //

f ′

A //

f

A′′ //

f ′′

0

0 // B′ // B // B′′ // 0

,

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1.2 δ-functors Prerequisites - Antoine Touze

we have a morphism of between the corresponding long exact se-quences, that is the following diagram commutes:

. . . // Gi(A′) //

Gi(f′)

Gi(A) //

Gi(f)

Gi(A′′) //

Gi(f′′)

Gi−1(A′)

Gi−1(f ′)

// . . .

. . . // Gi(B′) // Gi(B) // Gi(B′′) // Gi−1(B′) // . . .

.

A morphism of δ-functors from (Gi, δi)i≥0 to (Hi, δi)i≥0 is a family of naturaltransformations θi : Gi → Hi, i ≥ 0 such that for each short exact sequencein A, the θi induce a morphism between the associated long exact sequences.

The name ‘homological’ δ-functor comes from the fact that the connect-ing morphisms decrease the degrees by one.

If (Gi, δi)i≥0 is a homological δ-functor, the ‘end’ of the long exact se-quences

· · · → G0(A′)→ G0(A)→ G0(A′′)→ 0

imply that G0 is a right exact functor. Furthermore, for all i ≥ 0, theexactness in Gi(A) of these long exact sequences implies that the functorsGi are additive, that is the canonical inclusion morphisms A → A ⊕ Band B → A ⊕ B induce an isomorphism F (A) ⊕ F (B) ' F (A ⊕ B)1. Inparticular, the assignment (Gi, δi)i≥0 7→ G0 defines a functor:

homological δ-functorsfrom A to B

→

right exact functors

from A to B

.

The operation of deriving functors described below, will give a functor goingin the other way.

Definition 1.2. A cohomological δ-functor is a family of functors F i : A →B, i ≥ 0, equipped with connecting homomorphisms δi : F i(A′′)→ F i+1(A′)defined for each short exact sequence 0→ A′ → A→ A′′ → 0, such that wehave long exact sequences:

0→ F 0(A′)→ F 0(A)→ F 0(A′′) δ0−→ F 1(A′)→ . . .

and such that each morphism of short exact sequences induces a morphismbetween the corresponding long exact sequences.

If (Fi, δi)i≥0 is a cohomological δ-functor, the functors F i are automati-cally additive, and F 0 is a left exact functor. So the assignment (F i, δi)i≥0 7→G0 defines a functor:

cohomological δ-functorsfrom A to B

→

left exact functors

from A to B

.

1To prove this, look at the long exact sequence associated to the short exact sequence0→ A→ A⊕B → B → 0.

4

1.3 Injectives and projectives Prerequisites - Antoine Touze

1.3 Injectives and projectives

Definition 1.3. LetA be an abelian category. An object P ofA is projectiveif for all f : P → B and for all epimorphism p : A B, there exists amorphism f : P → A such that the following diagram commutes:

A

p

P

ef ??~~

~~ f // B

.

If A is the category of R-modules over a ring R (or over a k-algebraR), we can understand the projective objects as direct summands of freeR-modules. Notice that a direct summand of a free R-module need not bea free R-module in general. For example if R = Z/pZ × Z/pZ, the directsummand Z/pZ×0 of R is not a free R-module: it does not have enoughelements.

Lemma 1.4. Let M be an R-module. The following two statements areequivalent. (a) P is projective, and (b) P is a direct summand of a freeR-module.

Proof. Let us prove (a) ⇒ (b). Let π : F P be a surjection from a freeR-module F to P . Such a map always exists, take for example F = RP thefree R-module with basis P and if (bx)x∈P is the basis of RP define π on thebasis by π(bx) = x. Then the identity map P → P lifts to a map ι : P → Fsuch that p ι = IdP . In other words: P is a direct summand of F .

Now let us prove (b)⇒ (a). If P is a direct summand of a free R-moduleF , extend f : P → B into g : F → B by letting g = 0 on a direct summandof P . Using the basis of F we can define a liting g : F → A of g. Therestriction of g to P gives a lifting f : P → A of f .

In the course of the proof of lemma 1.4, we have actually proved thatfor a R-module A one can find a projective (and even free) R-module P andan epimorphism P A. Such a nice property is not always satisfied by anarbitrary abelian category, so we make a definition.

Definition 1.5. An abelian category A has enough projectives if for allA ∈ A, there is an epimorphism P A, where P is projective.

If A has enough projectives, one can build projective resolutions of ob-jects of A, that is chain complexes with projective objects Pi:

· · · → Pidi−→ Pi−1 → · · · → P1

d1−→ P0

which are exact in each Pi, i > 0 (i.e. the kernel of di−1 is the image of di),and such that the cokernel of d1 equals A.

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1.4 Derived functors. Prerequisites - Antoine Touze

Definition 1.6. Let A be an abelian category. An object J of A is injectiveif for all f : A → J and for all monomorphism p : A → B, there exists amorphism f : B → J such that the following diagram commutes:

B ef?

??

?

A?

OO

f // J

.

Remark 1.7. The definition of injective objects is the same as the definitionof projective objects but with the arrows reversed. Otherwise speaking, anobject J in injective in A if and only if it is projective in the oppositecategory Aop, where Aop is the category with the same objects as A, withHomAop(A,B) := HomA(B,A) and the composition in Aop is defined byf g := g A f , where A denotes the composition in A.

Definition 1.8. An abelian category A has enough injectives if for all Athere exists a monomorphism A → J with J injective.

If an abelian category has enough injectives, one can build injectiveresolutions J0 → J1 → . . . for every objectA ofA. Not all abelian categorieshave enough injectives. With a little work, one can prove:

Proposition 1.9. Let R be a k-algebra. The category of R-modules hasenough injectives.

Injective R-modules are somewhat more complicated than projectiveones. For example, the injective objects in abelian groups (i.e. in the cat-egory of Z-modules) are the direct summands of an arbitrary number of aproduct of copies of the abelian group Q/Z. In general, if R is a k-algebra,injective R-modules are direct summands of products of R-modules of theform Homk(R, J), where J is an injective k-module and the action of R onthis k-module is given by (rf)(s) := f(sr), cf. exercises 5.2 and 5.3.

1.4 Derived functors.

If (Gi, δi)i≥0 is a homological δ-functor, then G0 is a right exact functor. Thederivation process goes the other way: it enables to construct a homologicalδ-functor from a right exact functor G. For the sake of concreteness, webegin by recalling one of the most common examples of a right exact functor.Other examples are given in section 2.

Example 1.10. Let R be an algebra over a commutative ring k. If and Mis right R-module and N is a left R-module, the tensor product M ⊗R N isthe k-module generated by the symbols m ⊗ n, m ∈ M , n ∈ N , subject to

6


the following relations:

m⊗ n+m′ ⊗ n = (m+m′)⊗ n ,m⊗ n+m⊗ n′ = (m+m′)⊗ n ,λ(m⊗ n) = (mλ)⊗ n = m⊗ (λn) for λ ∈ k ,(mr)⊗ n = m⊗ (rn) for r ∈ R .

Tensor products yield a functor:

M⊗R : left R-modules → k-modulesN 7→ M ⊗R N

.

If M is a projective R-module then M⊗R is exact. (Indeed, by additivity,one reduces the proof of exactness to the case M = R. In the latter case,M⊗R sends a R-module to the underlying k-module, hence is exact). Butfor general M , M⊗R is only right exact. Similarly, tensor products yield aright exact functor (which is exact when N is projective)

⊗RN : right R-modules → k-modulesM 7→ M ⊗R N

.

Theorem 1.11. Let A and B be two abelian categories, and let G : A → Bbe a right exact functor. Assume that A has enough projectives. There existsa homological δ-functor (LiG, δi) : A → B such that:

(i) L0G = G,

(ii) for all i > 0, LiG(P ) = 0 if P is projective.

Such a δ-functor is unique up to isomorphism. Moreover, if G and G′ areright exact functors, a natural transformation θ : G → G′ extends uniquelyinto a morphism of homological δ-functors θi : LiG→ LiG

′ such that θ0 = θ.

Definition 1.12. The functors LiG, i ≥ 0 from theorem 1.11 are called theleft derived functors of G.

Before sketching the proof of theorem 1.11, let us give two examples.

Example 1.13. If G is exact, then (Gi, δi)i≥0 with G0 = G, Gi = 0 if i > 0,and δi = 0, is a homological δ-functor satisfying conditions (i) and (ii) oftheorem 1.11. Hence, by uniqueness we obtain: LiG = 0 for i > 0.

Example 1.14. Let G be the functor defined on abelian groups (or Z-modules) by

G(M) = Z/pZ⊗Z M = M/pM .

7


For all short exact sequence 0→M ′ →M →M ′′ → 0 we have a commuta-tive diagram where the vertical maps are multiplication by p:

0 // M ′ //

×p

M //

×p

M ′′ //

×p

0

0 // M ′ // M // M ′′ // 0

.

Hence by the snake lemma, we get a (not so) long exact sequence:

0→ pM′ → pM → pM

′′ δ−→M ′/pM ′ →M/pM →M ′′/pM ′′ → 0 .

Let (Gi, δi)i≥0 be the homological δ-functor with G0 = G, G1 = p− andGi = 0 for i ≥ 2, with connecting morphism δ1 provided by the snake lemma,and other δi zero. Then (Gi, δi)i≥0 satisfies the conditions of theorem 1.11.Hence by uniqueness L1G is the p-torsion functor and LiG = 0 for i ≥ 2.

Proof of theorem 1.11 (sketch). Extension of natural transformations followsfrom a ‘dimension shifting argument’. To be more specific, let θi : Gi → G′ibe a morphism of δ-functors between homological δ-functors satisfying con-ditions (i) and (ii). For all M ∈ A we can find a projective P and anepimorphism P M . Let K be the kernel of this epimorphism. The longexact sequences of Gi and G′i provide commutative diagrams:

0 // G1(M)

θ1M

δ1 // G0(KM )

θ0K

// . . .

0 // G1(M)δ′1 // G0(KM ) // . . .

, and Gi+1(M)

θi+1M

δi+1

'// Gi(M)

θiK

G′i+1(M)δ′i+1

'// Gi(M)

i ≥ 1.

So the natural transformations θi, i ≥ 0 are completely determined by θ0.Whence the uniqueness of extensions of natural transformations. For theexistence, one checks that taking θ1M as the restriction of θK to G1(M) andθi+1M = (δ′i+1)−1θiK δi+1 produces a well defined morphism of δ-functorsextending θ.

The uniqueness of (LiG, δi)i≥0 satisfying (i) and (ii) follows by a similarargument (with G = G′).

