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 transporter category as the corresponding Grothendieck construction. Transporter categories are generalizations 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 quotient categories 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 finitedimensional algebra kC, over a chosen field k. This algebra has a distinguished module, the trivial module \underline{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}^{∗}(\underline{k}, \underline{k}) = ⊕_{i≥0}Ext_{kC}^{i}(\underline{k}, \underline{k}) so that there is a ring isomorphism

Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(BC, k),

in which BC is the classifying space of C. This ring is called the ordinary cohomology ring of kC. In general such a ring is not finitely generated, but for various reasons we would like to have 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 the action by a group G, and proved that both the ordinary cohomology ring and the Hochschild cohomology ring of k(G ∝ P) are finitely generated. In this article, we continue to show that for any 𝔐, 𝔑 ∈ k(G ∝ P)-mod, Ext_{k(G∝P)}^{∗}(𝔐, 𝔑) is finitely generated over Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}), which generalizes the well known theorem of Evens and Venkov on group cohomology [2, 14]. Our main result is the following.

Theorem 1.1. Let G be a finite group and P a finite G-poset. Suppose k(G ∝ P) is the transporter category algebra. Then for any 𝔐, 𝔑 ∈ k(G ∝ P)-mod, Ext_{k(G∝P)}^{∗}(𝔐, 𝔑) is finitely generated over the Noetherian ring Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}).

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

In this article, we shall first recall the definition of a transporter category as well as basic facts about representations and cohomology of category algebras. We explain why transporter categories are interesting by examples in group representations and cohomology. These will be in Sections 2 and 3. Particularly we provide a generalized Frobenius Reciprocity in Section 3. The finite generation theorem is proved in Section 4, which itself is an evidence of the interests on transporter 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 a mistake 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 the Ministry of Science and Innovation of Spain.

2. preliminaries

In this section, we recall the definition of a transporter category and some background in category algebras. Throughout this article we will only consider finite categories, in the sense that they 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 transporter categories are mostly considered as auxiliary constructions before passing to various quotient categories of them. Here we want to stress on the perhaps unique property, among various categories constructed from a group, that transporter categories admit natural functors to the group 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 and subgroups, we follow a definition which is well known to some algebraic topologists. It is intended to be phrased explicitly in a way such that an algebraist, say a representation theorist, may readily accept as an entirely algebraic or categorical construction.

We deem a group as a category with one object, usually denoted by •. The identity of a group G 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 group homomorphism G → Aut(P). The Grothendieck construction on F will be called a transporter category. In the following explicit definition, the morphisms in a poset are customarily denoted by ≤.

Definition 2.1. Let G be a group and P a G-poset. The transporter category G ∝ P has the same objects as P, that is, Ob(G ∝ P) = Ob P. For x, y ∈ Ob(G ∝ P), a morphism from x to y is 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 Hom_{G∝P}(x, y) ≠ ∅ then both Aut_{G∝P}(x) and Aut_{G∝P}(y) act freely on Hom_{G∝P}(x, y).

The symbol G ∝ P is used because this particular Grothendieck construction resembles a semidirect product, yet is different. From the definition one can easily see that there is a natural embedding ι_{P} : P ↪ G ∝ P via (x ≤ y) ↦ (e, x ≤ y). On the other hand, the transporter category admits a natural functor π_{P} : G ∝ P → G, given by x ↦ • and (g, gx ≤ y) ↦ g. Thus we always have a sequence of functors

P↪^{ι_{P}}G ∝ P⟶^{π_{P}}G

such that π_{P} ∘ ι_{P}(P) is the trivial subgroup or subcategory of G. For convenience, in the rest of this article we often neglect the subscript P and write ι = ι_{P}, π = π_{P}. Topologically it is well known that B(G ∝ P) ≃ EG×_{G} BP. Passing to classifying spaces, we obtain a fibration sequence

BP⟶^{Bι}EG ×_{G} BP⟶^{Bπ}BG.

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

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

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/H which can be regarded as a G-poset: G acts via left multiplication. The transporter category G ∝ (G/H) is a connected groupoid whose skeleton is isomorphic to H. In this way one can recover all subgroups of G, up to category equivalences.

