FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS CHRISTINE VESPA

CHRISTINE VESPA

Abstract. This text is a preliminary version of material used for a course at the University of Tokyo, April-June 2019.

Keywords: functor categories; polynomial functors; functor homology; stable homology.

Contents

1. Categories of functors 1

1.1. Functor categories 2

1.2. Properties of the precomposition functor 3

1.3. Projective generators 5

1.4. Tensor products 6

2. Polynomial functors 7

2.1. Definition of polynomial functors with cross-effects 7

2.2. Basic properties and examples 8

2.3. Equivalent definitions 10

2.4. Description of polynomial functors 11

2.5. Exponential functors 13

3. Homology of functors: definitions and properties 14

3.1. Definition of Tor and Ext 14

3.2. Properties 14

4. Methods to compute functor homology 16

4.1. Functor homology and adjunction 16

References 18

1. Categories of functors

The following examples of small categories will be particularly interesting in these lectures.

Example 1.1. (1) LetGbe a group. We can define a category with a single object and where the endomorphisms of this object is the underlying set ofG. The composition of morphisms in this category is given by the binary operation on the groupG. The identity morphism is the identity element in G. This category associated to the group G will be denoted by G. Note that any morphism is an isomorphism since each element inGhas an inverse.

(2) LetFinbe (the skeleton of ) the category of finite sets with objectsn={1, . . . , n}and morphisms arbitrary functions of finite sets and F I the category of finite sets and morphisms injective maps.

(3) Let Γbe (the skeleton of ) the category of finite pointed sets with objects[n] ={0,1, . . . , n}with basepoint 0 and morphisms functions of finite sets preserving basepoint (i.e. sending0 to0).

(4) ForRa ring, letR-modbe (a skeleton of ) the category of finitely generated free leftR-modules.

The category Z-mod of finitely generated free abelian groups is also denoted byab.

(5) Let gr be (a skeleton of ) the category of finitely generated free groups.

Date: May 20, 2019.

1

1.1. Functor categories. A categoryC is called ”small” if both objects and morphisms are actually sets and not proper classes. In these lecturesC denotes a small category. Forc and c⁰ two objects of C, the set of morphisms fromc toc⁰ inC will be denoted byC(c, c⁰).

Forka commutative ring, we denote byk-Modthe category of modules overk.

ForCa small category and ka commutative ring, we denote byF(C,k) the category of all functors from C to k-Mod having natural transformations as morphisms. An object of F(C,k) is called a C-module.

Here are some examples of interesting objects in F(C,k) for C one of the categories considered in Example 1.1.

Example 1.2. (1) ForCa small category we denote by kthe functor inF(C,k)which is constant and equal to k. We denote also by k the functor inF(C^op,k)defined similarly.

(2) For C = k-mod, we denote by Id : k-mod → k-Mod the forgetful functor and by Tⁿ : k-mod→k-Modthe n-th tensor product functor (i.e. Tⁿ(G) =G^⊗n). The symmetric group Sn acts naturally on Tⁿ by permutation of the factors. We denote bySⁿ:k-mod→k-Mod the functor obtained taking the coinvariants of Tⁿ by the action ofSn. This functor is called the n-th symmetric power. We denote by Γⁿ : k-mod→k-Mod the functor obtained taking the invariants of Tⁿ by the action ofS_n. This functor is called the n-th divided power. The n-th exterior power functor Λⁿ :k-mod→ k-Mod is defined by: for V ∈k-mod, Λⁿ(V) is the quotient of Tⁿ(V)by the relationsv₁⊗. . .⊗v_n = 0if there existsiandjsuch thatv_i=v_j. (3) For C = gr and k = Z we denote by a : gr → Ab the abelianization functor. One can

postcompose a with any functor given in the previous example (fork=Z).

(4) For Ga group, we denote byI(G) the augmentation ideal of Z[G](i.e. the kernel of the map : Z[G] → Z given by (P

g∈G

αgg) = P

g∈G

αg). Let Qn : gr → Ab be the functor given by Qn(G) =I(G)/Iⁿ⁺¹(G)(this functor is called sometimes the n-th Passi functor). Note that Q1'a.

Here are some examples of morphisms in F(k-mod,k).

Example 1.3. (1) By definition, we have natural transformationsTⁿ→Sⁿ,Γⁿ →Tⁿ andTⁿ→ Λⁿ.

(2) The norm homomorphism defines a natural transformation N :Sⁿ→Γⁿ. Ifn!is invertible in k (in particular ifk is a field of characteristic zero) thenN is a natural isomorphism.

A ringRis the same as a preadditive category (i.e. a category which is enriched over the monoidal category of abelian groups) having one object. A covariant (resp. contravariant) additive functor (i.e.

an enriched functor over the monoidal category of abelian groups) from the preadditive categoryR to Z-Modis a left (resp. right) module. Therefore, functor categories can be viewed as a generalization to several objects of modules over a ring.

In particular, the usual notions used in modules theory can also be defined for functors. For example, a functorF ∈ F(C,k) is asubfunctor of a functorG∈ F(C,k) if for allC∈ C,F(C) is a subk-module ofG(C).

Example 1.4. The functorΓⁿ∈ F(k-mod,k)is a subfunctor ofTⁿ.

A functorS ∈ F(C,k) issimple if it contains no nonzero proper subfunctors.

Example 1.5. The functorTⁿ is not simple whereas the functorΛⁿ is simple.

A functor F ∈ F(C,k) isindecomposable if there is no nonzero subfunctors F₁, F₂ such that F is the direct sumF₁⊕F₂.

A sequence

0→F →G→H →0 is an exact sequence inF(C,k) if, for allC∈ C

0→F(C)→G(C)→H(C)→0 in exact in k-Mod.

Example 1.6. (1) InF(k-mod,k)we have a short exact sequence 0→Γ²→T²→Λ²→0.

If char(k) = 2this sequence does not split. Ifchar(k)6= 2this sequence has a sections: Λ²→ T² given by sV(x∧y) =¹₂(x⊗y−y⊗x)forV ∈k-modandx, y∈V.

(2) ForG∈gr, the following short exact sequence of abelian groups:

(1) 0 ,2IⁿG/Iⁿ⁺¹G ,2IG/Iⁿ⁺¹G ,2IG/IⁿG ,20.

gives a non split short exact sequence in F(gr,Z):

(2) 0 ,2Tⁿ◦a ,2Qn ,2Q_n−1 ,20.

Proposition 1.7. The categoryF(C,k)is abelian.

Proof. The limits and colimits inF(C,k) are computed pointwise andk-Modis an abelian category.