So it remains to construct the homological δ-functor (LiG, δi)i≥0. Thisis done by the following recipe. We first fix for each object M a projectiveresolution PM of M . Then LiG is defined on objects by:

LiG(M) = Hi(G(PM ))

To define LiG on morphisms, we use the following fundamental lemma,whose proof is an easy exercise on the definition of projective objects.

8


Lemma 1.15. Let M,N be objects of an abelian category A. Let PM andPN be projective resolutions of M and N in A. Then for all f : M → N ,there exists a chain map f : PM → PN such that H0(f) = f . Such a chainmap is unique up to homotopy2.

Each map f : M → N induces a chain map f : PM → PN , and wedefine LiG(f) as the map

Hi(G(f)) : Hi(G(PM ))→ Hi(G(PN )) .

Since f is unique up to homotopy and G is additive (as already noticed, rightexact functors are always additive), G(f is unique up to homotopy, hencedifferent choices of f induce the same morphism on homology. So Hi(G(f))is well defined. To finish the proof theorem 1.11 it remains to check thatthe functors LiG actually form a δ-functor. This verification is rather longso we refer the reader to the classical references [ML, W].

Derivation of functors has an analogue for left exact functors. Typicalleft exact functors are the ‘Hom-functors’

HomA(M,−) : A → Z-modN 7→ HomA(M,N)

,

HomA(−, N) : Aop → Z-modM 7→ HomA(M,N)

.

One checks from the definition of injective and projective objects that thefunctor HomA(−, N) is exact if and only if N is injective, and similarlythat HomA(M,−) is exact if and only if M is projective. If F : A → B isleft exact, the right derived functors RiF are the functors provided by thefollowing theorem.

Theorem 1.16. Let A and B be two abelian categories, and let F : A → Bbe a left exact functor. Assume that A has enough injectives. There existsa cohomological δ-functor (RiG, δi) : A → B such that:

(i) R0F = F ,

(ii) for all i > 0, RiF (J) = 0 if J is injective.

Such a δ-functor is unique up to isomorphism. Moreover, if F and F ′ areleft exact functors, a natural transformation θ : F → F ′ extends uniquelyinto a morphism of δ-functors θi : RiF → RiF ′ such that θ0 = θ.

2A homotopy between two morphisms of chain complexes f, g : C → D is a collectionof maps hi : Ci → Di+1 satisfying fi − gi = si−1 dC + dD si. Homotopic chain mapsinduce the same morphisms on homology.

9


Example 1.17. By the argument of example 1.13, if F is exact, its rightderived functors RiF are zero if i ≥ 0. Also, example 1.14 shows that theright derived functors of the p-torsion functor (which sends a Z-module Mto pM = a ∈ A ; pa = 0) are given by:

R1F (M) = Z/pZ⊗Z M = M/pM , RiF = 0 if i > 1.

The proof of theorem 1.16 is analogous to the proof of theorem 1.11, sowe won’t repeat the arguments. Let us just mention that the right derivedfunctors RiF are defined on objects by the formula:

RiF (M) = H i(F (JM )) ,

where JM is an injective resolution of M .

2 Examples of derived functors

2.1 Tor and Ext

Among the most common examples of derived functors are the functors Torand Ext. Basic references for this are [ML, III] and [W, Chap 3].

2.1.1 Tor

Let k be a commutative ring and let R be a k-algebra (if k = Z, a k-algebrais nothing but an ordinary ring). Let M be a right R-module and N be aleft R-module. We have right exact functors recalled in example 1.10:

M⊗R : left R-modules→ k-modules ,⊗R N : right R-modules→ k-modules .

The Tor-functors are defined by the following formulas (recall the the cate-gory of left or right R-modules has enough projectives so that left derivedfunctors are well defined).

TorRi (−, N) = Li(⊗RN) , TorRi (M,−) = Li(M⊗R) .

Remark 2.1. The notations for the Tor-functors could seem inappropriateat first sight, because the k-module TorRi (M,N) has two different meanings:on the one hand it is the value on M of Li(⊗RN) and on the other hand it isthe value of Li(M⊗R) on N . However, we shall see in example 4.6 that theycoincide. Thus, the k-module TorRi (M,N) can be indifferently be computedas the i-th homology of the complex PM ⊗RN or of the complex N ⊗R PN ,where PX denotes a projective resolution of X.

The name ‘Tor’ is the abreviation of ‘Torsion’, and it comes from thecase of abelian groups. Indeed, if M is an abelian group, we have seen inexample 1.14 that TorZ

i (Z/pZ,M) = 0 if i > 1 and TorZ1 (Z/pZ,M) = pM is

the p-torsion part of M .

10

2.1 Tor and Ext Prerequisites - Antoine Touze

2.1.2 Ext

Let A denote an abelian category. Assume that A is enriched over a com-mutative ring k (one also says that A is k-linear). This means that Homgroups in A are k-modules, and composition in A is k-bilinear. For example,the category of left (or right) modules over a k-algebra R are of that kind,but we shall see other examples below. Then for M , N in A we have leftexact functors:

HomA(M,−) : A → k-modules ,HomA(−, N) : Aop → k-modules .

If A has enough injectives, we define Ext-functors by the formula:

ExtiA(M,−) = Ri(HomA(M,−)) .

If Aop has enough injectives (this is equivalent to the fact that A has enoughprojectives by remark 1.7) then we define Ext-functors by the formula

ExtiA(M,−) = Ri(HomA(−, N)) .

Remark 2.2. Assume that A has enough injectives and enough projec-tives. Then, as in the case of Tor, the k-modules ExtiA(M,N) have twodifferent meanings but we can prove that they coincide. So the k-modulesExtiA(M,N) can be indifferently computed as the i-th homology of thecochain complex HomA(PM , N) where PM is a projective resolution of Mor as the i-th homology of the cochain complex HomA(M,JN ) where JN isan injective resolution of N .

The name Ext is the abreviation of ‘Extension groups’. If A,B areobjects of A, an extension of A by B in A is a short exact sequence 0 →B → C → A→ 0. Two extensions are isomorphic if we have a commutativediagram (in this case f is automatically an isomorphism by the five lemma):

0 // A // C //

f

B // 0

0 // A // C ′ // B // 0

An extension is trivial when it is isomorphic to 0→ A→ A⊕B → B → A.We let E(A,B) be the set of isomorphism classes of extensions of A by B,pointed by the class of trivial extensions. The following proposition is thereason for the name ‘Ext’.

Proposition 2.3. There is a bijection of pointed sets (on the left hand sideExt1A(A,B) is pointed by 0):

Ext1A(A,B) ' E(A,B) .

11

2.2 (Co)homology of discrete groups Prerequisites - Antoine Touze

Actually much more can be said: we can describe explicitly the abeliangroup structure in terms of operations on E(A,B), and higher Ext also havean interpretation. We refer the reader to [ML, III] for more details on thesetopics.

2.2 Homology and cohomology of discrete groups

As a basic reference for the (co)homology of groups, the reader can consultfor example [W, Chap 6], [ML, IV], [Br] or [Ben, Chap 2].

2.2.1 Definitions

Let G be a group, and let k be a commutative ring. A k-linear representationof G is a k-module M , equipped with an action of G by k-linear morphisms.A G-equivariant morphism f : M → N is a k-linear map which commuteswith the action of G, i.e. f(gm) = gf(m) for all m ∈M and all g ∈ G.

We denote by kG the group algebra over k. It is the free k modulewith basis (bg)g∈G, with product defined by bgbg′ := bgg′ . The unit of thek-algebra kG is the element 1 = be corresponding to the identity elemente ∈ G. A k-linear representation of G is the same as a left kG-module anda G-equivariant map is the same as a kG-linear map.

Let ktriv denote the trivial kG-module, that is ktriv is a free k-module ofrank one, acted on by G by gx = x. If M is a representation of G, the fixedpoints (or invariants) under the action of G is the k-module MG defined by:

MG = m ∈M , gm = m ∀g ∈ G = HomkG(ktriv,M).

The coinvariants is the k-module MG defined by:

MG = M/〈gm−m , m ∈M, g ∈ G〉 = ktriv ⊗kGM .

The (co)homology of G is defined by deriving the functor of (co)invariants.

H i(G,−) = Ri(−G) = ExtikG(ktriv,−) .

Hi(G,−) = Li(−G) = TorkGi (ktriv,−) .

Remark 2.4. We don’t mention the ring k in the notations H i(G,M) andHi(G,M). Since a kG-module is also a ZG-module, H i(G,M) could meanExtikG(ktriv,M) as well as ExtiZG(Ztriv,M). Fortunately, one can prove thatthe latter is endowed with a k-module structure, and that the two k-modulesare naturally isomorphic, see exercise 5.8.

2.2.2 The standard complex

There is a remarkable projective resolution of the trivial module ktriv in thecategory of kG-modules, called the bar resolution and denoted by B(kG).

12


The bar resolution is defined as follows. As a k-module, we have (tensorproducts are taken over k):

B(kG)n := kG⊗ (kG)⊗n .

An element x0 ⊗ · · · ⊗ xn ∈ B(kG) is denoted by x0[x1| . . . |xn], and anelement of B(kG)0 is denoted by x0[ ]. This handy notation with bars isresponsible for the name ‘bar resolution’. The action of kG on B(kG) isdefined for all x ∈ kG by the formula:

x · (x0[x1, . . . , xn]) := (xx0)[x1, . . . |xn] .

Hence B(kG)n is a free kG module (with basis the elements 1[bg1 | . . . |bgn ],for all (g1, . . . gn) ∈ G×n). The differential d : B(kG)n → B(kG)n−1 is thekG-linear map defined by

d(x0[x1, . . . , xn]) = x0x1[x2| . . . |xn]+∑n−1

i=1 (−1)ix0[x1| . . . |xixi+1| . . . |xn]+(−1)nx0[x1| . . . |xn−1]

.

A direct computation shows that B(kG) is actually a complex of left kG-modules. Moreover we have the following result.

Proposition 2.5. B(kG) is a projective resolution of ktriv in the categoryof left kG-modules.

Proof. We have already seen that B(kG) is a complex of projective kG-modules. So it remains to prove that Hi(B(kG)) = ktriv if i = 0 and zerootherwise. Let us consider ktriv as a complex of kG-modules concentratedin degree zero. The augmentation map

ε : kG = B(kG)0 → ktriv∑λgbg 7→

∑λg

induces a morphism of complexes of kG-modules B(kG) ε−→ ktriv. To proveproposition 2.5, it suffices to prove that ε is a quasi-isomorphism of com-plexes of kG-modules, i.e. that it induces an isomorphism of kG-moduleson the level of homology.