A category equivalence D → C induces a Morita equivalence between their category algebras-to be 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 module categories), or between BH and B(G ∝ (G/H)). Hence it makes sense if we deem transporter categories as generalized subgroups for a fixed finite group.

As two other motivating examples, transporter categories were implicitly considered by Mark Ronan and Steve Smith [11] in the 1980s for constructing group modules, and later on played a key role 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

\xymatrix{ & G ∝ P \ar[dl]_{π} \ar[dr]^{ρ} \\ G &  & C}

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

2.2. Category algebras, representations and cohomology. We recall some facts about category algebras. The reader is referred to [15, 17] for further details. Let C be a finite category and k a field. One can define the category algebra kC, which, as a vector space, has a basis the set of all 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-modules are usually written as M, N etc, while the modules of a (non-group) category algebra kC are denoted by 𝔐, 𝔑 etc., except those special modules, namely \underline{k} and κ_{M}, which are restrictions of kG-modules (to be defined shortly).

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

V ect_{k}^{C} ≃ kC-mod.

In the module category, there is a distinguished module \underline{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 ect_{k} is a symmetric monoidal category, V ect_{k}^{C} inherits this structure. It means there exists an (internal) tensor product, or the pointwise tensor product, written as ˆ⊗, such that for any two kC-modules 𝔐, 𝔑, (𝔐 ˆ⊗𝔑)(x) := 𝔐(x) ⊗_{k} 𝔑(x). Let α ∈ Mor C be a base element of kC. Then α acts on 𝔐 ˆ⊗𝔑 via α ⊗ α. Obviously \underline{k} is the identity with respect to ˆ⊗ and 𝔐 ˆ⊗𝔑 ≅ 𝔑 ˆ⊗𝔐. We can also construct a function object, the internal hom, Hom(𝔐, 𝔑) ∈ kC-mod such that

Hom_{kC}(𝔏 ˆ⊗𝔐, 𝔑) ≅ Hom_{kC}(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 is a group, we recover the usual construction − ˆ⊗− = − ⊗_{k} − and Hom(−, −) = Hom_{k}(−, −).

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

∪ : Ext_{kC}^{i}(𝔐, 𝔑) ⊗ Ext_{kC}^{j}(𝔐′, 𝔑′) → Ext_{kC}^{i+j}(𝔐 ˆ⊗𝔐′, 𝔑 ˆ⊗𝔑′).

In particular Ext_{kC}^{∗}(\underline{k}, \underline{k}) is a graded commutative ring and we have an isomorphism Ext_{kC}^{∗}(\underline{k}, \underline{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, but it is so when C = G ∝ P is finite (implicitly in Venkov’s proof of the finite generation of group cohomology).

One concept that we will refer to is the bar resolution, a combinatorally constructed projective resolution B_{∗}^{C}, of \underline{k} ∈ kC-mod.

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

Res_{τ} : kC-mod → kD-mod.

If we regard a kC-module as a functor, then its restriction is the precomposition of it with τ. If we consider the functor π : G ∝ P → G, then any kG-module M restricts to a k(G ∝ P)-module, written as κ_{M} = Res_{π} M, with only one exception \underline{k} = κ_{k} = Res_{π} k. The functor Res_{τ} is equipped with two adjoints: the left and right Kan extensions along τ

LK_{τ} , RK_{τ} : kD-mod → kC-mod.

The definition of the left and right Kan extensions [7] depend on the so-called over-categories and under-categories, respectively. For each x ∈ Ob C, one can define an over-category τ/x and an under-category x\τ. These concepts are frequently used in the representations and cohomology of category algebras. For example the bar resolution is constructed by using a certain collection of overcategories. The analysis of over- and undercategories associated with π leads to very interesting results, as they reveal the structures of various prominent kG-modules and complexes of 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 in group representations and cohomology. The purpose is to show that transporter categories should be further investigated, for better understanding of groups, or as prototypes for research in other local categories.

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

\xymatrix{ & G ∝ P \ar[dl]_{π} \ar[dr]^{ρ} \\ G &  & C}

for any given local category C.