1.2. Properties of the precomposition functor. Comparison of functor categories is one of the important tools used in the study of these categories. In this section we give some basic facts concerning the precomposition functor.

LetCandC⁰be small categories andF :C → C⁰be a functor. We denote byF^∗:F(C⁰,k)→ F(C,k) the functor obtained by precomposition byF (i.e. forM ∈ F(C⁰,k),F^∗(M) =M◦F).

Exercise 1.8. Show that the functorF^∗ is exact.

This section is concerned with the following question:

When the functor F has property P what can we deduce for the functor F^∗?

IfF is an equivalence of categories,F^∗is also an equivalence of categories. We will study below the previous question for conditions P weaker that to be an equivalence of category.

We recall that a functor F :C → C⁰ isessentially surjective if for each C⁰ ∈ C⁰ there existsC ∈ C such that F(C) =C⁰; F isfaithful if, for allC, D in C the mapf_C,D :C(C, D)→ C⁰(F(C), F(D)) is injective andF isfull if for all C, Din Dthe mapfC,D is surjective. A functor which is fully faithful and essentially surjective is an equivalence of category.

Proposition 1.9. IfF is essentially surjective, thenF^∗ is faithful.

Proof. As the functorF^∗ is exact, it is sufficient to prove that, ifM is an object ofF(C⁰,k) such that M ◦F = 0, thenM = 0. For an objectC ofC, there exists an objectC⁰ ofC⁰ such thatF(C⁰) =C as F is essentially surjective. So

M(C)'M ◦F(C⁰) = 0.

Proposition 1.10. If F is full and essentially surjective, thenF^∗ is fully faithful.

Proof. Exercise.

Exercise 1.11. Show that ifS ∈ F(C⁰,k)is simple thenF^∗(S)∈ F(C,k)is simple.

Recall thatF :C → C⁰ is left adjoint toG:C⁰→ C (orGis right adjoint of F) if for anyC∈ C and C⁰ ∈ C⁰ there is an isomorphism:

C⁰(F(C), C⁰)' C(C, G(C⁰))

which is natural inCandC⁰. Equivalently,F is left adjoint toGif there are two natural transformation µ:IdC →G◦F (unit of the adjunction) andν :F◦G→IdC⁰ (counit of the adjunction) such that

(3) (F −−→^{F µ} F GF −−→^νF F) =Id_F

(4) (G−−→^µG GF G−−→^Gν G) =IdG.

Proposition 1.12. If F is left adjoint toG: C⁰ → C then G^∗ :F(C,k)→ F(C⁰,k)is left adjoint to F^∗.

Proof. We denote byµ:Id_C →G◦F the unit of the adjunction betweenF andGandν:F◦G→Id_C⁰ its counit. We define

µ⁰ :Id_F(C,k)→F^∗◦G^∗

by the following way: forM ∈ F(C,k) the natural transformationµ⁰_M :M →F^∗◦G^∗(M) =M◦G◦F is defined, forC∈ C, by (µ⁰_M)_C =M(µ_C) :M(C)→M(G◦F(C)).

Similarly, we define

ν⁰:G^∗◦F^∗→Id_F(C0,k)

by the following way: forM⁰∈ F(C⁰,k) the natural transformationν_M⁰ 0 :G^∗◦F^∗(M⁰) =M⁰◦F◦G→ M⁰ is defined, forC⁰∈ C⁰, by (ν_M⁰ 0)_C⁰ =M⁰(ν_C⁰) :M⁰((F◦G)(C⁰))→M⁰(C⁰).

Relations (3) and (4) are satisfy by µ⁰ and ν⁰ since they are satisfy by µ and ν. (For example, applyingM to relation (3) forµandν gives relation (3) forµ⁰ andν⁰).

Example 1.13. Inclusion-projection adjunction. Let C be a small category. We denote objects of C by ai, where i∈N. For0≤j≤d we consider the functorπj :C^d+1→ C^d given on objects by the projection

πj(a0, . . . , ad) = (a0, . . . ,aˆj, . . . , ad).

• If C has a terminal object (denoted byT) we consider the functorij:C^d→ C^d+1 given on objects by i_j(a₀, . . . , a_d−1) = (a₀, . . . , a_j−1, T, a_j, . . . , a_d−1).

Proposition 1.14. If C has a terminal object, the functor π_j is left adjoint toi_j. Proof. On the one hand

C^d(πj(a0, . . . , ad),(b0, . . . , bd−1))' C(a0, b0)×. . .× C(aj−1, bj−1)× C(aj+1, bj)×. . .× C(ad, bd−1) and the other hand

C^d+1((a0, . . . , ad), ij(b0, . . . , b_d−1))' C(a0, b0)×. . .×C(a_j−1, b_j−1)×C(ai, T)×C(aj+1, bj)×. . .×C(ad, b_d−1).

SinceT is a terminal objectC(a_i, T) is a set having one element. The adjunction follows.

Applying Proposition 1.12 we obtain.

Corollary 1.15. If C has a terminal object, the functor(ij)^∗ is left adjoint to(πj)^∗.

• If C has an initial object (denoted byI) we consider the functor ij :C^d →C^d+1 given on objects by

i⁰_j(a₀, . . . , a_d−1) = (a₀, . . . , a_j−1, I, a_j, . . . , a_d−1).

Proposition 1.16. If C has an initial object, the functori⁰_j is left adjoint toπj. Applying Proposition 1.12 we obtain.

Corollary 1.17. If C has an initial object,the functor(π_j)^∗ is left adjoint to(i⁰_j)^∗.

The following adjunction is particularly important. It will be used in the proof of Pirashvili’s cancellation result.

Example 1.18. The sum-diagonal and product-diagonal adjunctions. Let C be a category having finite coproduct denoted byq. We denote also byq:C × C → Cthe functor given byq(C, C⁰) = CqC⁰. To define the functor on morphisms we use the universal property of coproduct: more concretely for f :C →C⁰,q(f) = fq(f). Consider the diagonal functor δ:C → C × C which assigns to each object C the ordered pair (C, C) and to each morphism f : C → D the pair (f, f). We have the following result.

Proposition 1.19 (Sum-diagonal adjunction). If C is a category having finite coproduct, then the functor q:C × C → C is left adjoint to the diagonal functorδ:C → C × C.

Proof. We have natural isomorphisms

Hom_C(q(C1, C₂), C⁰) =Hom_C(C₁qC2, C⁰)'Hom_C×C((C₁, C₂),(C⁰, C⁰)) =Hom_C×C((C₁, C₂), δ(C⁰)) where the isomorphism comes from the fact that qis the coproduct ofC.