To prove this, we can forget the action of G, and prove that ε is a quasiisomorphism of complexes of k-modules. We are actually going to provethat ε is a homotopy equivalence of complexes of k-modules. The inverse ofε is the map η : ktriv → B(kG) which sends an element λ ∈ k to λ1 ∈ kG =B(kG)0 (Notice that η is not kG-linear, so it only is a morphism of complexesof k-modules). Then ε η = Id, and η ε is homotopic to the identity viathe homotopy h defined by h(x0[x1| . . . |xn]) = 1[x0|x1| . . . |xn].

13


Remark 2.6. Every left kG-module can be converted into a right kG-module. Specifically, the right action ·r of G on M is defined from the leftaction ·` of G on M by the formula m ·r g := g−1 ·` m. With this trick, wecan make B(kG) into a projective resolution of the trivial module ktriv inthe category of right kG-modules.

Thanks to proposition 2.5, the following complexes of k-modules com-pute the homology, resp. the cohomology, of a group G with coefficients ina kG-module M :

B(kG)⊗kGM , and HomkG(B(kG),M) .

We call them ‘standard complexes’. They are remarkable in the sense thatno secific information on M or on the structure of the group G is needed towrite them down explicitely (but of course, without any further informationon G and M , it is impossible to compute their homology). The cochaincomplex HomkG(B(kG),M) can be rewritten in a very nice way.

Proposition 2.7. The standard complex HomkG(B(kG),M) is isomorphicto the complex of k-modules C∗(G,M) defined by

Cn(G,A) = Map(G×n,M) ,

where Map(G×n,M) means the maps from the set G×n to the k-module M ,and G×0 should be understood as a set with one element, and with differential∂ : Cn(G,A)→ Cn+1(G,A) defined by:

(∂f)(g1, . . . , gn+1) := g1f(g2, . . . , gn+1)+∑n

i=1(−1)if(g1, . . . , gigi+1, . . . , gn+1)+(−1)n+1f(g2, . . . , gn)

.

Proof. There is an isomorphism of k-modules:

Map(G×n,M) ' HomkG(kG⊗ (kG)⊗n,M)

which sends a map f to the kG-linear morphism f defined by:

f(bg0 [bg1 | . . . |bgn ] := g0f(g1, . . . , gn) .

We check that this isomorphism is compatible with the differentials.

2.2.3 Cohomology algebras

If A is a kG-algebra, the cohomology H∗(G,A) is equipped with a so-calledcup product, which makes it into a graded k-algebra. Moreover, if A iscommutative, then H∗(G,A) is graded commutative, i.e. the cup productof homogeneous elements x and y satisfies

x ∪ y = (−1)deg(x) deg(y)y ∪ x .

14

2.3 Cohomology of linear algebraic groups Prerequisites - Antoine Touze

This cup product may be defined in many different ways (there are fourdifferent definitions in [ML]). We present here the definition relying on thestandard complex.

Let us denote by ·A the product of the kG algebra A. We define the cupproduct on the standard complex C∗(G,A) by:

Cn(G,A)× Cm(G,A) → Cm+n(G,A)f1, f2 7→ f1 ∪ f2

where

(f1 ∪ f2)(g1, . . . , gn+m) := f1(g1, . . . , gn) ·A [(g1 . . . gn)f2(gn+1, . . . , gn+m)] .

This products satisfies the following compatibility relation with the differ-ential ∂ of C∗(G,A):

∂(f1 ∪ f2) = (∂f1) ∪ f2 + (−1)deg(f1)f1 ∪ (∂f2) .

That is, (C∗(G,A),∪, ∂) is a differential graded algebra. In particular, itinduces a product on H∗(G,A) by the formula (where the brackets standfor the cohomology class represented by a cycle):

[f1] ∪ [f2] := [f1 ∪ f2] .

Remark 2.8. If A is commutative, the differential graded algebra C∗(G,A)is not graded commutative, so it is not clear with our definition whyH∗(G,A)should be graded commutative. One has to show that the difference f1 ∪f2 − (−1)deg(f1) deg(f2)f2 ∪ f1 is a boundary in C∗(G,A), which is left as anexercise to the courageous reader.

2.3 Cohomology of linear algebraic groups

The basic references for the theory of linear algebraic groups are [Bo],[H]and [S]. The reference for their cohomology is [J].

2.3.1 Linear algebraic groups and representations

Let k be an algebraically closed field. A linear algebraic group G over k isa subgroup of some Mn(k) which is Zariski closed in Mn(k), i.e. which isdefined as the zero set of a family of polynomials with coefficients in k andwith variables in the entries of the matrices [xi,j ] ∈Mn(k).

Example 2.9. Besides finite groups, we have the following common exam-

15


ples of algebraic groups:

SLn(k) = M ∈Mn(k) ; detM = 1 ,

GLn(k) =[∗ 00 M

]∈ SLn+1(k)

,

Ga =[

1 ∗0 1

]∈M2(k)

,

On(k) = M ∈Mn(k) ; M tM = I ,Spn(k) = M ∈M2n(k) ; M tΩM = I ,

where Ω is the matrix in the canonical basis (ei) of k2n of the bilinear formωn defined by ωn(ei, ej) = 1 if j = i + n, ωn(ei, ej) = −1 if i + n = j, andωn(ei, ej) is zero otherwise.

A rational (or algebraic) representation of an algebraic group G is a k-vector space V , equipped with an action ρ : G → Endk(V ), satisfying thefollowing condition.

(C) If V is finite dimensional, choose a basis of V and let ρk,`(g) be thecoordinates of ρ(g) in the corresponding basis of Endk(V ). Then werequire that the maps g 7→ ρk,`(g) are polynomials in the entries of thematrices [gij ] ∈ G.

(C’) If V is infinite dimensional, we require that V is the union of finitedimensional sub-representations satisfying (C).

Example 2.10. 1. If V is a k-vector space, the representation V triv (i.e.G acts trivially) is a rational representation.

2. If G ⊂ GLn(k), then kn acted on by G by matrix multiplcation is arational representation.

3. If G ⊂ GLn(k) is an algebraic group, let k[G] be the regular functionson G. That is, k[G] is the quotients of the algebra k[xi,j ]1≤i,j≤n by theideal of the polynomials vanishing on G. The action of G on k[G] byleft translations (gf)(x) := f(g−1x) is a rational action.

4. If V and W are rational representations of G, the diagonal action onV ⊗k W (that is g(v ⊗ w) := (gv) ⊗ (gw) makes it into a rationalrepresentation of G.

We denote by rat-G-mod the category whose objects are rational represen-tations of G and morphisms are G-equivariant maps. Thus, rat-G-mod is afull subcategory of kG-mod. It is not hard to check that direct sums, sub-representations and quotients of rational representations are rational repre-sentations, so rat-G-mod is actually a full abelian subcategory of kG-mod.

16


2.3.2 Cohomology of algebraic groups

As regards homological algebra, the category rat-G-mod of rational repre-sentations of G has the following properties [J, I,Chap 3].

Proposition 2.11. The category rat-G-mod is an abelian category withenough injectives. The injective kG-modules are (direct summands of) directsums of copies of k[G].

As in the case of discrete groups, there is a fixed point functor

−G : rat-G-mod → k-vectV 7→ V G = v ∈ V ; gv = v ∀g ∈ G

= Homrat-G-mod(ktriv, V )

Since the category rat-G-mod has enough injectives, the right derived func-tors of the fixed point functor are well defined and we call them the rationalcohomology of G (or simply the cohomology of the algebraic group G).

H irat(G,−) = Ri(−G) = Extirat-G-mod(ktriv,−) .

Remark 2.12. The category of rational G-modules has usually not enoughprojectives [J, I, section 3.18]. So in general we cannot derive the functor ofcoinvariants to define a notion of rational homology of an algebraic group.

2.3.3 Algebraic groups versus discrete groups

Let G be an algebraic group over k and let V be a rational representa-tion of G. We have two cohomology theories at our disposal, namely thecohomology of the algebraic group G with coefficients in V :

H irat(G,V ) = Extirat-G-mod(ktriv, V )

and the cohomology of the discrete group G with coefficients in V :

H i(G,V ) = ExtikG(ktriv, V ) .

These cohomologies are not the same. For example, the multiplicativegroup GL1(k) is semi-simple, so for all rational representation M of GL1(k),H i

rat(GL1(k),M) = 0 if i > 0, cf. exercise 5.10. On the other hand, ac-cording to proposition 2.3, H1(GL1(C),Ctriv) is in bijection with the iso-morphism classes of extensions of Ctriv by Ctriv in CGL1(C)-mod. So thefollowing example shows that H1(GL1(C),Ctriv) 6= 0.

Example 2.13. Let GL1(C) act on C2 by the action

ρ : GL1(C) → End(C2)

z 7→[

1 ln(|z|)0 1

].

17


This representation of GL1(C) is an extension of Ctriv by Ctriv, and it is nottrivial (if the extension was trivial then the matrix ρ(z) would be diagonal-izable, hence conjugate to the identity matrix, hence it would actually bethe identity matrix, which is obviously not the case).

2.3.4 Hochschild complex and cohomology algebras

There is a remarkable complex of k-vector spaces computing the rationalhomology H∗rat(G,V ) of an algebraic group with coefficients in V , called theHochschild complex.

This Hochschild complex is similar to the standard complex for discretegroups described in section 2.2.2, but in the framework of algebraic groups,functions from G×n to V are replaced by regular maps. Regular functionsfrom G×n to V are functions whose coordinates can be written as polyno-mials in the entries of the matrices ([g1

i,j ], . . . , [gni,j ]) ∈ G×n. Recall that k[G]

denotes the regular maps from G to k. There is an isomorphism of k-vectorspaces:

V ⊗ (k[G]⊗n) '

regular functions G×n → V.

Specifically, this isomorphism sends a tensor product v⊗f1⊗· · ·⊗fn to theregular map whose value on (g1, . . . , gn) equals f1(g1) . . . fn(gn)v.

We are now ready to define the Hochschild complex. It is the complexof k-modules C∗(G,M) with:

Cn(G,M) = M ⊗ (k[G]⊗n) ,

and the differential ∂ : Cn(G,M) → Cn+1(G,M) sends a regular functionf to the regular function ∂f defined by

(∂f)(g1, . . . , gn+1) := g1f(g2, . . . , gn+1)+∑n

i=1(−1)if(g1, . . . , gigi+1, . . . , gn+1)+(−1)n+1f(g2, . . . , gn)

.