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

\xymatrix{ & k(G ∝ P)-mod \ar[dl]_{LK_{π},RK_{π}} \ar[dr]^{LK_{ρ},RK_{ρ}} \\ kG-mod \ar[ur]_{_{Res_{π}}} &  & kC-mod. \ar[ul]^{_{Res_{ρ}}}}

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 category of G ∝ P as in the preceding diagrams. Let M, N ∈ kG-mod and 𝔐, 𝔑 ∈ kC-mod. Then

(1) Hom_{kG}(M, RK_{π} Res_{ρ} 𝔑) ≅ Hom_{kC}(LK_{ρ} Res_{π} M, 𝔑);

(2) Hom_{kG}(LK_{π} Res_{ρ} 𝔐, N) ≅ Hom_{kC}(𝔐, RK_{ρ} Res_{π} N).

By direct calculations, these particular Kan extensions in Proposition 3.1 are simplified: • LK_{π} ≅ \underrightarrow{lim}_{P}, RK_{π} ≅ \underleftarrow{lim}_{P} (used by Ronan-Smith. See also [18]); • LK_{ρ} ≅↑_{k(G∝P)}^{kC} (the induction), RK_{ρ} ≅⇑_{k(G∝P)}^{kC} (the co-induction), since ρ induces an algebra homomorphism k(G ∝ P) → kC when C is a quotient category. (One can easily prove that a functor τ : D → C induces a k-linear map τ : kD → kC, and moreover this map 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 Res_{ρ} 𝔑 as 𝔑 because they share the same underlying vector space. Recall that we have set κ_{M} = Res_{π} M. Then the Frobenius reciprocity can be rewritten as

(1’) Hom_{kG}(M, \underleftarrow{lim}_{P} 𝔑) ≅ Hom_{kC}(κ_{M} ↑_{k(G∝P)}^{kC}, 𝔑); (2’) Hom_{kG}(\underrightarrow{lim}_{P} 𝔐, N) ≅ Hom_{kC}(𝔐, κ_{N} ⇑_{k(G∝P)}^{kC}).

When P = G/H for some subgroup H, we have natural equivalences \underleftarrow{lim}_{G/H} ≅ \underrightarrow{lim}_{G/H} ≅↑_{H}^{G}.

Then the above isomorphisms certainly become the usual adjunctions between ↑_{H}^{G} and ↓_{H}^{G} (the usual Frobenius Reciprocity) with C = G ∝ (G/H) and ρ = Id, in light of the Morita equivalence between 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 Gmap 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 algebra homomorphism from k(G ∝ P) to k(G ∝ Q). Hence their Kan extensions along ρ cannot be interpreted as induction and coinduction.

The functors ↑_{k(G∝P)}^{kC} and ⇑_{k(G∝P)}^{kC} admit interesting interpretations when G ∝ P → C is part of an extension (or an opposite extension) sequence of categories. Under the circumstance ↑_{k(G∝P)}^{kC} and ⇑_{k(G∝P)}^{kC} on certain k(G ∝ P)-modules can be very well understood. We shall discuss it in Section 5.

4. Finite generation of cohomology

The functor π is very useful to study cohomology of transporter categories. The reader can find more applications in [19] where we construct the Becker-Gottlieb transfer map. We do not put the construction here because it requires substantial analysis of various over- and under-categories associated 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 functor from the transporter category G ∝ P to G, regarded as a category with one object •. The next result is a direct generalization of the fact that the two obvious kG-module structures on P ⊗M are isomorphic, for P, M ∈ kG-mod with P projective. It reveals a connection between representations of 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 · 1_{x} because 1 = ∑_{x∈Ob C} 1_{x} is an orthogonal decomposition by idempotents. A direct summand kC · 1_{x} is often written as k Hom_{C}(x, −) [15].

Theorem 4.1. Let 𝔓 ∈ k(G ∝ P)-mod be a projective module and κ_{M} = Res_{π} M for some M ∈ kG-mod. Then 𝔓 ˆ⊗κ_{M} is a projective k(G ∝ P)-module. Consequently B_{∗}^{G∝P} ˆ⊗κ_{M} → \underline{k} ˆ⊗κ_{M} = κ_{M} → 0 is a projective resolution.

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

We define a k-linear map φ : k Hom_{G∝P}(x, −) ⊗ M → k Hom_{G∝P}(x, −) ˆ⊗κ_{M} as follows. On base elements φ((g, gx ≤ y) ⊗ m) = (g, gx ≤ y) ⊗ (g, gx ≤ y)m, where the latter m is considered as an element in κ_{M}(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)].

