on local categories of finite groups · on local categories of finite groups feixu abstract. let g...

ON LOCAL CATEGORIES OF FINITE GROUPS

FEI XU

Abstract. Let G be a finite group. Over any finite G-poset P we may define a transportercategory as the corresponding Grothendieck construction. Transporter categories are general-izations of subgroups of G, and we shall demonstrate the finite generation of their cohomology.We record a generalized Frobenius reciprocity and use it to examine some important quotientcategories of transporter categories, customarily called local categories.

1. Introduction

Let C be a finite category in the sense that Mor C is finite. Then we can define a finite-dimensional algebra kC, over a chosen field k. This algebra has a distinguished module, the trivialmodule k, which is the tensor identity of the closed symmetric monoidal category kC-mod [18].The tensor structure allows us to define a cup product on Ext∗kC(k, k) = ⊕i≥0Ext

ikC(k, k) so that

there is a ring isomorphismExt∗kC(k, k)

∼= H∗(BC, k),

in which BC is the classifying space of C. This ring is called the ordinary cohomology ring ofkC. In general such a ring is not finitely generated, but for various reasons we would like tohave a good class of finite categories with their ordinary cohomology rings finitely generated. In[18] we discussed the so-called transporter categories G ∝ P, defined over a poset P with theaction by a group G, and proved that both the ordinary cohomology ring and the Hochschildcohomology ring of k(G ∝ P) are finitely generated. In this article, we continue to show that forany M,N ∈ k(G ∝ P)-mod, Ext∗k(G∝P)(M,N) is finitely generated over Ext

∗k(G∝P)(k, k), which

generalizes the well known theorem of Evens and Venkov on group cohomology [2, 14]. Our mainresult is the following.

Theorem 1.1. Let G be a finite group and P a finite G-poset. Suppose k(G ∝ P) is the transportercategory algebra. Then for any M,N ∈ k(G ∝ P)-mod, Ext∗k(G∝P)(M,N) is finitely generated overthe Noetherian ring Ext∗k(G∝P)(k, k).

The proof is different from that for group algebras, or more generally for finite-dimensionalcocommutative Hopf algebras [6], where the internal hom plays a significant role. Here the proofis based on functor cohomology and is very similar to Venkov’s proof for finite groups. Howeverinterestingly it depends on Hochschild cohomology.

In this article, we shall first recall the definition of a transporter category as well as basicfacts about representations and cohomology of category algebras. We explain why transportercategories are interesting by examples in group representations and cohomology. These will be inSections 2 and 3. Particularly we provide a generalized Frobenius Reciprocity in Section 3. Thefinite generation theorem is proved in Section 4, which itself is an evidence of the interests ontransporter categories. The last section looks into various cases of the Frobenius Reciprocity.

I would like to thank the anonymous referee for helpful suggestions and for pointing out amistake in an earlier version.

2010 Mathematics Subject Classification. Primary 20C05; Secondary 20J99.The author (徐斐) was supported in part by a Beatriu de Pinós research fellowship from the government

of Catalonia of Spain, and a Grant MTM2010-20692 “Analisis local en grupos y espacios topologicos" from theMinistry of Science and Innovation of Spain.

1

2 FEI XU

2. preliminaries

In this section, we recall the definition of a transporter category and some background incategory algebras. Throughout this article we will only consider finite categories, in the sense thatthey have finitely many morphisms. Thus a group G, or a G-poset P, is always finite.

2.1. Transporter categories as Grothendieck constructions. In the literature the trans-porter categories are mostly considered as auxiliary constructions before passing to various quo-tient categories of them. Here we want to stress on the perhaps unique property, among variouscategories constructed from a group, that transporter categories admit natural functors to thegroup itself. It singles out this particular type of categories and is the starting point of this article.Here in order to emphasize the similarities and connections between transporter categories andsubgroups, we follow a definition which is well known to some algebraic topologists. It is intendedto be phrased explicitly in a way such that an algebraist, say a representation theorist, may readilyaccept as an entirely algebraic or categorical construction.

We deem a group as a category with one object, usually denoted by •. The identity of a groupG is always named e. We say a poset P is a G-poset if there exists a functor F from G to sCats,the category of small categories, such that F (•) = P. It is equivalent to say that we have a grouphomomorphism G → Aut(P). The Grothendieck construction on F will be called a transportercategory. In the following explicit definition, the morphisms in a poset are customarily denotedby ≤.

Definition 2.1. Let G be a group and P a G-poset. The transporter category G ∝ P has thesame objects as P, that is, Ob(G ∝ P) = ObP. For x, y ∈ Ob(G ∝ P), a morphism from x to yis a pair (g, gx ≤ y) for some g ∈ G.

If (g, gx ≤ y) and (h, hy ≤ z) are two morphisms in G ∝ P, then their composite is (hg, (hg)x ≤z). One can check directly that if HomG∝P(x, y) 6= ∅ then both AutG∝P(x) and AutG∝P(y) actfreely on HomG∝P(x, y).