As in the case of discrete groups, one can prove [J, I section 4.16]:

Proposition 2.14. The homology of C∗(G,M) is equal to H∗rat(G,M).

Let A be a rational kG-algebra. By this we mean that A is a k-algebra,equipped with a rational action of G by algebra automorphisms. Then therational cohomology of G with coefficients in A is equipped with a cupproduct. This cup product can be defined on the level of the Hochschildcomplex exactly as in the case of discrete groups. Namely, if ·A denotes themultiplication in the algebra A, we define the cup product of f1 ∈ Cn(G,A)and f2 ∈ Cm(G,A) by:

(f1 ∪ f2)(g1, . . . , gn+m) := f1(g1, . . . , gn) ·A [(g1 . . . gn)f2(gn+1, . . . , gn+m)] .

18

2.4 (Co)homology of categories Prerequisites - Antoine Touze

This product makes the Hochschild complex into a differential graded al-gebra, hence it induces a product on homology. As in the case of discretegroups, we can prove that if A is commutative, then the graded algebraH∗(G,A) is graded commutative.

2.4 Homology and cohomology of categories

2.4.1 From groups to categories

A group G can be thought of as a category with one object, say ∗, withHom(∗, ∗) = G and the composite of morphisms g and h is the product gh.Let us denote by ∗G the category corresponding to the group G. With thisdescription of a group, a k-linear representation of G is the same as a functor

F : ∗G → k-mod .

The k-module on which G acts is F (∗) and an element g ∈ G acts on F (∗)by the k-linear endomorphism F (g). Keeping this in mind, the followingdefinition is quite natural.

Definition 2.15. Let C be a small3 category and let k be a commutativering. A (left) k-linear representation of C is a functor F : C → k-mod.A morphism of representations θ : F → G is a natural transformation offunctors.

Since the (co)homology of groups was defined in terms of Tor and Extfunctors in the category of representations of G, the (co)homology of cate-gories should be similarly defined in terms of Tor and Ext functors in thecategory of representations of C. This motivates the study of functor cate-gories.

2.4.2 The structure of functor categories

Let C be a small category and let k be a commutative ring. We denote byFun(C, k-mod) the category4 whose objects are functors F : C → k-mod,and whose morphisms are natural transformations of functors.

Direct sums, products, kernels, cokernel, quotients of functors are definedin the target category. Specifically, this means that the direct sum of twofunctors F,G is the functor F ⊕ G defined by (F ⊕ G)(c) = F (c) ⊕ G(c)for all c ∈ C. Similarly, the kernel of a natural transformation θ : F → Gis defined by (ker θ)(c) := ker[θc : F (c) → G(c)] and so on. The followinglemma is an easy check.

3A small category is a category whose objects form a set. For example, the categoryof all sets is not a small category (cf. the classical paradox of the set of all sets).

4Actually we are cheating a bit here. It is a priori not clear that Fun(C, k-mod) is agenuine category. Indeed, it is not clear that the collection of natural transformationsbetween two functors form a set. However, this is the case when the source category C issmall, and this is exactly why we impose this condition on C.

19


Lemma 2.16. The category Fun(C, k-mod) is an abelian category.

Short exact sequences of functors are the diagrams of functors 0→ F ′ →F → F ′′ → 0 such that for all c ∈ C, evaluation on c yields a short exactsequence of k-modules:

0→ F ′(c)→ F (c)→ F ′′(c)→ 0 .

Since we consider functors with values in k-mod, a linear combina-tion of natural transformations is a natural transformation. So, the Hom-set in Fun(C, k-mod) are actually k-modules. Moreover, composition ofnatural transformations is bilinear. In other words, the functor categoryFun(C, k-mod) is enriched over k (we also say a k-linear category).

2.4.3 Homological algebra in functor categories

We want to do homological algebra in Fun(C,k-mod). So we have to deter-mine what the projective objects are, and if there are enough projectives.

We start by examining an elementary but important example of functor.Fix an element x ∈ C. The assignment c 7→ HomC(x, c) defines a functorfrom C to the category of sets. To obtain a functor from C to k-mod, we canmodify this functor in the following way. If X is a set, we denote by kX thefree k-module with basis (bx)x∈X indexed by the elements of X. We denoteby Px ∈ Fun(C,k-mod) the functor defined by

Px(c) := k HomC(x, c) .

The following fundamental lemma is proved exactly in the same fashion asthe traditional Yoneda lemma [MLCat].

Lemma 2.17 (k-linear Yoneda lemma). Let x ∈ C and let F ∈ Fun(C,k-mod).Let us denote by 1x ∈ kEndC(x) the basis element indexed by Idx ∈ EndC(x).The following map is an isomorphism of k-modules, natural with respect toF and x:

HomFun(C,k-mod)(Px, F ) → F (x)θ 7→ θx(1x)

.

Proof. The inverse map sends an element α ∈ F (x) to the natural transfor-mation α : Px → F defined by

αc : Px(c) = k HomC(x, c) → F (c)∑λfbf 7→

∑λfF (f)(α)

.

We leave to the reader the verification that the family α = (αc)c∈C actuallyis a natural transformation, and that the k-linear map α 7→ α is an inverseto the evaluation on 1x.

20


Let us spell out the consequences of the Yoneda lemma. First, for all F ∈Fun(C, k-mod) we can easily compute the k-module HomFun(C,k-mod)(Px, F ):it is isomorphic to the k-module F (x). The second important consequenceis the projectivity of the functors Px.

Lemma 2.18. Let x ∈ C. The functor Px is projective.

Proof. The projectivity of Px is equivalent to the exactness of the functor

HomFun(C,k-mod)(Px,−) : Fun(C,k-mod)→ k-mod .

By the Yoneda lemma, the latter functor is isomorphic to the functor F 7→F (x) which is exact by the definition of the notion of short exact sequencesof functors.

As a third consequence of the Yoneda lemma, the category Fun(C,k-mod)has enough projectives. To be more specific, the Px, x ∈ C form a projectivegenerator : this means that any functor F is a quotient of a direct sum ofsuch functors.

Lemma 2.19. The family of functors (Px)x∈C is a projective generator ofFun(C, k-mod).

Proof. By the Yoneda lemma, for all x ∈ C and all α ∈ F (x), there isa unique natural transformation α : Px → F which sends 1x ∈ Px(x) toα ∈ F (x). Taking the sum of all the α we obtain a surjective naturaltransformation:

⊕x∈C,α∈F (x) Px F .

The category Fun(C,k-mod) also has enough injectives, but as in thecase of the categories of R-modules, the injectives have a slightly more com-plicated form. An injective cogenerator is provided by the family of functorsIx,J , for x ∈ C and J an injective module of k-mod, where Ix,J is the functordefined by

Ix,J(c) := Homk (k HomC(c, x), J) .

2.4.4 Homology and cohomology of categories

Let C be a small category, let k be a commutative ring and let F : C → k-modbe a k-linear representation of C. Let us denote by k the constant functorwith value k. The cohomology of C with coefficients in F is the gradedk-module H∗(C, F ) defined by

H∗(C, F ) := Ext∗Fun(C,k-mod)(k, C) .

To define the homology H∗(C, F ) of C with coefficients in the repre-sentation F , we first need to introduce a generalization of tensor prod-ucts to the framework of functors. Let G ∈ Fun(Cop,k-mod), and let

21


F ∈ Fun(C, k-mod). The tensor product G ⊗C F is the k-module gener-ated by the symbols m ⊗ n, for m ∈ G(x) and n ∈ F (x), for all x ∈ C,subject to the relations:

m⊗ n+m′ ⊗ n = (m+m′)⊗ n ,m⊗ n+m⊗ n′ = (m+m′)⊗ n ,λ(m⊗ n) = (mλ)⊗ n = m⊗ (λn) for λ ∈ k ,(G(f)(m))⊗ n = m⊗ (F (f)(n)) for all morphisms f in C .

Tensor product yield right exact functors:

G⊗C : Fun(C, k-mod) → k-modF 7→ G⊗C F

⊗CF : Fun(Cop, k-mod) → k-modG 7→ G⊗C F

As in the case of modules over an algebra, the derived functors of thesefunctors are called Tor functors:

TorCi (G,−) = Li(G⊗C) , TorCi (−, F ) = Li(⊗CF ) .

As in the case of Tor in module categories, the notation TorCi (G,F ) has twointerpretations which coincide: it means Li(G⊗C)(F ) as well as Li(⊗CF )(G),cf. exercise 5.18. The homology of a small category F with coefficients in ak-linear representation F is the graded k-module H∗(C, F ) defined by

H∗(C, F ) := TorC(k, F ) .

22


3 Spectral sequences

Spectral sequences are a powerful tool to compute derived functors. Theycan be thought of as an optimal way to organize long exact sequences incomputations.

3.1 A quick overview

Let A be an abelian category, e.g. abelian groups, or modules over a ring.Spectral sequences appear in the following typical situation. We want tocompute a graded object in A, which we denote by K∗ (e.g. the singularhomology of a topological space, derived functors, etc.). We don’t know K∗but we can break it into smaller pieces which we understand. The spectralsequence is a kind of algorithm, which takes the pieces of K∗ as an input,and which computes K∗. Let us state some formal definitions.

Definition 3.1. Let r0 be a positive integer.A homological spectral sequence in A is a sequence of chain complexes

in A, (Er∗ , dr)r≥r0 , such that for all r ≥ r0, Er+1

∗ = H(Er∗).A cohomological spectral sequence in A is a sequence of cochain com-

plexes in A, (E∗r , dr)r≥r0 such that for all r ≥ r0, E∗r+1 = H(E∗r ).

In this section, we concentrate on homological spectral sequences. Weleave to the reader the translation for cohomological spectral sequences. Theterm Er∗ of a homological spectral sequence is called the r-th page of thespectral sequence, and dr is called the differential of the r-th page. Theterm Er0∗ is called the initial term of the spectral sequence.

In the sequel, we shall often omit to mention the category A in whichthe spectral sequence lives. Thus spectral sequences in A will simply calledspectral sequences, etc. This will cause no confusion since the category A istransparent in all the definitions, we only need the fact that it is abelian.

Definition 3.2. A spectral sequence (Er∗ , dr)r≥0 is convergent if for all k ∈

Z, there is an integer r(k) such that for r ≥ r(k) we have Erk = Er(k)k .

If a spectral sequence (Er∗ , dr)r≥0 is convergent, we define its E∞-page

as the ‘limit’ of the spectral sequence, i.e. E∞k := Er(k)k for all k ∈ Z.