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 Hom_{G∝P}(x, y) non-empty, both Aut_{G∝P}(x) and Aut_{G∝P}(y) act freely on Hom_{G∝P}(x, y). This implies that k Hom_{G∝P}(x, y) is a free kAut_{G∝P}(x)- or kAut_{G∝P}(y)-module. Consequently φ restricts on each y to the classical kAut_{G∝P}(y)-isomorphism (see [1, 3.1.5])

kHom_{G∝P}(x, y) ⊗ M → (kHom_{G∝P}(x, −) ˆ⊗κ_{M})(y) = kHom_{G∝P}(x, y) ⊗ M.

Furthermore because as vector spaces

kHom_{G∝P}(x, −) ˆ⊗κ_{M} = ⨁_{y∈Ob(G∝P)} kHom_{G∝P}(x, y) ⊗ M = kHom_{G∝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)}^{∗}(\underline{k}, \underline{k}) is finitely generated, and for an arbitrary 𝔑 ∈ k(G ∝ P)-mod, Ext_{k(G∝P)}^{∗}(\underline{k}, 𝔑) is finitely generated over the Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}).

The proof is entirely analogous to Venkov’s proof for group cohomology [2, 14]. The main difference here is the use of the Grothendieck spectral sequence, instead of the Leray-Serre spectral sequence, 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)}^{∗}(κ_{M}, 𝔑) is finitely generated over Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{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)}^{∗}(κ_{M}, 𝔑) ≅ Ext_{k(G∝P)}^{∗}(\underline{k}, Hom(κ_{M}, 𝔑)).

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

However in general Ext_{k(G∝P)}^{∗}(𝔐, 𝔑) ≅ E̸xt_{k(G∝P)}^{∗}(\underline{k}, Hom(𝔐, 𝔑)), showing the limit of internal hom. It is striking that Hochschild cohomology occurs when we solve the finite generation problem in the next section.

4.2. Finite generation through Hochschild cohomology. In [17], we proved that the ordinary cohomology ring Ext_{kC}^{∗}(\underline{k}, \underline{k}) is closely related to the Hochschild cohomology ring Ext_{kC^{e}}^{∗}(kC, kC). In order to see the connection we have to rely on F (C) (another finite category), the category of factorizations in C. There are natural functors and a commutative diagram

\xymatrix{F (C) \ar[rr]^{∇} \ar[dr]_{t} &  & C^{e} = C × C^{op} \ar[dl]^{p} \\  & C & .}

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

(1) Ext_{kC^{e}}^{∗}(kC, 𝔐) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{∇}𝔐).

for any 𝔐 ∈ kC^{e}-mod [17]. Thus our result is a generalization of a well know theorem in group cohomology. In fact when C = G is a finite group, G^{e} ≅ G × G and F (G) ≃ ΔG. We comment that although F (C) and C usually are not equivalent as categories, their classifying spaces are homotopy equivalent. Consequently

(2) Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{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, Hom_{k}(M, N)),

where Hom_{k}(M, N) is the internal hom. This is not the case for category cohomology as we mentioned earlier. Fortunately we find a replacement for the above isomorphism. The trick is that we must replace C with F (C). To this end we recall a theorem of Cartan-Eilenberg [4, Chapter IX, 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_{A^{e}}^{∗}(A, Hom_{k}(M, N)).

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

Ext_{kC}^{∗}(𝔐, 𝔑) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{∇}Hom_{k}(𝔐, 𝔑)).

Proof. By (3) and (1)

Ext_{kC}^{∗}(𝔐, 𝔑) ≅ Ext_{kC^{e}}^{∗}(kC, Hom_{k}(𝔐, 𝔑)) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{∇}Hom_{k}(𝔐, 𝔑)).
□

Thus in order to prove the finite generation of Ext_{kC}^{∗}(𝔐, 𝔑), we only need a result like Proposition 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)}^{∗}(\underline{k}, \underline{k}) is finitely generated, and for any 𝔑 ∈ kF (G ∝ P)-mod, Ext_{kF (G∝P)}^{∗}(\underline{k}, 𝔑) becomes a finitely generated Ext_{kF (G∝P)}^{∗}(\underline{k}, \underline{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 functors F (G ∝ P) → G ∝ P → G, which allows us to show that the spectral sequence has only finitely many rows.