We deduce from Proposition 1.12 that δ^∗ is left adjoint to q^∗ i.e. for B ∈ F(C × C,k) and F ∈ F(C,k), we have natural isomorphisms

Hom_F(C,k)(δ^∗(B), F)'Hom_F(C×C,k)(B,q^∗F).

Similarly, if C is a category having finite product denoted by×, we define the functor×:C × C → C by ×(C, C⁰) =C×C⁰).

Proposition 1.20 (Product-diagonal adjunction). If C is a category having finite product, then the functor ×:C × C → C is right adjoint to the diagonal functorδ:C → C × C.

We deduce from Proposition 1.12 that δ^∗ is right adjoint to ×^∗ i.e. for B ∈ F(C × C,k) and F ∈ F(C,k), we have natural isomorphisms

Hom_F(C,k)(F, δ^∗(B))'Hom_F(C×C,k)(×^∗F, B).

We can also iterate the construction to define δⁿ:C → C^×n.

If C has finite coproductqⁿ :C^×n → C is left adjoint to δⁿ. We deduce from Proposition 1.12 that (δⁿ)^∗ is left adjoint to(qⁿ)^∗ i.e. for M ∈ F(C^×n,k)andF ∈ F(C,k), we have natural isomorphisms (5) Hom_F(C,k)((δⁿ)^∗(M), F)'Hom_F(C×n,k)(M,(qⁿ)^∗F).

and if C has finite product×ⁿ:C^×n → C is right adjoint toδⁿ

1.3. Projective generators. Recall that a set of generators in an abelian category A is a setE of objects of Asuch that for allA∈ Athere exists an epimorphism from a direct sum of object in E to A.

Definition 1.21. For any C∈ C, the functorP_C^C ∈ F(C,k)is defined by:

P_C^C(X) =k[C(C, X)]

where k[−] :Set→k-Mod is thek-linearization functor (i.e. the left adjoint to the forgetful functor k-Mod→Set).

We will prove that these functors form a set of projective generators inF(C,k). The proof relies on the Yoneda lemma that we recall.

Lemma 1.22. • Set-theoretic version

For all C∈ C andF :C →Ens we have a natural bijection HomF unc(C,Ens)(C(C,−), F)'F(C).

• k-linear version

For all C∈ C andF :C →k-Modwe have a natural bijection Hom_F(C,k)(P_C^C, F)'F(C).

The following corollary shows that F(C,k) has enough projective objects (i.e. for every functor F :C →k-Modthere is an epimorphismP →F where P is projective).

Corollary 1.23. The set{P_C^C}C∈C is a set of projective generators of the categoryF(C,k).

Proof. Let α: A →B be an epimorphism in F(C,k) andf : P_C^C →B. SinceHom_F(C,k)(P_C^C, B)' B(C),Hom_F(C,k)(P_C^C, A)'A(C) andA(C)→B(C) is surjective we deduce that there existsg:P_C^C → A such thatα◦g=f.

LetF :C →k-Mod. We consider the natural transformation M

C∈C c∈F(C)

P_C^C →F

where the componentP_C^C →F indexed byc∈F(C) is the morphism corresponding tocby the Yoneda isomorphism. This map is surjective so the set{P_C^C}C∈C is a set of generators in the categoryF(C,k).

In particular, any functorF :C →k-Modadmits a resolution by direct sums of projective generators P_C^C. So we can do homological algebra in the categoryF(C,k) as in the category ofk-Mod(see section 3).

1.4. Tensor products. The aim of this section is to define three different tensor product functors on functor categories and to give their basic properties.

• Pointwise tensor product.

Definition 1.24. ForF :C →k-ModandG:C →k-Mod,F⊗G:C →k-Modis defined by (F⊗G)(C) =F(C)⊗G(C)

for all C inC.

Proposition 1.25. If the category C admits coproduct denoted byt then P_C^C⊗P_C^C00 'P_CtC^C 0.

Proof. Direct consequence of the definition of coproduct.

•Exterior tensor product. Exterior tensor product of functors is the analogue in the setup of functor categories of exterior product of modules over different rings.

Definition 1.26. LetCandC⁰ be small categories,F :C →k-ModandF⁰:C⁰ →k-Modbe functors, the exterior tensor product ofF and F⁰ is the functorFF⁰:C × C⁰ →k-Modsuch that forC∈ C, C⁰ ∈ C⁰, we have:

(FF⁰)(C, C⁰) =F(C)⊗F⁰(C⁰).

Lemma 1.27. We have an isomorphism

P_C^C P_C^C00 'P_(C,C^C×C00)

which is natural in C∈ C and in C⁰ ∈ C⁰.

ForF, G, H in F(C,k), the sum-diagonal adjunction (see Proposition 1.19) can be reformulated by the following natural isomorphisms:

Hom_F(C,k)(F⊗G, H)'Hom_F(C×C,k)(FG,q^∗H).

• Tensor product over a category.

Definition 1.28. Let F :C →k-ModandG:C^op→k-Mod, we define G⊗

C F∈k-Modby G⊗

C

F =Coeq L

f∈C(c,c⁰)

G(c⁰)⊗F(c)

f^∗ ,2

f_∗ ,2 L

c∈C

G(c)⊗F(c)

where for f ∈ C(c, c⁰), x ∈ G(c⁰) and y ∈ F(c), f^∗(x⊗y) = x⊗F(f)(y) ∈ G(c⁰)⊗F(c⁰) and f∗(x⊗y) =G(f)(x)⊗y∈G(c)⊗F(c).

Remark 1.29. The previous definition of tensor product over a category is a particular case of coends (see [ML98, IX 6]).

Remark 1.30. The enriched tensor product of such a contravariant and covariant functor is exactly the classical tensor product of a left and a right module overR.

Example 1.31. Let k be the constant functor andF:C →k-Mod. We have k⊗

C F =colim

C F.

(Replacing Gbykin Definition 1.28, we recover the description of colim

C F in terms of coproduct and coequalizer. See for example [ML98, Chapter V]).

Proposition 1.32. ForF :C →k-ModandG:C^op→k-Mod, we have natural isomorphisms P_C^Cop⊗

C F 'F(C) and G⊗

C P_C^C 'G(C).

Another way to characterize the tensor product over a category is by the following adjunction.

Proposition 1.33. ForF :C →k-Mod, G:C^op→k-Mod andM ∈k-Mod we have the following isomorphism

k-Mod(G⊗

C

F, M)' F(C,k)(F,Hom_k(G, M)) where Hom_k(G, M) :C →k-Modis given byC7→k-Mod(G(C), M).