Let K∗ be a graded object. We say that a spectral sequence (Er∗ , dr)r≥0

converges to K∗ if (i) the spectral sequence is convergent, and (ii) thereexists a bounded below filtration of each Kk:

0 = FmKk ⊂ · · · ⊂ Fp−1Kk ⊂ FpKk ⊂ · · · ⊂(⋃

p≥m FpKk

)= Kk ,

and a graded isomorphism: Gr(K∗) ' E∞∗ .

Typical theorems for spectral sequences assert the existence of a spectralsequence with an explicit initial term, say E2

∗ , converging to what we want

23

3.2 Bigraded spectral sequences Prerequisites - Antoine Touze

to compute: K∗. When such a spectral sequence exists, if we know theinitial page, we can run the algorithm provided by the spectral sequence(i.e. compute the successive homologies) to obtain the E∞-page, henceGr(K∗), hence K∗.

There are many other possible uses of spectral sequences. For example,we can use spectral sequences backwards: we know the abutment K∗ andwe want to compute the initial object E2

∗ , so we use the spectral sequence toobtain informations on E2

∗ . Spectral sequences can also be used to propagateproperties. For example, let us take the case of a spectral sequence ofabelian groups (Er∗ , d

r). Assume that the graded abelian group E2∗ is finitely

generated in each degree. Then all its subquotients are finitely generatedin each degree, so E∞∗ is finitely generated in each degree. Thus we deduce(under slight hypotheses on the filtration of K∗) that the abutment K∗ isalso finitely generated in each degree.

3.2 Bigraded spectral sequences

In practice, most spectral sequences are bigraded. In this section, we rewritethe definitions of the previous section in this more complicated setting.

Definition 3.3. A homological spectral sequence is a sequence (Er∗,∗, dr)r≥r0

of bigraded objects Er∗,∗, equipped with differentials dr of bidegree (−r, r −1)5, i.e. dr restricts to maps

dr : Erp,q → Erp−r,q+r−1 .

The terms of the spectral sequence are required to satisfy Er+1∗,∗ = H(Er∗,∗).

Let Erk =⊕

p+q=k Erp,q be the summand of total degree k of the r-th

page. Then the differential of the spectral sequence lowers the total degreeby 1. So bigraded spectral sequences are a refinement of definition 3.1.

Bigraded spectral sequences are often depicted by a diagram of the fol-lowing type (here we represent the second page, the summands E2

p,q of thesecond page are represented by dots, and the dots on the same dashed lines

5Other conventions on the bidegrees of the differentials are possible, but this conventionis the most common one.

24

3.2 Bigraded spectral sequences Prerequisites - Antoine Touze

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• • • • • • •

q OO E2p,q

Definition 3.4. A spectral sequence is convergent if for all p, q ∈ Z2 thereis an integer r(p, q) such that Erp,q = E

r(p,q)p,q for r ≥ r(p, q). For a convergent

spectral sequence, we define E∞p,q := Erp,q for r 0.

Bigraded spectral sequences converge to graded objects. Roughly speak-ing, a convergent bigraded spectral sequence converges to K∗ if for all k,Kk is filtered, and there is an isomorphism E∞k =

⊕p+q=k E

∞p,q ' Gr(Kk).

Thus, only the total degree in the spectral sequence is really meaningful torecover the abutment. The bidegree should be considered as a technical re-finement of the total degree, useful for intermediate computations6. Havingthis explained, we can now state the full definition of the convergence of aspectral sequence.

Definition 3.5. Let K∗ be a graded object. A spectral sequence (Er∗,∗, dr)

converges to K∗ if (i) it is convergent, and (ii) for all k there is a filtration :

0 = FmKk ⊂ · · · ⊂ Fp−1Kk ⊂ FpKk ⊂ · · · ⊂(⋃

p≥m FpKk

)= Kk ,

together with isomorphisms: E∞p,q ' FpKp+q/Fp−1Kp+q .

The convergence of a spectral sequence with initial page r0 is often writenin the symbolic way:

Er0p,q =⇒ Kp+q .

6In definition 3.5, the bigrading on E∞∗,∗ gives additional information on the isomor-phism E∞∗ ' Gr(K∗). But in many cases, this additional information is useless, so thereader can keep definition 3.2 in mind.

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3.3 A practical example Prerequisites - Antoine Touze

3.3 A practical example

The archetypal theorem for spectral sequences is the following theorem7.

Theorem 3.6 (Leray-Serre). Let f : X → B be a fibration between topolog-ical spaces. Assume that the base space B is simply connected, and denoteby F = f−1(b) the fiber over an element b ∈ B. There is a homologicalspectral sequence of abelian groups

E2p,q = Hp(B,Hq(F )) =⇒ Hp+q(X) .

In the statement, the spectral sequence is bigraded, and starts at thesecond page. The bidegree of the differential is not indicated (so the readershould think that the r-th differential dr has bidegree (−r, r − 1) as it ismost often the case). The summands E2

p,q are non zero only if p ≥ 0 andq ≥ 0: such a spectral sequence is called a first quadrant spectral sequence.

Thus, assume for example that we know H∗(B) and H∗(F ), hence thebigraded abelian group H∗(B,H∗(F )). Then we can use the Leray-Serrespectral sequence as an algorithm to compute H∗(X). Unfortunately, weimmediately encounter three practical problems.

3.3.1 Problem 1: does the spectral sequence stop ?

To compute the E∞-page of the Leray-Serre spectral sequence, we muststart from the initial page, compute its homology E3

p,q, then compute thehomology E4

p,q of the third page, and so on. Since the spectral sequence isconvergent, we know that for all index (p, q) the process stops at some point:Erp,q = E∞p,q for r big enough. But it may happen that the process does notstop for all the indices at the same time. This is a problem, because in thatcase we need infinitely many successive computations to compute completelyE∞∗,∗ (doing these computations would exceed a mathematician’s lifetime).

Definition 3.7. A spectral sequence (Er∗,∗, dr) collapses (or ‘stops’) at the

s-th page if for all r ≥ s, the differential dr is zero. (Thus E∞∗,∗ = Es∗,∗.)

In some cases, collapsing follows from the shape of the initial page ofthe spectral sequence. For example, assume that F and B are topologicalspaces with bounded singular homology. Then there exists integers p0 andq0 such that E2

p,q = 0 in the second page of the Leray-Serre spectral sequenceif p ≥ p0 or if q ≥ q0. Thus, if s := min(p0 + 1, q0), then for all r ≥ s eitherthe target or the source of dr is zero (the differentials dr are ‘too long’). Sothe spectral sequence stops at the s-th page. There are more elaboratedways to prove that spectral sequences stop, cf. exercise 5.15.

7In this theorem, the letter H refers to the singular homology of a topological space.The reader unfamiliar with algebraic topology can think of a fibration X → B with fiberF as a ‘topological space X which is an extension of B by F ’. The hypothesis that B issimply connected is not necessary, but in the general case, the statement is slightly morecomplicated.

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3.3 A practical example Prerequisites - Antoine Touze

3.3.2 Problem 2: differentials are not explicit.

Theorem 3.6 asserts that the spectral sequence exist, but does not give anyformula for the differential! So, even if we know what the second page is, itis not possible to compute the third page in general. It is the main andrecurrent problem for spectral sequences.

To bypass this problem, one can use additional algebraic structure onthe spectral sequence (see section 3.4). Alternatively, some differentials,or some algebraic maps related to the spectral sequence might have a ge-ometric interpretation, which can be used to compute them. For example,the full version of the Leray-Serre spectral sequence comprises the followingstatement in addition to theorem 3.6.

Theorem 3.8 (Leray-Serre - continued). If eB : E∞p,0 → E2p,0 denotes the

canonical inclusion, the composite

Hp(X) FpHp(X)/Fp−1Hp(X) ' E∞p,0eB−→ E2

p,0 ' Hp(B,H0(F )) ' Hp(B)

is equal to the map Hp(f) : Hp(X)→ Hp(B). Similarly, if eF : E20,q E∞0,q

denotes the canonical surjection, the composite

Hq(F ) ' H0(B,Hq(F )) ' E20,q

eF−→ E∞0,q ' F0Hn(X) → Hn(X)

equals the map Hn(F )→ Hn(E) induced by the inclusion F ⊂ X.

The composites appearing in the second part of theorem 3.8 are calledthe edge maps. Edge maps are a common feature of first quadrant spectralsequences, and the ‘geometric’ interpretation of edge maps can sometimesbe used to compute some differentials. For example, let us assume that fhas a section, i.e. there exists a map s : B → X such that f s = IdB.Then H∗(f) (hence eB) is onto. Thus, eB is an isomorphism. Therefore, allthe elements of E2

∗,0 survive to the E∞ page. That is, all the differentialsstarting from E2

∗,0 must be zero.

3.3.3 Problem 3: extension problems.

Assume now that we have (finally) succeeded in computing E∞∗,∗. We havenot finished yet! Indeed the E∞-page of the spectral sequence is isomorphicto Gr(H∗(X)), not to H∗(X). This can make a big difference. For example,if E∞k equals Z/2Z⊕Z/2Z, then Hk(X) could equal Z/2Z⊕Z/2Z or Z/4Z.

In the case of spectral sequences of k-vector spaces (or more generallyif the E∞-page is a bigraded projective k-module) this problem vanishesthanks to the following lemma.

Lemma 3.9. Let M be a k-module, equipped with a filtration

0 = FmM ⊂ · · · ⊂ Fp−1M ⊂ FpM ⊂ · · · ⊂(⋃

p≥m FpM)

= M .

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3.4 Additional structure Prerequisites - Antoine Touze

Assume that Gr(M) is a projective k-module. Then there is an isomorphismof k-modules M ' Gr(M).

Proof. Let 0 → M ′ → M → M ′′ → 0 be a short exact sequence of k-modules. If M ′′ is projective, we can find a section of the map M → M ′′

so M splits as a direct sum: M ' M ′ ⊕M ′′. We build the isomorphismM ' Gr(M) by iterative uses of this fact.

To solve extension problems in the general case, we often use additionalstructure on the spectral sequence, or additional information on the abut-ment H∗(X), obtained independently from the spectral sequence.

3.4 Additional structure on spectral sequences

Many spectral sequences bear more structure than what is stated in def-inition 3.3. This additional structure is usually of great help in effectivecomputations. In this section we present two properties among the mostfrequent and useful ones, namely algebra structures (the pages of the spec-tral sequences are bigraded algebras), and naturality (the spectral sequencedepends fonctorially on the object it is built from).