Theorem 4.6. Suppose 𝔐, 𝔑 are two k(G ∝ P)-modules. Then the module Ext_{k(G∝P)}^{∗}(𝔐, 𝔑) is finitely generated over Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}).

Proof. By Proposition 4.4

Ext_{k(G∝P)}^{∗}(𝔐, 𝔑) ≅ Ext_{kF (G∝P)}^{∗}(\underline{k}, Res_{∇}Hom_{k}(𝔐, 𝔑)).

Hence by Proposition 4.5 and (2), Ext_{kF (G∝P)}^{∗}(\underline{k}, Res_{∇} Hom_{k}(M, N)) is finitely generated over the ring Ext_{kF (G∝P)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}). We are done. □

Because of the above theorem, we can define the variety for a k(G ∝ P)-module 𝔐 to be the maximal ideal spectrum of Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k})/I_{𝔐}, where I_{𝔐} is the annihilator of the following map

− ˆ⊗𝔐 : Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}) → Ext_{k(G∝P)}^{∗}(𝔐, 𝔐).

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

5. The functor ρ: invariants and coinvariants

Here we return to the Frobenius Reciprocity and take the opportunity to write out explicitly the formulas in some interesting cases. In practice one often finds that the functor G ∝ P → C is part of an extension (or an opposite extension) sequence of categories. (Among examples are various orbit categories, Brauer categories and Puig categories.) It means that for such a quotient category C there exists a category K which is a disjoint union of subgroups K_{x} ⊂ Aut_{G∝P}(x), x running over Ob P = Ob(G ∝ P), such that we can add K into the picture

\xymatrix{K \ar[dr]^{ι} \\  & G ∝ P \ar[dl]_{π} \ar[dr]^{ρ} \\ G &  & C}

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

To this end we need to understand the functor ρ : G ∝ P → C, where C is a quotient category of G ∝ P. In general situation it seems hard to make group theoretic interpretation of ↑_{k(G∝P)}^{kC} and ⇑_{k(G∝P)}^{kC}. However we can do so when we have certain quotient categories, which are part of some 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) Ob K = 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 ∈ Hom_{C}(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}(1_{x}) for all 1_{x} ∈ 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 K^{op} ↪ E^{op} ↠ C^{op} 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 opposite extension) is that it enables us to provide a good characterization of the left (or right) Kan extension. Indeed it is the case for many familiar category constructions in representation theory and 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 LK_{ρ}𝔐 ≅ 𝔐_{K}, where 𝔐_{K} as a functor over C is given by 𝔐_{K}(x) = 𝔐(x)_{K(x)} (K(x)-coinvariants of the kAut_{E}(x)-module 𝔐(x)), for

any x ∈ Ob C = Ob E = Ob K.

(2) Suppose K → E → C is an opposite extension. Then RK_{ρ}𝔐 ≅ 𝔐^{K}, where 𝔐^{K} as a functor over C is given by 𝔐^{K}(x) = 𝔐(x)^{K(x)} (K(x)-invariants of the kAut_{E}(x)-module

𝔐(x)), for any x ∈ Ob C = Ob E = Ob K.

In the above lemma, there is another way to express the Kan extensions. Under the same assumptions, they are 𝔐_{K} = H_{0}(K; 𝔐) and 𝔐^{K} = H^{0}(K; 𝔐) respectively.

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

(1’) Hom_{kG}(M, \underleftarrow{lim}_{P} 𝔑) ≅ Hom_{kC}(LK_{ρ}κ_{M}, 𝔑); (2’) Hom_{kG}(\underrightarrow{lim}_{P} 𝔐, N) ≅ Hom_{kC}(𝔐, RK_{ρ}κ_{N}).

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

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

Hom_{O_{P}} (P, Q) = Q\N_{G}(P, Q).