2. Polynomial functors

The definition of cross-effects and polynomial functors comes from the work of Eilenberg and Mac Lane on homology of spaces thereafter linked to their names [EML54]. In this paper the authors take for C a category of finitely generated free modules over a ring. This definition can easily be extended to a small monoidal category where the unit 0 is the null object ofC. In a functor category there are very huge functors which are often out of control. The polynomial property should be viewed as a way to measure the complexity of a functor. We will see that polynomial functors are easier to understand that general functors. For example, if C is an additive category, the reduced polynomial functors of degree one correspond to additive functors and forC=R-modadditive functors have been described by Eilenberg and Watts in the 60’s. Polynomial functors should be thought as being an analogue for functors of polynomial functions for functions.

2.1. Definition of polynomial functors with cross-effects. Let (C,⊕,0) be a small monoidal category where the unit 0 is the null object ofC.

Example 2.1. (1) Any pointed category with finite coproducts is an example of such category. In particular, the following categories introduced in Example 1.1 are such categories.

• The categoryΓ is pointed by[0]and its coproduct is given by the wedge product of pointed sets[m]q[n] = [m+n].

• The category k-mod is pointed by0 and its coproduct is the direct sum of modules⊕.

• The category gr is pointed by0 and its coproduct is the free product of groups∗.

(2) The category of finite pointed noncommutative sets Γ(Ass) having the same objects asΓ and where a morphism from[m] to[n]is a morphism inΓtogether with a total ordering onf⁻¹(j) for all j ∈ {1, . . . , n}. The composition is induced by the ordered union of ordered sets. The category Γ(Ass)is pointed by [0]but has no coproduct. However the wedge product of pointed sets provides a symmetric monoidal structure onΓ(Ass). This category has been considered by Pirashvili[Pir02] and is a particular case of categories of pointed sets associated to an operad (here the associative operad). See [HPV15]for more details.

Remark 2.2. All the previous examples give rise to symmetric monoidal categories.

LetF :C →k-Modbe a functor. The functor F is saidreduced ifF(0) = 0. We have a canonical decomposition F 'F(0)⊕F¯ whereF(0) is the constant functor equal toF(0) and

F(C) =¯ ker(F(C)→F(0))'coker(F(0)→F(C)).

The functor ¯F is reduced and is called the reduced functor associated toF.

FoX1∈ CandX2∈ C, we denote byp1the compositionp1:X1⊕X2 1X1⊕t

−−−−→X1⊕0'X1wheretis the unique element inC(X2,0) (0 is terminal by hypothesis). We define similarlyp2:X1⊕X2→X2. Definition 2.3. The n-th cross-effect of F is a functor crnF : C^×n →k-Mod (or a multi-functor) defined inductively by

cr1F(X) =ker(F(0) :F(X)→F(0))

cr2F(X1, X2) =ker((F(p1), F(p2))^t:F(X1⊕X2)→F(X1)⊕F(X2)) and, for n≥3, by

crnF(X1, . . . , Xn) =cr2(cr_n−1F(−, X3, . . . , Xn))(X1, X2).

In other words, to define then-th cross-effect ofF we consider the (n−1)-st cross-effect, we fix the n−2 last variables and we consider the second cross-effect of this functor.

More generally, forX1, . . . , Xn∈ C we denote byrⁿ_k the composition rⁿ_k :X1⊕. . .⊕Xk⊕. . .⊕Xn

1⊕...⊕t⊕...⊕1

−−−−−−−−−→X1⊕. . .⊕0⊕. . .⊕Xn'X1⊕. . .⊕Xˆk⊕. . .⊕Xn. We have the following alternative definition of cross-effect.

Proposition 2.4. ForF :C →k-Mod, then-th cross-effect of F is equal to the kernel of the natural homomorphism

r^F :F(X1⊕. . .⊕Xn)→ ⊕ⁿ

k=1

F(X1⊕. . .⊕Xˆk⊕. . .⊕Xn) where r^F is the map (F(rⁿ₁), . . . , F(rⁿ_n))^t.

Cross-effect should be viewed as being derivations for functors. This analogy motivates the following definition.

Definition 2.5. A functor F : C →k-Mod is said to be polynomial of degree lower or equal to n if cr_n+1F = 0.

We denote by Pol_n(C,k) the full subcategory of F(C,k) consisting of reduced polynomial functors of degree lower or equal ton. We have a filtration of categories

. . . ,→ Pol_n−1(C,k),→ Pol_n(C,k),→. . . ,→ F(C;k).

A reduced polynomial functor of degree 1 is calledlinear and a reduced polynomial functor of degree 2 is calledquadratic.

2.2. Basic properties and examples. The following decomposition result is particularly important.

Proposition 2.6. LetF :C →k-Modbe a reduced functor. Then there is a natural decomposition F(X1⊕. . .⊕Xn)'

n

M

k=1

M

1≤i1<...<i_k≤n

crkF(Xi₁, . . . , Xi_k).

Proof. The homomorphism r^F in Proposition 2.4 splits. So we can prove the statement by induction.

For example this proposition gives the following decompositions:

F(X₁⊕X₂)'F(X₁)⊕F(X₂)⊕cr₂F(X₁, X₂)

F(X1⊕X2⊕X3)'F(X1)⊕F(X2)⊕F(X3)⊕cr2F(X1, X2)⊕cr2F(X1, X3)⊕cr2F(X2, X3)⊕cr3F(X1, X2, X3).

Example 2.7. (1) The functorId:k-mod→k-Modis reduced andId(U⊕V) =Id(U)⊕Id(V).

By the previous proposition we deduce that cr₂(Id)(U, V) = 0, soIdis polynomial of degree1.

(2) The abelianization functorais reduced (a(0) = 0) anda(G∗H)'a(G)⊕a(H). By the previous proposition we deduce that cr₂(a)(G, H) = 0, soa is polynomial of degree1.

(3) The functor T²:k-mod→k-Modis reduced and we have:

T²(U⊕V) = (U⊕V)⊗(U⊕V) =T²(U)⊕T²(V)⊕(U⊗V ⊕V ⊗U).

By the previous proposition we deduce that cr2(T²)(U, V) =U⊗V ⊕V ⊗U. Furthermore, we have:

T²(U⊕V⊕W) = (U⊕V⊕W)⊗(U⊕V⊕W) =T²(U)⊕T²(V)⊕T²(W)⊕cr2T²(U, V)⊕cr2T²(U, W)⊕cr2T²(V, W).

We deduce from the previous proposition thatcr₃T²(U, V, W) = 0. SoT²is a polynomial func- tor of degree 2.