The symbol G ∝ P is used because this particular Grothendieck construction resembles asemidirect product, yet is different. From the definition one can easily see that there is a naturalembedding ιP : P ↪→ G ∝ P via (x ≤ y) 7→ (e, x ≤ y). On the other hand, the transportercategory admits a natural functor πP : G ∝ P → G, given by x 7→ • and (g, gx ≤ y) 7→ g. Thuswe always have a sequence of functors

P ιP↪→G ∝ P πP−→G

such that πP ◦ ιP(P) is the trivial subgroup or subcategory of G. For convenience, in the rest ofthis article we often neglect the subscript P and write ι = ιP , π = πP . Topologically it is wellknown that B(G ∝ P) ' EG×GBP. Passing to classifying spaces, we obtain a fibration sequence

BP Bι−→EG×G BPBπ−→BG.

Forming the transporter category over a G-poset eliminates the G-action, and thus is the algebraicanalogy of introducing a Borel construction over a G-space.

This neat but seemingly abstract definition can be easily seen to give the usual transportercategories. For example, when P = Sp is the poset of non-trivial p-subgroups, we get G ∝ Sp =Trp(G), the p-transporter category of G. The advantage of taking our approach is that we maygive group-theoretic interpretations to transporter categories, which is shown by the upcomingexamples, where each subgroup of G is identified as a transporter category, up to a categoryequivalence.

Example 2.2. If G acts trivially on P, then G ∝ P = G× P.

Example 2.3. Let G be a finite group and H a subgroup. We consider the set of left cosets G/Hwhich can be regarded as a G-poset: G acts via left multiplication. The transporter categoryG ∝ (G/H) is a connected groupoid whose skeleton is isomorphic to H. In this way one canrecover all subgroups of G, up to category equivalences.

ON LOCAL CATEGORIES OF FINITE GROUPS 3

A category equivalence D → C induces a Morita equivalence between their category algebras-tobe recalled in Section 2.2, kD ' kC as well as a homotopy equivalence BD ' BC (see [15]).It means there is no essential difference between kH and k(G ∝ (G/H)) (and their modulecategories), or between BH and B(G ∝ (G/H)). Hence it makes sense if we deem transportercategories as generalized subgroups for a fixed finite group.

As two other motivating examples, transporter categories were implicitly considered by MarkRonan and Steve Smith [11] in the 1980s for constructing group modules, and later on played a keyrole in Bill Dwyer’s work [5] on homology decomposition of classifying spaces. Roughly speaking,one often finds in various situations a diagram of categories and functors

G ∝ Pπ

{{

ρ

##G C

for C a quotient category of G ∝ P and π the canonical functor. (The quotient categories we havein mind are orbit categories, Brauer categories, Puig categories or even transporter categoriesthemselves.) Dwyer used this diagram to establish connections among various homotopy colimits(e.g. classifying spaces), while Ronan and Smith constructed kG-modules via representations ofG ∝ P (using the language of G-presheaves on P).

2.2. Category algebras, representations and cohomology. We recall some facts about cat-egory algebras. The reader is referred to [15, 17] for further details. Let C be a finite category andk a field. One can define the category algebra kC, which, as a vector space, has a basis the set ofall morphisms in C, and in which multiplication is determined by compositions of base elements.When C is a group, kC is the group algebra. As a convention, throughout this article, kG-modulesare usually written as M,N etc, while the modules of a (non-group) category algebra kC aredenoted by M,N etc., except those special modules, namely k and κM , which are restrictions ofkG-modules (to be defined shortly).

A k-representation of C is a covariant functor from C to V ectk, the category of finite dimensionalk-vector spaces. All representations of C form the functor category V ectCk . By a theorem of B.Mitchell, the finitely generated left kC-modules are the same as the k-representations of C, in thesense that there exists a natural equivalence

V ectCk ' kC-mod.

In the module category, there is a distinguished module k, sometimes called the trivial module,which can be defined as a constant functor taking k as its value at every object of C. Since V ectk isa symmetric monoidal category, V ectCk inherits this structure. It means there exists an (internal)tensor product, or the pointwise tensor product, written as ⊗̂, such that for any two kC-modulesM,N, (M⊗̂N)(x) := M(x) ⊗k N(x). Let α ∈ Mor C be a base element of kC. Then α acts onM⊗̂N via α⊗ α. Obviously k is the identity with respect to ⊗̂ and M⊗̂N ∼= N⊗̂M. We can alsoconstruct a function object, the internal hom, Hom(M,N) ∈ kC-mod such that

HomkC(L⊗̂M,N) ∼= HomkC(L,Hom(M,N)),

for any L ∈ kC-mod, and the isomorphism is natural in L,M and N, see [12, 18]. When C = G isa group, we recover the usual construction −⊗̂− = −⊗k − and Hom(−,−) = Homk(−,−).