Then we have an extension sequence

S ↪ G ∝ P ↠ O_{P},

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

When 𝔐 = κ_{M} for some M ∈ kG-mod, (LK_{ρ}κ_{M})(P) = M_{P} for any P ∈ Ob P. We denote such a kO_{P}-module by M_{S} := (κ_{M})_{S} = LK_{ρ}κ_{M}. Since giving a morphism (g,^{g}P ≤ Q) is the same as giving a group homomorphism P → Q, the conjugation induced by g, there is a natural way to construct a map M_{P} → M_{Q}, identical to the natural map H_{0}(P; M) = k ⊗_{kP} M → k ⊗_{kQ} M = H_{0}(Q; M). Hence we know how kO_{P} acts on M_{S}.

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

Proposition 5.2. Let M ∈ kG-mod and 𝔑 ∈ kO_{P}-mod. Then

Hom_{kG}(M, \underleftarrow{lim}_{P}𝔑) ≅ Hom_{kO_{P}} (M_{S}, 𝔑),

where M_{S} is as above.

Corollary 5.3. With the same notations as above, we have \underleftarrow{lim}_{P}𝔑 ≅ Hom_{kO_{P}} (kG_{S}, 𝔑) and Hom_{kG}(k, \underleftarrow{lim}_{P}𝔑) ≅ Hom_{kO_{P}} (\underline{k}, 𝔑).

As an example we let H be a subgroup of G and P the subgroups that are conjugate to H. The size of the discrete poset P is G/N_{G}(H). Note that both G ∝ P and O_{P} are connected groupoids, the former equivalent to N_{G}(H) and the latter N_{G}(H)/H. Thus the isomorphism in Proposition 5.2 can be interpreted as

Hom_{kG}(M, Hom_{kN_{G}(H)}(kG, N)) ≅ Hom_{kN_{G}(H)}(M, N)
≅ Hom_{k(N_{G}(H)/H)}(k(N_{G}(H)/H) ⊗_{kN_{G}(H)} M, N)
≅ Hom_{k(N_{G}(H)/H)}(M_{H}, N),

where M ∈ kG-mod and N ∈ k(N_{G}(H)/H)-mod.

5.2. Brauer categories, fusion and linking systems. Suppose b is a p-block of the group algebra kG and P_{b} is the poset of b-Brauer pairs [13, Section 47]. Then for any G-subposet P ⊂ P_{b} we can introduce the Brauer category B_{P} as the quotient category of G ∝ P such that

Hom_{B_{P}} (P, Q) = Hom_{G∝P}(P, Q)/C_{G}(P).

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

C_{G} ↪ (G ∝ P)^{op} → B_{P}^{op}

is an extension sequence, given that C_{G} is the disjoint union of all C_{G}(P), P ∈ Ob P. Dual to the extension situation we examined before, now we are able to describe the right Kan extension of modules.

If 𝔐 = κ_{M} for some M ∈ kG-mod, we denote by M^{C_{G}} the kF_{P}-module RK_{ρ}κ_{M}. Since a morphism (g,^{g}P ≤ Q) provides a group homomorphism P → Q and thus induces an injection c_{g^{−1}} : C_{G}(Q) → C_{G}(P), we obtain an injection M^{C_{G}(P)} → M^{C_{G}(Q)}. This leads to the kB_{P}-action on M^{C_{G}}.

Combining the Frobenius Reciprocity (2’) and Lemma 5.1 (2), we obtain the following isomorphism.

Proposition 5.4. Let 𝔐 ∈ kB_{P}-mod and N ∈ kG-mod. Then

Hom_{kG}(\underrightarrow{lim}_{P}𝔐, N) ≅ Hom_{kB_{P}} (𝔐, N^{C_{G}}),

where N^{C_{G}} is as above.

As an example we assume b is the principal block b_{0} and P is the conjugacy class of a fixed p-subgroup H. Then the discrete poset P has |G/N_{G}(H)| objects. Both G ∝ P and B_{P} are connected groupoids, the former equivalent to N_{G}(H) and the latter N_{G}(H)/C_{G}(H). Thus the above isomorphism can be interpreted as

Hom_{kG}(kG ⊗_{kN_{G}(H)} M, N) ≅ Hom_{kN_{G}(H)}(M, N)

≅ Hom_{k(N_{G}(H)/C_{G}(H))}(M, Hom_{kN_{G}(H)}(k(N_{G}(H)/C_{G}(H)), N))

≅ Hom_{k(N_{G}(H)/C_{G}(H))}(M, N^{C_{G}(H)}),

where N ∈ kG-mod and M ∈ k(N_{G}(H)/C_{G}(H))-mod.