Proposition 2.8. If F : C → k-Mod and G: C →k-Mod are polynomial functors, thenF ⊕G is polynomial and

deg(F⊕G) =M ax{deg(F), deg(G)}.

Proof. Suppose that F is polynomial of degree m andG is polynomial of degree nwith m ≤n. By Proposition 2.6 we have

(F⊕G)(X1⊕. . .⊕Xn+1) =F(X1⊕. . .⊕Xn+1)⊕G(X1⊕. . .⊕Xn+1) '

n+1

M

k=1

M

1≤i1<...<i_k≤n+1

cr_kF(X_i1, . . . , X_ik)⊕

n+1

M

k=1

M

1≤i1<...<i_k≤n+1

cr_kG(X_i1, . . . , X_ik)

=

m

M

k=1

M

1≤i1<...<i_k≤n+1

cr_kF(X_i1, . . . , X_ik)⊕

n

M

k=1

M

1≤i1<...<i_k≤n+1

cr_kG(X_i1, . . . , X_ik)

where the last equality comes from the fact that deg(F) = m and deg(G) = n. We deduce that crn+1(F⊕G) = 0 and thatcrn(F⊕G)6= 0 sodeg(F⊕G) =n.

In the following propositions we study the polynomiality of the tensor product of two polynomial functors. The proofs of these propositions are inspired by [Tou18, Fact C.11 C.12].

Proposition 2.9. If F : C → k-Mod and G: C →k-Mod are polynomial functors, then F⊗G is polynomial and

deg(F⊗G)≤deg(F) +deg(G).

Proof. Suppose thatF is polynomial of degreemandGis polynomial of degreen. By Proposition 2.6 we have

(F⊗G)(X₁⊕. . .⊕X_m+n+1) =F(X₁⊕. . .⊕X_n+1)⊗G(X₁⊕. . .⊕X_m+n+1) '

m+n+1

M

k=1

M

1≤i1<...<ik≤m+n+1

crkF(Xi₁, . . . , Xi_k)⊗

m+n+1

M

k=1

M

1≤i1<...<ik≤n+1

crkG(Xi₁, . . . , Xi_k)

=

m

M

k=1

M

1≤i1<...<i_k≤m+n+1

crkF(Xi1, . . . , Xi_k)⊗

n

M

k=1

M

1≤i1<...<i_k≤m+n+1

crkG(Xi1, . . . , Xi_k) where the last equality comes from the fact that deg(F) = m anddeg(G) = n. So we have a direct sum of functors in several variables where the number of variables is between 1 and m+n. On the other hand

(F⊗G)(X₁⊕. . .⊕X_m+n+1)'

m+n+1

M

k=1

M

1≤i1<...<i_k≤m+n+1

cr_k(F⊗G)(X_i1, . . . , X_ik).

Sincecr_m+n+1(F⊗G)(X_i1, . . . , X_ik) hasm+n+1 variables, we deduce by identification thatcr_m+n+1(F⊗

G) = 0 sodeg(F⊗G)≤m+n.

Remark 2.10. The inequality in the previous proposition can be strict. For example, consider F : Z-mod→Z-Modgiven byF(−) =− ⊗Q/Z. This functor is polynomial of degree 1 butF⊗F = 0 sinceQ/Z⊗Q/Z= 0.

Proposition 2.11. Ifkis an integral domain andF :C →k-ModandG:C →k-Modare polynomial functors taking torsion free values, then F⊗Gis polynomial and

deg(F⊗G) =deg(F) +deg(G).

The proof of this proposition relies on the following lemma.

Lemma 2.12. LetRbe a commutative domain,M andN be torsion freeR-modules, thenM⊗

R

N = 0 iff M = 0or N= 0.

Proof. Assume thatM 6= 0 andN 6= 0. LetKbe the field of fractions ofR. We haveM⊗

R

K=S⁻¹M whereS=R\{0}andS⁻¹Mdenotes the localisation. SinceM is torsion free we have: ^m₁ = 0⇔m= 0.

Then the morphismφ:M →S⁻¹M given byφ(m) =^m₁ is injective soM⊗

R

KcontainsM. We deduce that M⊗

R

K andN⊗

R

K are non zero so (M ⊗

R

K)⊗

K

(N⊗

R

K)6= 0. Since

(M⊗

R

K)⊗

K

(N⊗

R

K)'(M ⊗

R

N)⊗

R

K6= 0

we deduce that M⊗

R

N 6= 0.

Proof of Proposition 2.11. Suppose thatF is polynomial of degreemandGis polynomial of degreen.

By Proposition 2.6 we have (F⊗G)(X₁⊕. . .⊕X_m+n)'

m

M

k=1

M

1≤i1<...<i_k≤m+n

cr_kF(X_i1, . . . , X_ik)⊗

n

M

k=1

M

1≤i1<...<i_k≤m+n

cr_kG(X_i1, . . . , X_ik)

'

m+n

M

k=1

M

1≤i1<...<i_k≤m+n

cr_k(F⊗G)(X_i1, . . . , X_ik).

So, the functor ofm+nvariables

C(X₁, . . . , X_m+n) =cr_mF(X₁, . . . , X_m)⊗cr_nG(X_m+1, . . . , X_m+n)

is a subfunctor ofcr_m+n(F⊗G)(X₁, . . . , X_m+n). Sincecr_mF andcr_nGare non-zero and take torsion free values we deduce from Lemma 2.12 that C is non zero, hencecr_m+n(F⊗G) is non zero. Hence deg(F⊗G)≥m+n. The other inequality is given in Proposition 2.9.

Example 2.13. For a commutative domain k, using Proposition 2.11 and Example 2.7(1) we can prove by induction that the functor Tⁿ:k-mod→k-Modis polynomial of degreen.

Proposition 2.14. The functor cr_n:F(C;k)→ F(C^×n;k)is exact for alln≥1.

Definition 2.15. A full subcategoryC⁰ of an abelian categoryC is thick (or is a Serre subcatrgory of C) if it contains 0 and is closed under extensions i.e. for every exact sequence 0→B →A→C→0 in C,A∈ C⁰ if and only if B andC are in C⁰.

Proposition 2.16. The subcategoryPoln(C,k)of F(C;k)is thick.

Proof. Immediate consequence of Proposition 2.14.

Example 2.17. Using the short exact sequence (2) in Example 1.6, the fact that a is polynomial of degree 1 and the previous proposition we can prove by induction that Q_n : gr→ Ab is a polynomial functor of degree n.