For any two kC-modules it makes sense to consider the Ext groups Ext∗kC(M,N). The tensorproduct ⊗̂ induces a cup product as follows

∪ : ExtikC(M,N)⊗ ExtjkC(M

′,N′)→ Exti+jkC (M⊗̂M′,N⊗̂N′).

In particular Ext∗kC(k, k) is a graded commutative ring and we have an isomorphism Ext∗kC(k, k)

∼=H∗(BC, k). This ring is called the ordinary cohomology ring of kC and it acts on Ext∗kC(M,N).The ordinary cohomology ring of a category algebra is usually far from finitely generated, butit is so when C = G ∝ P is finite (implicitly in Venkov’s proof of the finite generation of groupcohomology).

4 FEI XU

One concept that we will refer to is the bar resolution, a combinatorally constructed projectiveresolution BC∗ , of k ∈ kC-mod.

Finally we record the basic tools for comparing category algebras and their modules. Whenτ : D → C is a functor between two finite categories. There are functors for comparing thererepresentations. The functor τ induces a restriction

Resτ : kC-mod→ kD-mod.

If we regard a kC-module as a functor, then its restriction is the precomposition of it with τ . Ifwe consider the functor π : G ∝ P → G, then any kG-module M restricts to a k(G ∝ P)-module,written as κM = ResπM , with only one exception k = κk = Resπ k. The functor Resτ is equippedwith two adjoints: the left and right Kan extensions along τ

LKτ , RKτ : kD-mod→ kC-mod.

The definition of the left and right Kan extensions [7] depend on the so-called over-categories andunder-categories, respectively. For each x ∈ Ob C, one can define an over-category τ/x and anunder-category x\τ . These concepts are frequently used in the representations and cohomologyof category algebras. For example the bar resolution is constructed by using a certain collectionof overcategories. The analysis of over- and undercategories associated with π leads to veryinteresting results, as they reveal the structures of various prominent kG-modules and complexesof kG-modules. The reader is referred to [19] for the work based on studying these categories,where we build the Becker-Gottlieb transfer map.

3. The role of transporter categories

From this section to the end of the article, we illustrate the role of transporter categories ingroup representations and cohomology. The purpose is to show that transporter categories shouldbe further investigated, for better understanding of groups, or as prototypes for research in otherlocal categories.

3.1. A diagram of categories. We have seen in the introduction that given a G-poset P thereexists a diagram of categories and functors

G ∝ Pπ

{{

ρ

##G C

for any given local category C.

3.2. Frobenius reciprocity. Applying the adjunctions, described in Section 2.2, to the firstdiagram, we obtain a diagram of module categories

k(G ∝ P)-modLKπ,RKπ

{{

LKρ,RKρ

##kG-mod

Resπ

;;

kC-mod.Resρ

cc

The adjunctions between the restrictions and Kan extensions have the following consequences.

Proposition 3.1 (Frobenius Reciprocity). Suppose P is a G-poset and C is a quotient categoryof G ∝ P as in the preceding diagrams. Let M,N ∈ kG-mod and M,N ∈ kC-mod. Then

(1) HomkG(M,RKπ ResρN) ∼= HomkC(LKρ ResπM,N);(2) HomkG(LKπ ResρM, N) ∼= HomkC(M, RKρ Resπ N).

By direct calculations, these particular Kan extensions in Proposition 3.1 are simplified:• LKπ ∼= lim−→P , RKπ

∼= lim←−P (used by Ronan-Smith. See also [18]);


• LKρ ∼=↑kCk(G∝P) (the induction), RKρ ∼=⇑kCk(G∝P) (the co-induction), since ρ induces an

algebra homomorphism k(G ∝ P) → kC when C is a quotient category. (One can easilyprove that a functor τ : D → C induces a k-linear map τ : kD → kC, and moreover thismap becomes an algebra homomorphism if and only if the functor is injective on objects.)

Remark 3.2. For any N ∈ kC-mod, we shall write the k(G ∝ P)-module ResρN as N becausethey share the same underlying vector space. Recall that we have set κM = ResπM . Then theFrobenius reciprocity can be rewritten as

(1’) HomkG(M, lim←−P N)∼= HomkC(κM ↑kCk(G∝P),N);

(2’) HomkG(lim−→PM, N)∼= HomkC(M, κN ⇑kCk(G∝P)).

When P = G/H for some subgroup H, we have natural equivalences lim←−G/H∼= lim−→G/H

∼=↑GH .Then the above isomorphisms certainly become the usual adjunctions between ↑GH and ↓GH (theusual Frobenius Reciprocity) with C = G ∝ (G/H) and ρ = Id, in light of the Morita equivalencebetween kC and kH, see Example 2.3.

Remark 3.3. Our Frobenius reciprocity is different from a similar result of Ronan-Smith, see [2,7.2.4], where they (implicitly) had a diagram of the same shape. However their C = G ∝ Q,not necessarily a quotient of G ∝ P, is another transporter category and ρ is induced by a G-map P → Q. This prohibits us from considering various quotients of transporter categories.Moreover since a G-map P → Q usually is not injective on objects, it does not induce an algebrahomomorphism from k(G ∝ P) to k(G ∝ Q). Hence their Kan extensions along ρ cannot beinterpreted as induction and coinduction.

