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Functor Homology: Theory and Applications

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FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 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) Let G be a group. We can define a category with a single object and where the endomorphisms of this object is the underlying set of G. The composition of morphisms in this category is given by the binary operation on the group G. 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 in G has an inverse. (2) Let Fin be (the skeleton of) the category of finite sets with objects n = f1; : : : ; ng and morphisms arbitrary functions of finite sets and FI the category of finite sets and morphisms injective maps. (3) Let Γ be (the skeleton of) the category of finite pointed sets with objects [n] = f0; 1; : : : ; ng with basepoint 0 and morphisms functions of finite sets preserving basepoint (i.e. sending 0 to 0). (4) For R a ring, let R-mod be (a skeleton of) the category of finitely generated free left R-modules. The category Z-mod of finitely generated free abelian groups is also denoted by ab. (5) Let gr be (a skeleton of) the category of finitely generated free groups. Date: May 20, 2019. 1 2 CHRISTINE VESPA 1.1. Functor categories. A category C is called "small" if both objects and morphisms are actually sets and not proper classes. In these lectures C denotes a small category. For c and c0 two objects of C, the set of morphisms from c to c0 in C will be denoted by C(c; c0). For | a commutative ring, we denote by |-Mod the category of modules over |. For C a small category and | a commutative ring, we denote by F(C; |) the category of all functors from C to |-Mod having natural transformations as morphisms. An object of F(C; |) is called a C-module. Here are some examples of interesting objects in F(C; |) for C one of the categories considered in Example 1.1. Example 1.2. (1) For C a small category we denote by | the functor in F(C; |) which is constant and equal to |. We denote also by | the functor in F(Cop; |) defined similarly. (2) For C = |-mod, we denote by Id : |-mod ! |-Mod the forgetful functor and by T n : |-mod ! |-Mod the n-th tensor product functor (i.e. T n(G) = G⊗n). The symmetric group n n Sn acts naturally on T by permutation of the factors. We denote by S : |-mod ! |-Mod n the functor obtained taking the coinvariants of T by the action of Sn. This functor is called the n-th symmetric power. We denote by Γn : |-mod ! |-Mod the functor obtained taking n the invariants of T by the action of Sn. This functor is called the n-th divided power. The n-th exterior power functor Λn : |-mod ! |-Mod is defined by: for V 2 |-mod, Λn(V ) is n the quotient of T (V ) by the relations v1 ⊗ ::: ⊗ vn = 0 if there exists i and j such that vi = vj. (3) For C = gr and | = Z we denote by a : gr ! Ab the abelianization functor. One can postcompose a with any functor given in the previous example (for | = Z). (4) For G a group, we denote by I(G) the augmentation ideal of Z[G] (i.e. the kernel of the map P P : Z[G] ! Z given by ( αgg) = αg). Let Qn : gr ! Ab be the functor given by g2G g2G n+1 Qn(G) = I(G)=I (G) (this functor is called sometimes the n-th Passi functor). Note that Q1 ' a. Here are some examples of morphisms in F(|-mod; |). Example 1.3. (1) By definition, we have natural transformations T n ! Sn, Γn ! T n and T n ! Λn. (2) The norm homomorphism defines a natural transformation N : Sn ! Γn. If n! is invertible in | (in particular if | is a field of characteristic zero) then N is a natural isomorphism. A ring R is 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 category R to Z-Mod is 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 functor F 2 F(C; |) is a subfunctor of a functor G 2 F(C; |) if for all C 2 C, F (C) is a sub |-module of G(C). Example 1.4. The functor Γn 2 F(|-mod; |) is a subfunctor of T n. A functor S 2 F(C; |) is simple if it contains no nonzero proper subfunctors. Example 1.5. The functor T n is not simple whereas the functor Λn is simple. A functor F 2 F(C; |) is indecomposable if there is no nonzero subfunctors F1, F2 such that F is the direct sum F1 ⊕ F2. A sequence 0 ! F ! G ! H ! 0 is an exact sequence in F(C; |) if, for all C 2 C 0 ! F (C) ! G(C) ! H(C) ! 0 in exact in |-Mod. FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 3 Example 1.6. (1) In F(|-mod; |) we have a short exact sequence 0 ! Γ2 ! T 2 ! Λ2 ! 0: If char(|) = 2 this sequence does not split. If char(|) 6= 2 this sequence has a section s :Λ2 ! 2 1 T given by sV (x ^ y) = 2 (x ⊗ y − y ⊗ x) for V 2 |-mod and x; y 2 V . (2) For G 2 gr, the following short exact sequence of abelian groups: (1) 0 ,2 InG=In+1G ,2 IG=In+1G ,2 IG=InG ,2 0: gives a non split short exact sequence in F(gr; Z): n (2) 0 ,2 T ◦ a ,2 Qn ,2 Qn−1 ,2 0: Proposition 1.7. The category F(C; |) is abelian. Proof. The limits and colimits in F(C; |) are computed pointwise and |-Mod is 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. Let C and C0 be small categories and F : C!C0 be a functor. We denote by F ∗ : F(C0; |) !F(C; |) the functor obtained by precomposition by F (i.e. for M 2 F(C0; |), F ∗(M) = M ◦ F ). Exercise 1.8. Show that the functor F ∗ 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 ∗? If F 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 is essentially surjective if for each C0 2 C0 there exists C 2 C 0 0 such that F (C) = C ; F is faithful if, for all C; D in C the map fC;D : C(C; D) !C (F (C);F (D)) is injective and F is full if for all C; D in D the map fC;D is surjective. A functor which is fully faithful and essentially surjective is an equivalence of category. Proposition 1.9. If F is essentially surjective, then F ∗ is faithful. Proof. As the functor F ∗ is exact, it is sufficient to prove that, if M is an object of F(C0; |) such that M ◦ F = 0, then M = 0. For an object C of C, there exists an object C0 of C0 such that F (C0) = C as F is essentially surjective. So M(C) ' M ◦ F (C0) = 0: Proposition 1.10. If F is full and essentially surjective, then F ∗ is fully faithful. Proof. Exercise. Exercise 1.11. Show that if S 2 F(C0; |) is simple then F ∗(S) 2 F(C; |) is simple. Recall that F : C!C0 is left adjoint to G : C0 !C (or G is right adjoint of F ) if for any C 2 C and C0 2 C0 there is an isomorphism: C0(F (C);C0) 'C(C; G(C0)) which is natural in C and C0. Equivalently, F is left adjoint to G if there are two natural transformation µ : IdC ! G ◦ F (unit of the adjunction) and ν : F ◦ G ! IdC0 (counit of the adjunction) such that F µ νF (3) (F −−! F GF −−! F ) = IdF µG Gν (4) (G −−! GF G −−! G) = IdG: Proposition 1.12.
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