2.3. Equivalent definitions. Polynomial functors can be defined in several other ways which can be more or less useful depending on the context. There are definitions in terms of cokernel instead of kernel, idempotent or the difference functor. We refer the reader to [DV15, Section 2.2] for the equivalence between these definitions when Chas a null object.

Remark 2.18. In [DV19] we extend the notion of polynomial functors from a symmetric monoidal category where the unit 0 is theinitialobject.

2.4. Description of polynomial functors. In this section we present several results concerning the description of polynomial functors. The polynomial functors of degree one are simple to describe but, in general, it is difficult to describe the categoriesPoln(C,k) forn >1. However, we will see in Section

?? that the quotient categoriesPoln(C,k)/Pol_n−1(C,k) have simple descriptions.

• Description of polynomial functors of degree1

If C is an additive category, for F : C → k-Mod a reduced polynomial functor of degree one, by Proposition 2.6 we have a natural isomorphism F(X ⊕Y)' F(X)⊕F(Y). So reduced polynomial functors of degree one on an additive category are equivalent to additive functors. For C an additive category, we denote byAdd(C,k) the full subcategory ofF(C,k) of additive functors.

IfC=R-modforRa ring, the Eilenberg-Watts theorem gives the following description of additive functors from R-modtok-Mod.

Theorem 2.19. [Eil60, Wat60]LetRandkbe two rings. The evaluation on Rinduces an equivalence of categories

Add(R-mod,k)−^'→(R^op⊗k)-Mod.

ForM ∈(R^op⊗k)-Mod, the quasi-inverse of this functor is given by M 7→M ⊗

R−.

LetC be a pointed category having finite coproducts denoted byq. For E ∈ C we denote by hEiC

the full subcategory ofC having as objects finite sums of copies of E.

Example 2.20. (1) ForC=R-ModandE=R, we have hEiC =R-mod.

(2) ForC=Gr andE=Z, we havehEiC =gr.

(3) ForC=Set_∗ (the category of pointed sets) andE= [1], we havehEi_C = Γ.

Eilenberg-Watts theorem has been generalized to linear functors onhEi_C in [HV11]. We need some notations before to give the statement.

A functorF :hEi_C →k-Modhas a greatest linear quotient functor denoted by ¯T1(F). This functor is given explicitely by the following formula

T¯₁(F)(V) =Coker(F(p₁) +F(p₂)−F(s) :F(V qV)→F(V))

where p1, p2, s:V qV →V are respectively, the first projection, the second projection and the sum (given by the universal property of coproduct). Let ΛC(E) := ¯T1(PE)(E) wherePE is the projective generator associated toE (see Definition 1.21).

Theorem 2.21. [HV11, Theorem 3.12]The evaluation on E induces an equivalence of categories Pol1(hEiC,Z)−^'→Λ_C(E)-Mod.

ForM ∈Λ_C(E)-Mod, the quasi-inverse of this functor is given byM 7→T¯1PE(−) ⊗

Λ_C(E)

M. Remark 2.22. This theorem can easily be extend to functors with values ink-modules.

• Description of polynomial functors of degreen

In general it is difficult to give a complete description of the categoriesPol_n(C,k).

The case of quadratic functors has been studied for several categories C. For C = ab quadratic functors have been described by Baues in [Bau94], for C=gr they have been described by Baues and Pirashvili in [BP99]. For C = Γ quadratic functors can be described thanks to the Dold-Kan type theorem of Pirashvili as we will explain below. All these cases are covered by the complete description of the categoryPol2(hEi_C,k) given in [HV11].

For higher degrees, a description of polynomial functors of degree n, for all n, has been given in [BDFP01] for C = ab and in [HPV15] for C = gr. Polynomials functor on the category Γ can be described thanks to the following theorem.

Theorem 2.23(Dold-Kan type theorem of Pirashvili). [Pir00]Let Ωbe the category of finite sets and surjections. The functor:

cr:F(Γ;k)→ F(Ω;k)

given by cr(F)(n) =cr_nF([1], . . . ,[1])for F ∈ F(Γ;k)is an equivalence of categories (where cr_nF is the n-th cross-effect of F, see Definition 2.3).

LetF(Ω;k)_≤n the full subcategory ofF(Ω;k) having as objects functors vanishing on setsX such that |X |> n.

Corollary 2.24. The functor:

cr:Poln(Γ;k)→ F(Ω;k)≤n

is an equivalence of categories.

Remark 2.25.The classical Dold-Kan theorem gives an equivalence between simplicial sets in an abelian categoryA(i.e. F(∆^op,A)) and the category of chain complexes ofA. The category of chain complexes can be viewed as the category of functors from an enriched category over Γ preserving nul morphism.

This justifies the name of the previous theorem.

Remark 2.26. In [DPV] we extend this result to the PROP associated to a set-operad. The previous theorem corresponding to the PROP associated to the operad Com.

Remark 2.27. Two rings are calledMorita equivalent if they have equivalent categories of left modules.

As functor categories are generalizations to several objects of the theory of modules over a ring this definition can be extended in the following form: two small categories C andC⁰ are Morita equivalent if the functors categories F(C,k) andF(C⁰,k) are equivalent.

IfCandC⁰ are equivalent they are obviously Morita equivalent but there are lots of examples of non equivalent categories which are Morita equivalent. For example Theorem 2.23 says that the categories Γ and Ω are Morita equivalent.

In [S lo04] and [LS15] the authors give general conditions in order to obtain equivalences of functor categories. Their methods cover the classical Dold-Kan theorem but also the Dold-Kan type theorem of Pirashvili and its extension to PROP associated to the associative set-operad.

• Description of polynomial functors of degreenmodulo polynomial functors of degree n−1

In order to understand polynomial functors of degree n we would like to describe the category of polynomial functors of degree ≤ n from the category of polynomial functors of degree ≤ n−1 and another category which morally measures the difference between the functors of degree ≤nand the functors of degree ≤ n−1. For this we present in this section a general way to study an abelian categoryCfrom ”smaller” categories. More precisely, forC⁰a subcategory ofChaving good properties, we define the quotient category C/C⁰. The study ofC can be reduced to the study ofC⁰ andC/C⁰.

The original reference for this section is [Gab62, Chapitre III].

Remark 2.28. A quotient category is a particular case of the localisation of Gabriel-Zisman [GZ67] of a category relatively to a set of morphisms (here, we inverse the morphisms whose kernel and cokernel are in the subcategory C⁰).

Proposition 2.29. Let F : C → A be a functor between abelian categories. If F is exact then the kernel ofF (i.e. the full subcategory ofC of objects which are sent to 0by F) is thick.