The functors ↑kCk(G∝P) and ⇑kCk(G∝P) admit interesting interpretations when G ∝ P → C is part

of an extension (or an opposite extension) sequence of categories. Under the circumstance ↑kCk(G∝P)and ⇑kCk(G∝P) on certain k(G ∝ P)-modules can be very well understood. We shall discuss it inSection 5.

4. Finite generation of cohomology

The functor π is very useful to study cohomology of transporter categories. The reader can findmore applications in [19] where we construct the Becker-Gottlieb transfer map. We do not putthe construction here because it requires substantial analysis of various over- and under-categoriesassociated with π and related functors, and accompanying permutation G-modules.

4.1. First attempt with internal hom. Suppose P is a G-poset and π is the natural functorfrom the transporter category G ∝ P to G, regarded as a category with one object •. The nextresult is a direct generalization of the fact that the two obvious kG-module structures on P⊗M areisomorphic, for P,M ∈ kG-mod with P projective. It reveals a connection between representationsof groups and of transporter categories.

Recall that for any finite category C, the regular module kC decomposes into a direct sum⊕x∈Ob C kC ·1x because 1 =

∑x∈Ob C 1x is an orthogonal decomposition by idempotents. A direct

summand kC · 1x is often written as kHomC(x,−) [15].

Theorem 4.1. Let P ∈ k(G ∝ P)-mod be a projective module and κM = ResπM for someM ∈ kG-mod. Then P⊗̂κM is a projective k(G ∝ P)-module. Consequently BG∝P∗ ⊗̂κM →k⊗̂κM = κM → 0 is a projective resolution.

Proof. We will prove P⊗̂κM ∼= P⊗M , with k(G ∝ P) acting on the latter via left multiplication.To this end, we assume P = kHomG∝P(x,−). The proof is entirely analogues to the case whenP = •, i.e. when G ∝ • = G.

We define a k-linear map ϕ : kHomG∝P(x,−) ⊗M → kHomG∝P(x,−)⊗̂κM as follows. Onbase elements ϕ((g, gx ≤ y) ⊗m) = (g, gx ≤ y) ⊗ (g, gx ≤ y)m, where the latter m is consideredas an element in κM (x). For any (h, hy ≤ z), we readily verify

(h, hy ≤ z)ϕ[(g, gx ≤ y)⊗m] = ϕ[(h, hy ≤ z)((g, gx ≤ y)⊗m)].

6 FEI XU

Thus ϕ is a homomorphism of k(G ∝ P)-modules.We remind the reader that, following definition, given any pair of x, y ∈ Ob(G ∝ P) with

HomG∝P(x, y) non-empty, both AutG∝P(x) and AutG∝P(y) act freely on HomG∝P(x, y). Thisimplies that kHomG∝P(x, y) is a free kAutG∝P(x)- or kAutG∝P(y)-module. Consequently ϕrestricts on each y to the classical kAutG∝P(y)-isomorphism (see [1, 3.1.5])

kHomG∝P(x, y)⊗M → (kHomG∝P(x,−)⊗̂κM )(y) = kHomG∝P(x, y)⊗M.

Furthermore because as vector spaces

kHomG∝P(x,−)⊗̂κM =⊕

y∈Ob(G∝P)

kHomG∝P(x, y)⊗M = kHomG∝P(x,−)⊗M,

the linear map ϕ is actually one-to-one and hence an isomorphism of k(G ∝ P)-modules. �

The following result from [18] is needed.

Proposition 4.2. Let G be a finite group and P a finite G-poset. Then Ext∗k(G∝P)(k, k) is finitelygenerated, and for an arbitrary N ∈ k(G ∝ P)-mod, Ext∗k(G∝P)(k,N) is finitely generated over theExt∗k(G∝P)(k, k).

The proof is entirely analogous to Venkov’s proof for group cohomology [2, 14]. The maindifference here is the use of the Grothendieck spectral sequence, instead of the Leray-Serre spectralsequence, in order to allow arbitrary functors as coefficients.

Proposition 4.3. For any M ∈ kG-mod and N ∈ k(G ∝ P)-mod, Ext∗k(G∝P)(κM ,N) is finitelygenerated over Ext∗k(G∝P)(k, k).

Proof. One can easily deduce from Theorem 4.1, together with the internal hom in Section 2.2,an Eckmann-Sharpiro type isomorphism

Ext∗k(G∝P)(κM ,N)∼= Ext∗k(G∝P)(k,Hom(κM ,N)).

Then we apply the finite generation result that we just quoted. �

However in general Ext∗k(G∝P)(M,N) 6∼= Ext∗k(G∝P)(k,Hom(M,N)), showing the limit of inter-

nal hom. It is striking that Hochschild cohomology occurs when we solve the finite generationproblem in the next section.