3.4.1 Spectral sequences of algebras

Definition 3.10. Let (Er∗,∗, dr) be a spectral sequence of R-modules, con-

verging to K∗. We say that it is a spectral sequences of algebras if thefollowing conditions are satisfied.

(1) For all r there is a bigraded product Erp1,q1 ⊗R Erp2,q2 → Erp1+p2,q2+q2 ,

satisfying a leibniz relation

dr(x1 · x2) = dr(x1) · x2 + (−1)p1+q1x1 · dr(x2) for xi ∈ Erpi,qi .

In particular H(Er∗,∗) is a bigraded algebra.

(2) The isomorphism H(Er∗,∗) ' Er+1∗,∗ is compatible with products.

(3) The abutment K∗ is a filtered graded algebra and the isomorphismE∞∗ ' Gr(K∗) is an isomorphism of graded algebras.

Spectral sequences of algebras are interesting for many purposes. First,one might very well be interested in computing the graded R-algebra K∗rather than the graded R-module K∗. In that case, we need a spectralsequence of algebras to do the job. But even if we are not interested inthe algebra structure of the abutment, the algebra structure on a spectralsequence (Er∗,∗, d

r) is an extremely useful tool to compute the differentials.Indeed, by the leibniz rule, it suffices to determine dr(x) for generators x ofEr∗,∗ to completely determine dr on Er∗,∗! Examples of spectral sequences ofalgebras are given in section 4.

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3.4 Additional structure Prerequisites - Antoine Touze

3.4.2 Naturality of spectral sequences

Let A be an abelian category. To define the category of homological spectralsequences in A, we need to define morphisms of spectral sequences.

Definition 3.11. Let (Er∗,∗, dr) and (E′r∗,∗, d

′r) be two spectral sequencesconverging to K∗ and K ′∗ respectively. A morphism of spectral sequencesis a sequence of bigraded maps f r : Er∗,∗ → E′r∗,∗, which commute withthe differentials: d′r(f(x)) = f(dr(x)), and such that H(f r) = f r+1. (Inparticular, the morphism induces a map f∞ : E∞∗,∗ → E′∞∗,∗.)

From a practical point of view, morphisms of spectral sequences f r :Er∗,∗ → E′r∗,∗ are very useful for explicit computations. Indeed, if the dif-ferentials in the spectral sequence Er∗,∗ are known, then one can use f r toprove that d′r(x) = 0, for x in the image of f r, or to prove that some somex in the image of f r are boundaries.

In practice, spectral sequences usually come from a functor

E : C → Spectral sequences ,

where C is a given category. One also says that the spectral sequence isnatural with respect to the objects C ∈ C. In this situation, morphisms inC induce maps of spectral sequences. The Leray-Serre spectral sequence isa typical example (C is the category of fibrations).

Theorem 3.12 (Leray-Serre - continued). The Leray-Serre spectral se-quence is natural with respect to the fibration f .

The precise meaning of theorem 3.12 is the following. If (gB, gX , gF ) isa morphism of fibrations, i.e. there is a commutative diagram:

F _

gF // F ′ _

X

f

gX // X ′

f ′

B

gB // B′

,

then (gB, gX , gF ) induces a morphism gr : Er∗,∗ → E′r∗,∗ of spectral sequencesbetween the two Leray-Serre spectral sequences. On the level of the secondpage, g2 coincides with the morphism H∗(gB, H∗(gF )). The map H∗(gX)preserves the filtrations on the abutments and Gr(H∗(gX)) equals the mapg∞ : E∞∗,∗ → E′∞∗,∗.

29


4 Examples of spectral sequences

In this section we give some of the most common examples of spectral se-quences. We shall not explain in details how these spectral sequences areconstructed, since their construction is quite involved and usually uselessfor practical computations. We refer the reader to [W, Chap. 5], [ML, XI],[Ben, Chap. 3] or [MC] for more information on their construction.

4.1 Filtered complexes

Let C be a chain complex in an abelian category A. A filtration of C is afamily of subcomplexes · · · ⊂ FpC ⊂ Fp+1C ⊂ . . . of C. The filtration isbounded below if for all k, there is an integer pk such that FpkCk = 0. It isexhaustive if

⋃p∈Z FpC = C.

If C is a filtered complex, we denote by GrpC the quotient complex:GrpC = FpC/Fp−1C. The filtration of C induces a filtration on its homology,defined by

FpH∗(C) := Im(H∗(FpC)→ H∗(C)

).

The following theorem provides a spectral sequence which recovers the ho-mology of C from the homology of the complexes GrpC.

Theorem 4.1. Let C be a filtered chain complex, whose filtration is boundedbelow and exhaustive. There is a homological spectral sequence

E1p,q(C) = Hp+q(GrpC) =⇒ Hp+q(C) .

More explicitly, there are isomorphisms

E∞p,q(C) ' FpHp+q(C)/Fp−1Hp+q(C) .

If f : C → C ′ is a chain map preserving the filtrations (i.e. it sends FpCinto FpC ′), then f induces a morphism of spectral sequences with E1

p,q(f) =Hp+q(Grpf), and E∞p,q(f) coincides on the abutment with GrpHp+q(f).

Example 4.2. Let us test theorem 4.1 on the simplest example. Let 0 →C ′ → C → C ′′ → 0 be a short exact sequence of chain complexes. This shortexact sequence is equivalent to saying that C is filtered with F0C = C ′ and

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4.1 Filtered complexes Prerequisites - Antoine Touze

F1C = C. Thus we have a spectral sequence whose first page is of the form:

0 H2(C ′)· · ·H3(C ′′)d1oo 0 0

0 H1(C ′) H2(C ′′)d1oo 0 0

0p//H0(C ′)

· · ·H1(C ′′)d1oo 0 0

q OO E1p,q

The shape of the E1-page implies that the differentials dr are zero for r > 1(these differentials are ‘too long’: their source or their target must be zero),so E2

p,q = E∞p,q. As a result we have exact sequences:

0→ E∞1,q → Hq+1(C ′′) d1−→ Hq(C ′)→ E∞0,q → 0

Now the convergence of the spectral sequence is equivalent to tha data ofshort exact sequences:

0→ E∞0,q+1 → Hq+1(C)→ E∞1,q → 0 .

Splicing all these exact sequences together, we recover the homology longexact sequence associated with the short exact 0→ C ′ → C → C ′′ → 0.

Theorem 4.1 brings no new information about the long exact sequencein homology (the fact that the differential d1 is not explicit in the spectralsequence corresponds to the fact that the connecting homomorphism ∂ inthe long exact sequence is not explicit). Thus, theorem 4.1 can really bethought of as a (huge) generalisation of homology long exact sequences.

So far, we have presented homological spectral sequences. But cohomo-logical spectral sequences also occur in practice. For the reader’s conve-nience, we formulate the cohomological analogue of theorem 4.1, which canbe obtained from theorem 4.1 by a suitable reindexing.

A cochain complex C is filtered when it is equipped with a family ofsubcomplexes · · · ⊂ F pC ⊂ F p−1C ⊂ . . . (beware of the indices, whichare decreasing for filtrations of cohomological complexes). The filtration isbounded below if for each k there is an integer pk such that F pkCk = 0,and it is exhaustive if

⋃p∈Z F

pC = C. The cohomology of a filtered cochaincomplex C is filtered by

F pH∗(C) := Im(H∗(F pC)→ H∗(C)

).

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4.2 Bicomplexes Prerequisites - Antoine Touze

Theorem 4.3. Let C be a filtered cochain complex, whose filtration isbounded below and exhaustive. There is a cohomological spectral sequence

Ep,q1 (C) = Hp+q(F pC/F p+1C) =⇒ Hp+q(C) .

To be more explicit, the differentials are maps dr : Ep,qr → Ep+r,q+1−rr and

there are isomorphisms:

Ep,q∞ (C) ' F pHp+q(C)/F p+1Hp+q(C) .

If f : C → C ′ preserves the filtrations, then f induces a morphism of spec-tral sequences with Ep,q1 (f) = Hp+q(Grpf), and Ep,q∞ (f) coincides on theabutment with GrpHp+q(f).

4.2 Bicomplexes

A first quadrant chain bicomplex is a bigraded object C =⊕

p≥0,q≥0Cp,q,together with differentials d : Cp,q → Cp−1,q and ∂ : Cp,q → Cp,q−1 whichcommute: d ∂ = ∂ d. The total complex associated such a bicomplexis the complex TotC with (TotC)n =

⊕p+q=nCp,q and whose differential

maps an element of x ∈ Cp,q to dx+ (−1)p∂x.

Example 4.4. Let C and D be nonnegative chain complexes of R-modules.Their tensor product C ⊗R D is equal to Tot (C R D), where C R Dis the first quadrant chain bicomplex defined by (C R D)p,q = Cp ⊗ Dq,d = dC ⊗ Id and ∂ = Id⊗ dD.

If C is a first quadrant bicomplex, we can obtain a spectral sequenceconverging to the homology of TotC as a particular case of theorem 4.1.Let us sketch the construction. We first define a filtration of TotC by

FnTotC :=⊕

0≤p, 0≤q≤nCp,q .

Thus F0C equals the 0-th column of the bicomplex C, with differential ∂,and more generally Fn is the ‘totalization’ of the sub-bicomplex of C formedby the objects of columns 0 to n. So there is an isomorphism of complexes:

FnTot (C)/Fn−1Tot (C) ' (Cn,∗−n, ∂) .

By theorem 4.1, the filtered complex TotC gives birth to a spectral sequencestarting with page E1

p,q(C) = Hq(Cp,∗, ∂), converging to Tot (C)p+q. Lookingat the construction of the spectral sequence of theorem 4.1, one finds that thefirst differential d1 : E1

p,q(C)→ E1p−1,q(C) is equal to the map Hq(Cp,∗, ∂)→

Hq(Cp−1,∗, ∂) induced by the differential d : Cp,q → Cp−1,q. Thus the secondpage is given by:

E2p,q(C) = Hp(Hq(C, ∂), d) .

This is summarized in the following statement.

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4.2 Bicomplexes Prerequisites - Antoine Touze

Theorem 4.5. Let (C, d, ∂) be a first quadrant chain bicomplex. There is ahomological spectral sequence

E2p,q(C) = Hp(Hq(C, ∂), d) =⇒ Hp+q(TotC) .

To be more specific, we have isomorphisms:

E∞p,q(C) ' FpHp+q(TotC)/Fp−1Hp+q(TotC) .