The converse of this proposition is given by the notion of quotient category defined below. More precisely, if C⁰ is a thick subcategory of C, we define a new abelian category C/C⁰ called ”quotient category” such that there exists an exact functor F :C → C/C⁰ whose kernel isC⁰.

Definition 2.30. LetC be an abelian category and C⁰ a thick subcategory of C. The quotient category C/C⁰ has as objects the objects ofC and forX andY two objects ofC

C/C⁰(X, Y) =colim Hom_C(X⁰, Y /Y⁰)

where the colimit runs through all subobjects X⁰ ⊂X, Y⁰ ⊂Y such thatX/X⁰ andY⁰ are objects in C⁰.

The composition of morphisms in C/C⁰ is given carefully in [Gab62].

Iff ∈Hom_C(X, Y) such thatKer(f)∈ C⁰ (resp. Coker(f)∈ C⁰) thenf is a monomorphism (resp.

epimorphism) in C/C⁰.

We have a quotient functor T :C → C/C⁰ given byT(X) =X forX ∈ C and forf ∈Hom_C(X, Y), T(f) is the image off in colimHom_C(X⁰, Y /Y⁰).

Proposition 2.31. The functorT :C → C/C⁰ is exact and its kernel isC⁰. Moreover, forDan abelian category, F :C → Dan exact functor which is trivial onC⁰, there exists a unique functorG:C/C⁰→ D such that G◦T =F.

By Proposition 2.16, the categoryPoln(C,k) is thick. We will describe below the quotient categories Pol_n(C,k)/Poln−1(C,k).

Proposition 2.32 (Pirashvili). The functor cr_n : Pol_n(ab,k) →k[S_n]-Mod, F 7→ cr_nF(Z, . . . ,Z) induces an equivalence of categories:

Pol_n(ab,k)/Pol_n−1(ab,k)'k[S_n]-Mod.

For n > 1 the categories Poln(ab,k) and Poln(gr,k) are not equivalent. However we have the following result.

Proposition 2.33. [DV15, Corollaire 3.6]The abelianization functora:gr→Ab induces an equiva- lence of categories:

Poln(gr,k)/Pol_n−1(gr,k)' Poln(ab,k)/Pol_n−1(ab,k) Combinig the last two propositions we obtain the equivalence of categories

Poln(gr,k)/Poln−1(gr,k)'k[Sn]-Mod.

Remark 2.34. In [DV15], we give more generally the description of Poln(hEiC,k)/Pol_n−1(hEiC,k) where C is a small pointed category with finite coproduct, E is a fixed object in C and hEiC is the full subcategory of Chaving as objects finite coproducts ofE. More precisely, we give therecollement diagram between the categoriesPol_n(hEi_C,k),Pol_n−1(hEi_C,k) and a category of modules.

2.5. Exponential functors. One of the most important property of exponential function is that the exponential of a sum is the product of exponentials. In this section we introduce functors satisfying a similar property. We will see in section??that functor homology of exponential functors has interesting properties. LetCbe a small category having finite coproducts denoted byt. Recall that (C,t,0) (where 0 denotes the initial object corresponding to the empty coproduct) is a symmetric monoidal category.

Definition 2.35. (1) An exponential functor E of F(C,k)is a symmetric monoidal functor from C tok-Mod. In particular,E is equipped with natural isomorphisms

E(CtC⁰)'E(C)⊗E(C⁰) forC andC⁰ inC.

(2) A graded functor E^• = (Eⁿ) where Eⁿ : C → k-Mod is exponential if we have a natural isomorphism:

Eⁿ(UtV)'

n

M

i=0

Eⁱ(U)⊗Eⁿ⁻ⁱ(V).

Example 2.36. The graded functors Λ^•, S^•,Γ^• are exponential functors. The graded functor T^• is not exponential but it is not far to be exponential in the sense that we have the following isomorphism

Tⁿ(U⊕V)'

n

M

i=0

(Tⁱ(U)⊗Tⁿ⁻ⁱ(V)) ⊗

S_i×Sn−i

Z[Sn].

Proposition 2.37. Let E^• = (Eⁿ) be a graded exponential functor such that E⁰ =k. Then Eⁿ is polynomial of degree ≤n.

Proof. By the exponential property and the hypothesis onE⁰we have natural isomorphisms: E¹(Ut V)'E¹(U)⊕E¹(V). Socr2(E¹) = 0.

By induction, suppose that for alli≤n,deg(Eⁱ)≤i. Using the exponential property and Proposi- tion 2.9 we obtain the result.

Example 2.38. The functors Γⁿ, Sⁿ,Λⁿ fromk-modtok-Modare polynomial of degree n.

3. Homology of functors: definitions and properties

The terms ”functor homology” denote homological algebra in functor categories. In this section we will explain in more details what does it mean, give some basic properties and several results concerning functor homology overgr.

3.1. Definition of Tor and Ext.

• Tor groups. The functors− ⊗

C F andG⊗

C −are right exact. They commute with colimits in each variables. We can derive these functors on the left.

Definition 3.1. ForF ∈ F(C,k) andG∈ F(C^op,k)we define:

T or^C_i(G, F) =H_i(G⊗

C

P_•) where P_• is a projective resolution of F ∈ F(C,k).

Remark 3.2. We could equally well resolveG.

Homology of a category

Definition 3.3. ForF ∈ F(C,k) we define the gradedk-module H_∗(C, F) =T or_∗^C(k, F).

Remark 3.4. We haveH0(C, F) =k⊗

C

F = colim(F) by Example 1.31.

Example 3.5. If G is the category associated to a group G (see example 1.1), a functor F : G → k-Modis ak[G]-module and the homology of the categoryG,H_∗(G, F), corresponds to the usual notion of homology of the group Gwith coefficients in ak[G]-module.

We give two simple examples of computation of homologies of categories.

Example 3.6. IfC has an initial object I thenH0(C,k) =k andH∗(C,k) = 0for∗>0.

In fact, k=P_I^C(−) =k[C(I,−)]sok is a projective object in F(C,k).

Example 3.7. IfC has a terminal object T thenH0(C;F) =F(T)andHi(C;F) = 0for∗>0.

In fact, k=P_T^Cop(−) =k[C(−, T)] sokis a projective object in F(C^op,k).

• Ext groups.

Definition 3.8. ForF ∈ F(C,k) andG∈ F(C,k) we define:

Extⁱ_F(C

k)(G, F) =H_i(Hom_F(C,k)(P_•, F)) where P• is a projective resolution of G∈ F(C,k).

3.2. Properties.

• Relation between Ext and Tor.