4.2. Finite generation through Hochschild cohomology. In [17], we proved that the ordi-nary cohomology ring Ext∗kC(k, k) is closely related to the Hochschild cohomology ring Ext

∗kCe(kC, kC).

In order to see the connection we have to rely on F (C) (another finite category), the category offactorizations in C. There are natural functors and a commutative diagram

F (C) ∇ //

t!!

Ce = C × Cop

p

yyC .This diagram induces various functors among the module categories of the category algebras ofthese three categories. We shall not recall the definitions of these categories and functors as we donot need them here. For applications in the present article, in that paper we demonstrated that

(1) Ext∗kCe(kC,M) ∼= Ext∗kF (C)(k,Res∇M).

for any M ∈ kCe-mod [17]. Thus our result is a generalization of a well know theorem in groupcohomology. In fact when C = G is a finite group, Ge ∼= G × G and F (G) ' ∆G. We commentthat although F (C) and C usually are not equivalent as categories, their classifying spaces arehomotopy equivalent. Consequently

(2) Ext∗kF (C)(k, k)∼= Ext∗kC(k, k).


Before moving to transporter categories, we remind the reader that in order to prove Evens-Venkov theorem, we depend on the following isomorphism

Ext∗kG(M,N)∼= Ext∗kG(k,Homk(M,N)),

where Homk(M,N) is the internal hom. This is not the case for category cohomology as wementioned earlier. Fortunately we find a replacement for the above isomorphism. The trick is thatwe must replace C with F (C). To this end we recall a theorem of Cartan-Eilenberg [4, ChapterIX, Corollary 4.4]. Let A be a finite-dimensional algebra over a field k and M,N ∈ A-mod. Then

(3) Ext∗A(M,N)∼= Ext∗Ae(A,Homk(M,N)).

Proposition 4.4. Suppose M,N are two kC-modules. Then we have

Ext∗kC(M,N)∼= Ext∗kF (C)(k,Res∇Homk(M,N)).

Proof. By (3) and (1)

Ext∗kC(M,N)∼= Ext∗kCe(kC,Homk(M,N)) ∼= Ext

∗kF (C)(k,Res∇Homk(M,N)).

�

Thus in order to prove the finite generation of Ext∗kC(M,N), we only need a result like Propo-sition 4.2, but for F (G ∝ P). This is also done in [18].

Proposition 4.5. Let G be a finite group and P a finite G-poset. Then Ext∗kF (G∝P)(k, k) is finitelygenerated, and for any N ∈ kF (G ∝ P)-mod, Ext∗kF (G∝P)(k,N) becomes a finitely generatedExt∗kF (G∝P)(k, k)-module.

The proof is similar to that of Proposition 4.2, using the Grothendieck spectral sequence.However it requires the understanding of undercategories associated to the composite of functorsF (G ∝ P) → G ∝ P → G, which allows us to show that the spectral sequence has only finitelymany rows.

Theorem 4.6. Suppose M,N are two k(G ∝ P)-modules. Then the module Ext∗k(G∝P)(M,N) isfinitely generated over Ext∗k(G∝P)(k, k).

Proof. By Proposition 4.4

Ext∗k(G∝P)(M,N)∼= Ext∗kF (G∝P)(k,Res∇Homk(M,N)).

Hence by Proposition 4.5 and (2), Ext∗kF (G∝P)(k,Res∇Homk(M,N)) is finitely generated over thering Ext∗kF (G∝P)(k, k) ∼= Ext

∗k(G∝P)(k, k). We are done. �

Because of the above theorem, we can define the variety for a k(G ∝ P)-module M to be themaximal ideal spectrum of Ext∗k(G∝P)(k, k)/IM, where IM is the annihilator of the following map

−⊗̂M : Ext∗k(G∝P)(k, k)→ Ext∗k(G∝P)(M,M).

It implies that there exists a support variety theory [2]. However we shall investigate it in anotherarticle [20].

5. The functor ρ: invariants and coinvariants

Here we return to the Frobenius Reciprocity and take the opportunity to write out explicitlythe formulas in some interesting cases. In practice one often finds that the functor G ∝ P → Cis part of an extension (or an opposite extension) sequence of categories. (Among examples arevarious orbit categories, Brauer categories and Puig categories.) It means that for such a quotient

8 FEI XU

category C there exists a category K which is a disjoint union of subgroups Kx ⊂ AutG∝P(x), xrunning over ObP = Ob(G ∝ P), such that we can add K into the picture

Kι

##G ∝ P

π

{{

ρ

##G C

and moreover K ↪→ G ∝ P � C satisfies some natural conditions. The explicit definition of acategory extension, due to G. Hoff, will be provided in Section 5. It will helps us to understandrelationship between kG-mod and kC-mod.