Moreover, the spectral sequence is natural with respect to morphisms of bi-complexes.

First quadrant chain bicomplexes are ‘symmetric’, that is the trans-posed bicomplex Ct defined by Ctp,q = Cq,p gives rise to another spectralsequence with second page E2

p,q(Ct) = Hq(Hp(C, d), ∂) and which converges

to Hp+q(Ct) ' Hp+q(C). So there are actually two spectral sequences asso-ciated to a bicomplex.

Example 4.6. Let us illustrate theorem 4.5 by an application. We aregoing that TorR∗ (M,N) is ‘well-balanced’, i.e. it can be computed usingindifferently a projective resolution of M or a projective resolution of N .

Let P (resp. Q) be a projective resolution of M (resp. N). We denoteby `TorR∗ (M,N) the homology of the complex P ⊗R N , which is the Torobtained by taking a projective resolution of the left variable M , and byrTorR∗ (M,N) the homology of M ⊗R Q. Let us make explicit the secondpage associated to the bicomplex

(C, d, ∂) = (P R Q, dP ⊗ Id, Id⊗ dQ) .

Since the objects of P are projectives, Hq(Pp RQ, ∂) equals Pp⊗RHq(Q),which equals Pp ⊗RM if q=0, and zero otherwise. So we obtain that:

E2p,q(C) = `Torp(M,N) if q = 0 and zero if q 6= 0.

The shape of the second page implies that the differentials dr, for r ≥ 2must be zero (their source or their target is zero), so E2

p,q(C) = E∞p,q(C) andsince the spectral sequence abuts to H∗(TotC), we obtain an isomorphism

`TorR∗ (M,N) ' H∗(TotC) .

The same argument applied to the transposed bicomplex Ct yields an iso-morphism:

rTorR∗ (M,N) ' H∗(TotCt) ' H∗(TotC) .

This proves that TorR∗ (M,N) is well-balanced.

Of course, theorem 4.5 has a cohomological analogue for first quadrantcochain bicomplexes (i.e. bigraded objects C =

⊕p≥0,q≥0C

p,q equippedwith differentials d : Cp,q → Cp+1,q and ∂ : Cp,q → Cp,q+1 which commute.)We give the statement for the convenience of the reader.

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4.3 Grothendieck spectral sequences Prerequisites - Antoine Touze

Theorem 4.7. Let (C, d, ∂) be a first quadrant cochain bicomplex. There isa cohomological spectral sequence

Ep,q2 (C) = Hp(Hq(C, ∂), d) =⇒ Hp+q(TotC) .

To be more specific, the r-th differential is a map Ep,qr → Ep+r,q+1−rr and we

have isomorphisms:

Ep,q∞ (C) ' F pHp+q(TotC)/F p+1Hp+q(TotC) .

Moreover, the spectral sequence is natural with respect to morphisms of bi-complexes.

4.3 Grothendieck spectral sequences

Let A, B, C be abelian categories, and let F : A → B and G : B → C be rightexact functors. On can wonder wether it is possible to recover the derivedfunctors Li(G F ) of the composition from the derived functors LiF andLjG. The Grothendieck spectral sequence answers this question under anacyclicity assumption. One says that N ∈ B is G-acyclic if LiG(N) = 0 fori > 0.

Theorem 4.8. Let A, B, C be abelian categories such that A and B haveenough projectives. Assume that we have right exact functors F : A → Band G : B → C such that F sends projective objects to G-acyclic objects.Then for all M ∈ A, there is a homological spectral sequence:

E2p,q = LpG LqF (M) =⇒ Lp+q(G F )(M) .

Of course, the Grothendieck spectral sequence has a cohomological ana-logue for right derived functors.

Theorem 4.9. Let A, B, C be abelian categories such that A and B haveenough injectives. Assume that we have left exact functors F : A → B andG : B → C such that F sends injective objects to G-acyclic objects. Thenfor all M ∈ A, there is a cohomological spectral sequence, with differentialsdr : Ep,qr → Ep+r,q+1−r

r :

Ep,q2 = RpG RqF (M) =⇒ Rp+q(G F )(M) .

Example 4.10. Let us illustrate the Grothendieck Spectral sequence on anexample from algebraic group cohomology. Let k be a field, let G be analgebraic group over k and let H be a closed subgroup. Then the restrictionfunctor:

resGH : rat. G-mod → rat. H-mod

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4.4 Filtered differential graded algebras Prerequisites - Antoine Touze

admits a right adjoint, namely the induction functor8

indGH : rat. H-mod → rat. G-mod .

Being a right adjoint, indGH is a left exact functor, and since resGH is an exactfunctor, indGH preserves injectives. Moreover, H∗(H,M) is the derived func-tor of M 7→ Homrat. H-mod(k,M) = Homrat. G-mod(k, indGHM) so theorem4.9 yields a spectral sequence:

Ep,q2 = Hp(G,RqindGH(M)) =⇒ Hp+q(H,M) .

In particular, if by chance the H-module M satisfies the vanishing con-dition RqindGH(M) = 0 for q > 0, the spectral sequence stops at the secondpage (since it is concentrated on the 0-th row) so there is an isomorphism:

H∗(G, indGHM) ' H∗(H,M) .

Kempf vanishing theorem, a cornerstone for the representation theory ofalgebraic groups, asserts that the vanishing condition is satisfied when G isreductive, H is a Borel subgroup of G and M = kλ is the one dimensionalrepresentation of H given by a character χλ of H, associated to a dominantweight λ9. In this case, the representation indGH(kλ) is called ‘the costandardmodule with highest weight λ’.

4.4 Filtered differential graded algebras

A differential graded algebra is a graded algebra A equipped with a differen-tial d satisfying the Leibniz rule: d(xy) = d(x)y+(−1)deg xxd(y). A filtrationof a differential graded algebra A is a filtration · · · ⊂ FpA ⊂ Fp+1A ⊂ . . . Aof the complex (A, d) compatible with products, i.e. the product sendsFpA⊗ FqA into Fp+qA.

If A is a filtered differential algebra, the product of A induces bigradedalgebra structures on

⊕p,qHq(GrpA) and

⊕p,q GrpHq(A). Thus, the first

page and the ∞-page of the spectral sequence of theorem 4.1 are bigradedalgebras. More is true: in this situation, the spectral sequence is a spectralsequence of algebras.

Theorem 4.11. Let A be a filtered differential graded algebra, whose fil-tration is bounded below and exhaustive. There is a homological spectralsequence of algebras

E1p,q(A) = Hp+q(GrpA) =⇒ Hp+q(A) .

8Let us consider the regular functions k[G] as a G ×H-module, where G acts by lefttranslations and H by right translations on G. The induction functor can be defined bythe formula: indGH(M) = (k[G]⊗M)H .

9For example, G = GLn(k), H = Bn(k) is the subgroup of upper trangular matricesand M = kλ is the one dimensional vector space k acted on by Bn(k) by multiplicationwith a scalar χλ([gi,j ]) =

Qgλii,i , where λ1 ≥ λ2 ≥ · · · ≥ λn.

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4.5 The LHS spectral sequence Prerequisites - Antoine Touze

More explicitly, there are bigraded algebra isomorphisms

E∞p,q(A) ' FpHp+q(A)/Fp−1Hp+q(A) .

If f : A → A′ is a morphism of differential graded algebras preserving thefiltrations (i.e. it sends FpA into FpA

′), then f induces a morphism ofspectral sequences with E1

p,q(f) = Hp+q(Grpf), and E∞p,q(f) coincides on theabutment with GrpHp+q(f).

Theorem 4.11 has an obvious cohomological analogue, whose formulationis left to the reader.

4.5 The Lyndon-Hochschild-Serre spectral sequence

Let G be a discrete group, let H be a normal subgroup of G, and let Mbe a kG-module (k is a commutative ring). The Lyndon-Hochschild-Serrespectral sequence allows to reconstruct the cohomologyH∗(G,M) from somecohomology groups of H and G/H. Let us be more specific. We can restrictthe action on M to G to obtain a kH-module, still denoted by M .

Lemma 4.12. The quotient group G/H acts on H i(H,M) for all i ≥ 0.

Proof. Let [g] denote the class of g ∈ G in G/H. Then G/H acts on MH bythe formula: [g]m := gm. Consider the cohomological δ-functor (F i, δi)i≥0,where the F i are the functors

kG-mod → k-modM 7→ H i(H,M)

.

Restriction from kG-modules to kH-modules preserves injectives (cf. exer-cise 5.7), so F i = Ri(F 0). Each [g] ∈ G/H defines a natural transformation[g] : F 0 → F 0 so it extends uniquely into a morphism of δ-functors from(F i, δi)i≥0 to (F i, δi)i≥0. By uniqueness, since the axioms of a group actionare satisfied on F 0(M), they are satisfied on all F i(M), i ≥ 0.

We are now ready to describe the Lyndon-Hochschild-Serre spectral se-quence.

Theorem 4.13. Let H be a normal subgroup of G, and let M be a kG-module. There is a first quadrant cohomological spectral sequence of k-modules, with differentials dr : Ep,qr → Ep+r,q+r−1

r :

Ep,q2 = Hp(G/H,Hq(H,M)) =⇒ Hp+q(G,M) .

This spectral sequence is natural with respect to M . Moreover, if M isa kG-algebra then the spectral sequence is a spectral sequence of algebras.(On the second page of the spectral sequence, the product is the cup productof the cohomology algebra H∗(G/H,A) where A is the cohomology algebraH∗(H,M).)

A similar Lyndon-Hochschild-Serre spectral sequence exists for the ho-mology of discrete groups, and for rational cohomology of algebraic groups.

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5 Exercises

5.1 Homological algebra without spectral sequences

Exercise 5.1. Let k be a commutative ring and let R be a k-algebra.Show that an arbitrary (i.e. not necessarily finite) direct sum of projec-tive R-modules is projective. Show that an arbitrary product of injectiveR-modules is injective.

Exercise 5.2 (Z-mod has enough injectives). 1. Prove that Q/Z is aninjective Z-module. (hint: begin by proving that if C ′ is a subgroup ofa cyclic group C, then all f : C ′ → Q/Z can be extended into a mapf : C → Q/Z.)

2. Fix an abelian group A. Prove that for all a ∈ A, there exists amap f : A → Q/Z such that f(a) 6= 0. Deduce that there exists amonomorphism A →

∏Q/Z, where the product is indexed by the set

HomZ(A,Q/Z).

Exercise 5.3 (R-mod has enough injectives). Let k be a commutative ringand R be a k-algebra. If N ∈ k-mod we make Homk(R,N) into a R-moduleby (r · f)(s) := f(sr).