Proposition 3.9. Let G:C^op→k-Mod,F :C →k-Mod andI an injective k-module then we have a natural graded isomorphism

T or_•^C(G, F)^∨'Ext^•_F(C)(F,Hom(G, I))

where V^∨=Hom_k(V, I)for ak-moduleV andHom(G, I) :C →k-Mod,C7→Hom_k-Mod(G(C), I).

Proof. We use the characterization of Ext groups given in [ML63, Chapter III, Theorem 10.1] for Exⁿ(F) =T or^C_n(G, F)^∨.

We have

Ex⁰(F) =T or₀^C(G, F)^∨= (G⊗

C F)^∨=Hom_k(G⊗

C F, I)'Hom_C(F,Hom_k(G, I)) by Proposition 1.33.

For a projective functorP we have

Exⁿ(P) =T or_n^C(G, P)^∨= 0

Since I is an injective k-module, Hom_k(−, I) is exact and Exⁿ send short exact sequence to long exact sequence.

We deduce thatExⁿ'Extⁿ(−, G^∗).

• Functoriality.

Lemma 3.10. Let C andD be two small category,ϕ:C → D,F :D →k-Mod and G:D^op→k-Mod be functors then ϕinduces a natural morphism

ϕ_• :T or^C_•(ϕ^∗(G), ϕ^∗(F))→T or^D_•(G, F).

Proof. To construct the maps ϕ_• we use the notion of universal δ-functors (see [Wei94, Section 2.1]

for the definition) and the following result (see [Wei94, Theorem 2.4.7 p47]) Let A andB be abelian categories. If Ahas enough projective objects, then for any right exact functorF :A → B, the derived functors LnF form a universalδ-functor.

The categories F(C,k) and F(D,k) are abelian by Proposition 1.7 and have enough projective objects by Corollary 1.23. For G : D^op → k-Mod, the functor G⊗

D− : F(D,k) →k-Mod is right exact. Since the functorϕ^∗is exact (see Exercise 1.8), the functorϕ^∗(G)⊗

C−:F(C,k)→k-Modis also right exact. We deduce that the functors T or^D_n(G,−) and T or^C_n(ϕ^∗(G),−) form universal δ-functors.

By the universal property of the δ-functorT or^C_n(ϕ^∗(G),−) the natural transformation ϕ0:ϕ^∗(G)⊗

C ϕ^∗(−)→G⊗

D− gives morphisms

ϕn:T or_n^C(ϕ^∗(G), ϕ^∗(F))→T or^D_n(G, F)

extendingϕ₀. Moreover these morphisms are unique if we require to have a morphism ofδ-functor.

Corollary 3.11. Let C andDbe two small category, ϕ:C → DandF :D →k-Mod be functors then ϕ induces a natural morphism

ϕ_•:H_•(C, ϕ^∗(F))→H_•(D, F).

Proof. Apply the previous lemma forGthe constant functor equal tok. One of the main ingredient of the method developed in [DV10] consists to compare functor homolo- gies of various categories. More precisely, we will see in Proposition ?? a criterion implying that the natural mapϕ_∗:H_∗(C, ϕ^∗(F))→H_∗(D, F) obtained in the previous corollary is an isomorphism.

The following lemma will be also useful.

Lemma 3.12. Let C and D be two small category, ϕ, ψ : C → D be functors, u : ϕ →ψ a natural transformation and F :D →k-Mod andG:D →k-Modbe functors, then the following diagram is commutative

T or^C_•(ψ^∗G, ϕ^∗F) ^{T or}

C

•(ψ^∗F,F◦u) ,2

T or^C_•(G◦u,ϕ^∗F)

T or^C_•(ϕ^∗G, ϕ^∗F)

ψ•

T or^C_•(ϕ^∗G, ϕ^∗F) _ϕ

• ,2T or^D_•(G, F).

4. Methods to compute functor homology In this section we present several useful tools to compute functor homology.

4.1. Functor homology and adjunction. We will prove that the adjunction between functor cat- egories coming from an adjunction of functors (see Proposition 1.12) can be derived to give natural isomorphisms between Ext.

We will need the following lemma

Lemma 4.1. Ifψ:C → C⁰ is left adjoint to ϕ:C⁰→ C then, for eachc in C ϕ^∗P_C^C 'P_ψ(C)^C0 .

In particularϕ^∗ preserves projective objects.

Proof. ϕ^∗P_C^C =K[Hom_C(C, ϕ(−))]'K[Hom_C⁰(ψ(C),−) =P_ψ(C)^C0 . Proposition 4.2. Ifψ:C → C⁰ is left adjoint to ϕ:C⁰→ C then we have natural isomorphisms

Ext^∗_F(C0,k)(ϕ^∗F, G)'Ext^∗_F(C,

k)(F, ψ^∗G) of graded k-modules forF ∈ F(C,k)andG∈ F(C⁰,k).

Proof. Let P_• → F be a projective resolution. Since ϕ^∗ is exact (see Exercice 1.8) and ϕ^∗ preserves projective objects by Lemma 4.1, ϕ^∗(P_•)→ϕ^∗(F) is also a projective resolution. Therefore, sinceϕ^∗ is left adjoint to ψ^∗ according to Proposition 1.12, we have:

Ext^∗_F(C0,k)(ϕ^∗F, G) =H^∗(Hom_F(C⁰_,k)(ϕ^∗(P_•), G))'H^∗(Hom_F(C,k)(P_•, ψ^∗G))'Ext^∗_F(C,k)(F, ψ^∗G).

Application 1: Base change. LetKandK⁰be two fields andK→K⁰be a finite extension of degree d. Consider the forgetful functor F :P(K⁰)→P(K) and the base change T :=K⁰⊗

K

−:K-mod→ K⁰-mod.

Proposition 4.3. The base change functorT is left adjoint and right adjoint to the forgetful functor.

Proposition 4.4. ForI_K :P(K)→AbandI_K⁰ :P(K)→Abthe forgetful functors we have:

Ext^∗_F(P(K0),k)(IK⁰, IK⁰)'Ext^∗_F(P(K),k)(IK, IK)^⊕d. Proof. We haveF^∗(I_K) =I_K⁰ andT^∗(I_K⁰)'K⁰⊗

K

I_K 'I_K^⊕d. By Proposition 4.2 we have:

Ext^∗_F(P(K0),k)(I_K⁰, I_K⁰) =Ext^∗_F(P(K0),k)(I_K⁰, F^∗(I_K))'Ext^∗_F(P(K),

k)(T^∗I_K⁰, I_K) 'Ext^∗_F(P(K),k)(I_K^⊕d, IK)'Ext^∗_F(P(K),k)(IK, IK)^⊕d.