To this end we need to understand the functor ρ : G ∝ P → C, where C is a quotient categoryof G ∝ P. In general situation it seems hard to make group theoretic interpretation of ↑kCk(G∝P)and ⇑kCk(G∝P). However we can do so when we have certain quotient categories, which are part ofsome category extension sequences in the sense of G. Hoff [8].

An extension E of a category C via a category K is a sequence of functors

K ι−→E ρ−→C,

which has the following properties:(1) ObK = Ob E = Ob C, ι is injective and ρ is surjective on morphisms;(2) if ρ(α) = ρ(β), for two morphisms α, β ∈ Mor(E), if and only if there is a unique g ∈

Mor(K) such that β = ι(g)α.The following properties can be deduced from the definition.(3) if αι(h) exists for α ∈ Mor(E) and h ∈ Mor(K), then there exists a unique h′ ∈ Mor(K)

such that ι(h′)α = αι(h);(4) for any α ∈ HomC(x, y), K(y) acts regularly on ρ−1(α).It is known to Hoff that K is a disjoint union of the groups ρ−1(1x) for all 1x ∈ Mor(C) (regarded

as categories), and can be identified with a functor K : E → Groups. Usually from the context,one can easily see when we take K to be a category and when it is regarded as a functor.

A sequence K ↪→ E � C is called an opposite extension if Kop ↪→ Eop � Cop is an extension.For a fuller discussion of category extensions, see [16, Section 4].

The advantage of considering π : G ∝ P → C which is part of an extension (or oppositeextension) is that it enables us to provide a good characterization of the left (or right) Kanextension. Indeed it is the case for many familiar category constructions in representation theoryand homotopy theory.

The following statement is a special case of [16, Lemma 4.2.1]

Lemma 5.1. Let K → E ρ→C a sequence of three EI-categories and M ∈ kE-mod.(1) Suppose K → E → C is an extension. Then LKρM ∼= MK, where MK as a functor overC is given by MK(x) = M(x)K(x) (K(x)-coinvariants of the kAutE(x)-module M(x)), forany x ∈ Ob C = Ob E = ObK.

(2) Suppose K → E → C is an opposite extension. Then RKρM ∼= MK, where MK as afunctor over C is given by MK(x) = M(x)K(x) (K(x)-invariants of the kAutE(x)-moduleM(x)), for any x ∈ Ob C = Ob E = ObK.

In the above lemma, there is another way to express the Kan extensions. Under the sameassumptions, they are MK = H0(K;M) and MK = H0(K;M) respectively.

In what follows, we shall apply the above statements to various local categories of G, in com-bination with the Frobenius reciprocity (see Proposition 3.1 and Remark 3.2)

(1’) HomkG(M, lim←−P N)∼= HomkC(LKρκM ,N);

(2’) HomkG(lim−→PM, N)∼= HomkC(M, RKρκN ).


In what follows, we shall bring up several local categories considered in modular representationtheory and homotopy theory of classifying spaces. Here the information coming from their cate-gorical structures is our chief concern. To this end, the reader does not have to understand thatmuch of these categories and their functions.

5.1. Orbit categories. Suppose P is a collection of subgroups of G, which is closed under conju-gation, and on which G acts by conjugation. Then it forms a G-poset and we can define an orbitcategory OP as the quotient category of G ∝ P by asking

HomOP (P,Q) = Q\NG(P,Q).Then we have an extension sequence

S ↪→ G ∝ P � OP ,where S is the disjoint union of all objects in P, regarded as a subcategory of G ∝ P.

When M = κM for someM ∈ kG-mod, (LKρκM )(P ) = MP for any P ∈ ObP. We denote sucha kOP -module by MS := (κM )S = LKρκM . Since giving a morphism (g, gP ≤ Q) is the same asgiving a group homomorphism P → Q, the conjugation induced by g, there is a natural way toconstruct a map MP → MQ, identical to the natural map H0(P ;M) = k ⊗kP M → k ⊗kQ M =H0(Q;M). Hence we know how kOP acts on MS .

Combining the Frobenius Reciprocity (1’) and Lemma 5.1 (1), we obtain the following isomor-phism.

Proposition 5.2. Let M ∈ kG-mod and N ∈ kOP -mod. ThenHomkG(M, lim←−PN)

∼= HomkOP (MS ,N),

where MS is as above.

Corollary 5.3. With the same notations as above, we have lim←−PN∼= HomkOP (kGS ,N) and

HomkG(k, lim←−PN)∼= HomkOP (k,N).

As an example we let H be a subgroup of G and P the subgroups that are conjugate to H. Thesize of the discrete poset P is G/NG(H). Note that both G ∝ P and OP are connected groupoids,the former equivalent to NG(H) and the latter NG(H)/H. Thus the isomorphism in Proposition5.2 can be interpreted as

HomkG(M,HomkNG(H)(kG,N))∼= HomkNG(H)(M,N)∼= Homk(NG(H)/H)(k(NG(H)/H)⊗kNG(H) M,N)∼= Homk(NG(H)/H)(MH , N),

where M ∈ kG-mod and N ∈ k(NG(H)/H)-mod.