1. Let M be a R-module. Show that there is a bijection, natural withrespect to M , N :

Homk(M,N) '−→ HomR(M,Homk(R,N)) .

Deduce that if J is an injective k-module, then Homk(R, J) is injective.

2. Let J be an injective k-module. Assume that there exists a k-linearinjection ι : M → J . Build a R-linear injection: M → Homk(R, J).

3. Assuming that Z-mod has enough injectives, prove that for all com-mutative ring k and all k-algebra R, the category R-mod has enoughinjectives.

Exercise 5.4. Let A, B be abelian categories with enough projectives andenough injectives. Let G : B → A be a right adjoint to F : A → B. Recallthat this means that there is a bijection, natural in x, y:

HomB(F (x), y) ' HomA(x,G(y)) .

1. Show that F is right exact and that G is left exact.

2. Show that G is exact if and only if F preserves the projectives (andsimilarly F is exact if and only if G preserves the injectives).

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5.1 Exercises Prerequisites - Antoine Touze

Exercise 5.5. Let A, B, C be abelian categories with enough injectives, andlet F : A → B and G : B → C be left exact functors.

1. Assume that G is exact. Prove that Ri(G F ) = G RiF .

2. Assume that F is exact and that F sends injective objects to G-acyclicobjects (i.e. to objects X such that RiG(X) = 0 for i > 0). Provethat Ri(G F ) = (RiG) F .

3. Find an example of functors F,G such that F is exact and does notsend injective objects to G-acyclic objects (hint: use section 2.3.3)

Exercise 5.6. Let k be a commutative algebra, and let R be a k-algebra.Assume that all R-module splits as a direct sum of simple R-modules. Showthat for all R-modules M and N , ExtiR(M,N) = 0 if i > 0.

Exercise 5.7 (Shapiro’s lemma). Let H be a subgroup of G. Let us denoteby resGH : kG-mod→ kH-mod the restriction functor. If M is a kH-module,we denote by indGHM the k-module kG⊗kHM , acted on by G by the formula:g(x⊗m) := gx⊗m. We denote by coindGHM the k-module HomkH(kG,M),acted on by G by the formula (gf)(x) := f(xg).

1. Prove that indGH (resp. coindGH) is left (resp. right) adjoint to resGH .

2. Prove that resGH is exact and preserves injectives and projectives, thatindGH is exact and preserves projectives, and that coindGH is exact andpreserves injectives (use exercise 5.4).

3. Prove Schapiro’s lemma, namely we have isomorphisms, natural withrespect to the kH-module N (use exercise 5.5):

H∗(H,N) ' H∗(G, indGHN) , H∗(H,N) ' H∗(G, coindGHN) .

Exercise 5.8. Let M ∈ kG-mod. Consider M as a ZG-module. Prove thatExtiZG(Ztriv,M) is actually a k-module, and that there is an isomorphismof k-modules, natural with respect to M :

ExtiZG(Ztriv,M) ' ExtikG(ktriv,M) .

(hint: use the standard complexes)

Exercise 5.9. Let G be a finite group. Prove that all kG-modules arerational representations of G. Deduce that the cohomology of the finitegroup G (in the sense of section 2.2) is the same as the cohomology of thealgebraic group G (in the sense of section 2.3).

Exercise 5.10 (rational representations of GL1(k)). Let k be an alge-braically closed field.

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1. Let ρ : GL1(k) → GLn(k) be a representation of the multiplicativegroup. Prove that there exists a family of n × n-matrices (Mk)k∈Zwhich are all zero but a finite number of them such that ρ(x) =∑xkMk. Show that the Mk satisfy the relations∑

Mk = I , MkM` = 0 if k 6= ` , M2k = Mk .

Deduce that the representation splits as a direct sum of simple repre-sentations of dimension 1.

2. Prove that for all rational representation V of GL1(k),

H irat(GL1(k), V ) = 0 if i > 0.

Exercise 5.11. Let C be a small category, and let x, y be objects of C.Write an explicit basis of the k-module HomFun(C,k-mod)(Px, Py).

Exercise 5.12 (k-linear functors). Let K be a k-linear category. A k-linearfunctor is a functor F : K → k-mod, such that for all x, y ∈ K the followingmap is k-linear:

HomK(x, y) → Homk(F (x), F (y))f 7→ F (f)

If C is a small category, let kC denote the category with the same ob-jects, with HomkC(x, y) = k HomC(x, y) and with composition defined by(∑λfbf ) (

∑µgbg) =

∑λfµgbfg.

1. Prove that the category Fun(C,k-mod) is equivalent to the full sub-category of Fun(kC,k-mod) whose objects are k-linear functors.

2. Prove that the k-linear Yoneda lemma reduces to the usual Yonedalemma in Fun(kC, k-mod).

Exercise 5.13. Let C be a small category.

1. Prove that for all x in C we have isomorphisms, natural in F and x:F ⊗C Px ' F (x). Deduce that the functor ⊗CPx is exact.

2. Prove that there is an isomorphism, natural in F : F ⊗C (⊕

iGi) '⊕i F ⊗C Gi. Deduce that if P is projective in Fun(C,k-mod) then

⊗CP is an exact functor.

5.2 Homological algebra with spectral sequences

Exercise 5.14. Let (Erp,q, dr)r≥2 be a homological spectral sequence. As-

sume that Erp,q = 0 if p is odd or if q is odd. Show that E2p,q = E∞p,q.

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Exercise 5.15. Let (Erp,q, dr)r≥2 be a spectral sequence of R-modules. As-

sume that R is a noetherian ring and that E2p,q is finitely generated over

R. Prove that the spectral sequence stops. (Hint: consider the sequences ofR-submodules of the second page:

B2 ⊂ B3 ⊂ · · · ⊂ Bi ⊂ . . . ⊂ · · · ⊂ Zi ⊂ · · · ⊂ Z3 ⊂ Z2 ,

where Z2 = ker d2, B2 = Im d2 and the Zi, Bi are defined inductivelyby Zi+1 = π−1(ker di+1), Bi+1 = π−1(Im di+1) where π is the map Zi Zi/Bi = Ei+1.)

Exercise 5.16. Consider the bicomplex of k-vector spaces depicted below(the dimension of each object is indicated between parentheses):

0 0 0

0 // Z0(2) //

OO

Z1(6) //

OO

Z2(4) //

OO

0

0 // Y 0(4) //

f

OO

Y 1(8) //

OO

Y 2(4) //

OO

0

0 // X0(2) //

OO

X1(6) //

OO

X2(1) //

OO

0

0

OO

0

OO

0

OO

.

1. Assume that the column of index 0 is exact at X0, that the columnof index 1 is exact at Y 1 and at Z1 and that the column of index 2 isexact at X2 and Y 2. Compute the dimensions of the H i(Tot(C)) infunction of the rank of f .

2. Assume furthermore that the middle row (the one with the Yis) isexact. Compute the rank of f .

Exercise 5.17. 1. Let R be an algebra over a commutative ring k. LetM be a right R-module and let P be a complex of projective leftR-modules. Show that there is a first quadrant homological spectralsequence of k-modules

E2p,q = Torp(M,Hq(P )) =⇒ Hp+q(M ⊗R P ) .

(hint: consider the spectral sequence associated to the bicomplexQM ⊗R P , where QM is a projective resolution of M)

40

REFERENCES Prerequisites - Antoine Touze

2. If k is a PID, show that Torki (M,N) = 0 for i > 1. Deduce that if P is

a complex of projective k-modules and if M is a k-module, there areshort exact sequences of k-modules:

0→M ⊗Hq(P )→ Hq(M ⊗ P )→ Tork1(M,Hq−1(P ))→ 0 .

Exercise 5.18. Let C be a small category. Show that TorC∗(G,F ) is wellbalanced, i.e. that we have isomorphisms (natural with respect to G,F ):

Li(G⊗C)(F ) ' Li(⊗CF )(G) .

Exercise 5.19. Let H be a normal subgroup of G and let M ∈ kG-mod (kis a commutative ring). Derive from the Lyndon-Hochschild-Serre spectralsequence the five terms exact sequence:

0→ H1(G/H,MH)→ H1(G,M)→ H1(H,M)G/H → H2(G/H,MH)→ H2(G,M) .

Exercise 5.20. 1. Let H be a normal subgroup of G, which is containedin the center of G. Let M ∈ kG-mod. Prove that the action of G/Hon H∗(H,M) is trivial.

2. Let C be an infinite subgroup, contained in the center of G. Letk denote the trivial representation. Show that there is a long exactsequence:

· · · → Hp(G/C,k) d−→ Hp+2(G/C,k)→Hp+2(G, k)→ Hp+1(G/C, k)d−→ Hp+3(G/C,k)→ . . .

3. Show that in the preceding long exact sequence, the maps d are givenby cup product with a class c ∈ H2(G/C, k).

References

[Ben] D. Benson, Representations and cohomology. II. Cohomology ofgroups and modules. Second edition. Cambridge Studies in AdvancedMathematics, 31. Cambridge University Press, Cambridge, 1998.

[Bo] A. Borel, Linear algebraic groups. Second edition. Graduate Texts inMathematics, 126. Springer-Verlag, New York, 1991.

[Br] K. Brown, Cohomology of groups. Graduate Texts in Mathematics, 87.Springer-Verlag, New York-Berlin, 1982.

[H] J. Humphreys, Linear algebraic groups. Graduate Texts in Mathemat-ics, No. 21. Springer-Verlag, New York-Heidelberg, 1975.

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REFERENCES Prerequisites - Antoine Touze

[J] J.C. Jantzen, Representations of algebraic groups. Second edition.Mathematical Surveys and Monographs, 107. American MathematicalSociety, Providence, RI, 2003.

[MC] J. McCleary, A user’s guide to spectral sequences. Second edition.Cambridge Studies in Advanced Mathematics, 58. Cambridge Univer-sity Press, Cambridge, 2001.

[ML] S. Mac Lane, Homology. Reprint of the 1975 edition. Classics in Math-ematics. Springer-Verlag, Berlin, 1995.

[MLCat] S. Mac Lane, Categories for the working mathematician. Secondedition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York,1998.

[S] T.A. Springer, Linear algebraic groups. Reprint of the 1998 second edi-tion. Modern Birkhuser Classics. Birkhuser Boston, Inc., Boston, MA,2009.

[W] C . Weibel, An introduction to homological algebra. Cambridge Studiesin Advanced Mathematics, 38. Cambridge University Press, Cambridge,1994.

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