5.2. Brauer categories, fusion and linking systems. Suppose b is a p-block of the groupalgebra kG and Pb is the poset of b-Brauer pairs [13, Section 47]. Then for any G-subposetP ⊂ Pb we can introduce the Brauer category BP as the quotient category of G ∝ P such that

HomBP (P,Q) = HomG∝P(P,Q)/CG(P ).

This gives us an opposite extension, which means that the following sequence

CG ↪→ (G ∝ P)op → BopPis an extension sequence, given that CG is the disjoint union of all CG(P ), P ∈ ObP. Dual to theextension situation we examined before, now we are able to describe the right Kan extension ofmodules.

If M = κM for some M ∈ kG-mod, we denote by MCG the kFP -module RKρκM . Since amorphism (g, gP ≤ Q) provides a group homomorphism P → Q and thus induces an injectioncg−1 : CG(Q)→ CG(P ), we obtain an injection MCG(P ) →MCG(Q). This leads to the kBP -actionon MCG .


10 FEI XU

Proposition 5.4. Let M ∈ kBP -mod and N ∈ kG-mod. Then

HomkG(lim−→PM, N)∼= HomkBP (M, NCG),

where NCG is as above.

As an example we assume b is the principal block b0 and P is the conjugacy class of a fixedp-subgroup H. Then the discrete poset P has |G/NG(H)| objects. Both G ∝ P and BP areconnected groupoids, the former equivalent to NG(H) and the latter NG(H)/CG(H). Thus theabove isomorphism can be interpreted as

HomkG(kG⊗kNG(H) M,N) ∼= HomkNG(H)(M,N)∼= Homk(NG(H)/CG(H))(M,HomkNG(H)(k(NG(H)/CG(H)), N))∼= Homk(NG(H)/CG(H))(M,NCG(H)),

where N ∈ kG-mod and M ∈ k(NG(H)/CG(H))-mod.

Corollary 5.5. We have HomkG(lim−→PM, k)∼= HomkBP (M, k) for any M ∈ kBP -mod.

Let Bb = BPb . If we fix a maximal object (S, eS) and take all objects (Q, eQ) with Q ⊂ S, thenthe full subcategory of Bb, consisting of all these objects, is called a fusion system, usually writtenas Fb or FS . The inclusion Fb ⊂ Bb is an equivalence. There is a general theory of (abstract)fusion systems and p-local finite groups introduced by Broto, Levi and Oliver[3]. We recall onlynecessary ingredients for our case. Let us take the full subcategory Fcb ⊂ Fb whose objects areself-centralizing b-Brauer pairs (Q, eQ) [13]. According to a newly established difficult theorem ofChermak [10] (true in general for all fusion systems), there exists a unique category, the centriclinking system Lc, fitting into the middle of a sequence

Z ↪→ Lc � Fc

which is an opposite extension. Here Z is the disjoint union of Z(Q) for all objects (Q, eQ) of Fcb .From Lemma 5.1 (2) we obtain the following isomorphism.

Proposition 5.6. Let N ∈ kLc-mod and M ∈ kFc-mod. Then

HomkLc(ResρM,N) ∼= HomkFc(M,NZ),

where NZ is defined by NZ(P ) = N(P )Z(P ).

Let Bcb be the full subcategory of Bb for a block b, consisting of self-centralizing b-Brauer pairs.Since Fcb naturally embeds into Bcb and it induces an equivalence, we similarly can consider anopposite extension

Z ↪→ L̃cb � Bcb .There exists a natural embedding Lcb → L̃cb inducing an category equivalence. By taking the largercategory L̃cb, we can write down

CG/Z ↪→ G ∝ Pcb � L̃cb,another opposite extension. Here CG/Z is the disjoint union of CG(P )/Z(P ) in which P runsover all F-centric subgroups. For the sake of convenience, we introduce a notation C ′G = CG/Zso that C ′G(P ) = CG(P )/Z(P ) ∼= PCG(P )/P for each P .

When M = κM for some M ∈ kG-mod, we write the kL̃cb-module RKρκM as MC′G . Given

a morphism (g, gP ≤ Q) it induces an injection C ′G(Q) → C ′G(P ) thus a morphism MC′G(P ) →

MC′G(Q). Hence we get the kL̃cb-action on MC

′G .


Proposition 5.7. Let M ∈ k(G ∝ Pcb )-mod and N ∈ kG-mod. Then

HomkG(lim−→PM, N)∼= HomkL̃cb(M, N

C′G),

where NC′G is defined as above.


In particular if M = ResρM′ for some M′ ∈ kBcb-mod then

HomkG(lim−→PM′, N) ∼= HomkBcb (M

′, NCG),

a special situation of Proposition 5.2.

Proof. The first part is a direct consequence of Corollary 5.3. As for the special case We need tonotice that lim−→P ResρM

′ ∼= lim−→PM′ as kG-modules. Then

HomkG(lim−→P ResρM′, N) ∼= HomkL̃cb(ResρM

′, NC′G)

∼= HomkBcb (M′, (NC

′G)Z)

∼= HomkBcb (M′, NCG).

�

5.3. Puig categories. If we take PA to be the poset of pointed subgroups on an interior G-algebra A, then analogues to the Brauer category for any G-subposet P ⊂ PA we can introducethe Puig category [13, Section 47] LP as a quotient category of G ∝ P such that

HomLP (Pγ , Qδ) = HomG∝P(Pγ , Qδ)/CG(P ).

Then some results in last section can be obtained accordingly.If M = κM for some M ∈ kG-mod, we denote by MCG the kLP -module RKρκM . Since a

morphism (g, gP ≤ Q) provides a group homomorphism P → Q and thus induces an injectioncg−1 : CG(Q)→ CG(P ), we obtain an injection MCG(P ) →MCG(Q). This leads to the kLP -actionon MCG .


Proposition 5.8. Let M ∈ kLP -mod and N ∈ kG-mod. ThenHomkG(lim−→PM, N)

∼= HomkLP (M, NCG),

where NCG is as above.

5.4. Orbit categories of fusion systems. This method also works for the orbit category of afusion system [9]. Since we are not going to recall the definition of an abstract fusion system, herewe only deal with a special case. Suppose Bb is a Brauer category and Fb is a fusion system asintroduced after Corollary 5.5. One may continue to define the orbit category OFb as a quotientcategory with

HomOFb (P,Q) = Q\HomFb(P,Q).Then we obtain an extension sequence

S ↪→ Fb � OFb ,where S is given by S(Q, eQ) = Q for any object (Q, eQ) ∈ ObFb. The above constructions stillwork if we replace Fb by Fcb . Lemma 5.1 (1) leads to the following statements.

Proposition 5.9. If M ∈ kFb-mod and N ∈ kOFb-mod, thenHomkFb(M,ResρN)

∼= HomkOFb (MS ,N).

Given M ∈ kFcb -mod, we get HomkG(lim−→PM, k)∼= HomkOFc

b(MS , k).

Proof. The first isomorphism comes directly from Lemma 5.1 (1). Replacing Fb and OFb byFcb and OFcb , respectively, we obtain HomkFcb (M,ResρN) ∼= HomkOFcb (MS ,N). Then we applyCorollary 5.5 to it and get

HomkG(lim−→PM, k)∼= HomkBcb (M, k)∼= HomkFcb (M, k)∼= HomkOFc

b(MS , k).

�

12 FEI XU

References

[1] D. Benson, Representations and Cohomology I, Cambridge Studies in Adv. Math. 30 (Cambridge UniversityPress 1998).

[2] D. Benson, Representations and Cohomology II, Cambridge Studies in Adv. Math. 31 (Cambridge UniversityPress 1998).

[3] C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003) 779-856.[4] H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press 1956.[5] W. G. Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997), 783-804.[6] E. Friendlander, A. Suslin, Cohomology of finite group schemes over a field, Inventiones Math. 127 (1997)

209-270.[7] P. Hilton, U. Stammbach, A Course in Homological Algebra (Second edition), GTM 4 (Springer 1997).[8] G. Hoff, Cohomologies et extensions de catégories, Math. Scand. 74 (1994), 191-207.[9] M. Linckelmann, Fusion category algebras, J. Algebra 277 (2004) 222-235..

[10] B. Oliver, Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory, preprint 2011.[11] M. A. Ronan, S. D. Smith, Sheaves on buildings and modular representations of Chevalley groups, J. Algebra

96 (1985), 319-346.[12] D. Swenson, The Steinberg Complex of an Arbitrary Finite Group in Arbitrary Positive Characteristic, PhD

thesis, University of Minnesota 2009.[13] J. Thévenaz, G-algebras and Modular Representation Theory, Oxford University Press 1995.[14] B. B. Venkov, Cohomology algebras for some classifying spaces, (Russian) Dokl. Akad. Nauk. SSSR 127 (1959)

943-944.[15] P. J. Webb, An introduction to the representations and cohomology of categories, in: Group Representation

Theory, (EPFL Press 2007), pp. 149-173.[16] F. Xu, On the cohomology rings of small categories, J. Pure Appl. Algebra 212 (2008) 2555-2569.[17] F. Xu, Hochschild and ordinary cohomology rings of small categories, Adv. Math. 219 (2008) 1872-1893.[18] F. Xu, Tensor structure on kC-mod and cohomology, to appear in Skye 2009 Conference Proceedings.[19] F. Xu, Becker-Gottlieb transfer for Hochschild cohomology, preprint 2011.[20] F. Xu, Support varieties for transporter category algebras, preprint 2011.

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain.E-mail address: [email protected]

on local categories of finite groups · on local categories of finite groups feixu abstract. let g...

Documents