Corollary 5.5. We have Hom_{kG}(\underrightarrow{lim}_{P}𝔐, k) ≅ Hom_{kB_{P}} (𝔐, \underline{k}) for any 𝔐 ∈ kB_{P}-mod.

Let B_{b} = B_{P_{b}}. If we fix a maximal object (S, e_{S}) and take all objects (Q, e_{Q}) with Q ⊂ S, then the full subcategory of B_{b}, consisting of all these objects, is called a fusion system, usually written as F_{b} or F_{S}. The inclusion F_{b} ⊂ B_{b} 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 only necessary ingredients for our case. Let us take the full subcategory F_{b}^{c} ⊂ F_{b} whose objects are self-centralizing b-Brauer pairs (Q, e_{Q}) [13]. According to a newly established difficult theorem of Chermak [10] (true in general for all fusion systems), there exists a unique category, the centric linking system L^{c}, fitting into the middle of a sequence

Z ↪ L^{c} ↠ F^{c}

which is an opposite extension. Here Z is the disjoint union of Z(Q) for all objects (Q, e_{Q}) of F_{b}^{c}. From Lemma 5.1 (2) we obtain the following isomorphism.

Proposition 5.6. Let 𝔑 ∈ kL^{c}-mod and 𝔐 ∈ kF^{c}-mod. Then

Hom_{kL^{c}}(Res_{ρ}𝔐, 𝔑) ≅ Hom_{kF^{c}}(𝔐, 𝔑^{Z}),

where 𝔑^{Z} is defined by 𝔑^{Z}(P) = 𝔑(P)^{Z(P)}.

Let B_{b}^{c} be the full subcategory of B_{b} for a block b, consisting of self-centralizing b-Brauer pairs. Since F_{b}^{c} naturally embeds into B_{b}^{c} and it induces an equivalence, we similarly can consider an opposite extension

Z ↪ L̃_{b}^{c} ↠ B_{b}^{c}.

There exists a natural embedding L_{b}^{c} → L̃_{b}^{c} inducing an category equivalence. By taking the larger category L̃_{b}^{c}, we can write down

C_{G}/Z ↪ G ∝ P_{b}^{c} ↠ L̃_{b}^{c},

another opposite extension. Here C_{G}/Z is the disjoint union of C_{G}(P)/Z(P) in which P runs over all F-centric subgroups. For the sake of convenience, we introduce a notation C′_{G} = C_{G}/Z so that C′_{G}(P) = C_{G}(P)/Z(P) ≅ P C_{G}(P)/P for each P.

When 𝔐 = κ_{M} for some M ∈ kG-mod, we write the kL̃_{b}^{c}-module RK_{ρ}κ_{M} as M^{C′_{G}}. Given a morphism (g,^{g}P ≤ Q) it induces an injection C′_{G}(Q) → C′_{G}(P) thus a morphism M^{C′_{G}(P)} → M^{C′_{G}(Q)}. Hence we get the kL̃_{b}^{c}-action on M^{C′_{G}}.

Combining the Frobenius Reciprocity (2’) and Lemma 5.1 (2), we obtain the following isomorphism.

Proposition 5.7. Let 𝔐 ∈ k(G ∝ P_{b}^{c})-mod and N ∈ kG-mod. Then

Hom_{kG}(\underrightarrow{lim}_{P}𝔐, N) ≅ Hom_{kL̃_{b}^{c}}(𝔐, N^{C′_{G}}),

where N^{C′_{G}} is defined as above.

In particular if 𝔐 = Res_{ρ} 𝔐′ for some 𝔐′ ∈ kB_{b}^{c}-mod then

Hom_{kG}(\underrightarrow{lim}_{P}𝔐′, N) ≅ Hom_{kB_{b}^{c}}(𝔐′, N^{C_{G}}),

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 to notice that \underrightarrow{lim}_{P} Res_{ρ} 𝔐′ ≅ \underrightarrow{lim}_{P} 𝔐′ as kG-modules. Then

Hom_{kG}(\underrightarrow{lim}_{P} Res_{ρ} 𝔐′, N) ≅ Hom_{kL̃_{b}^{c}}(Res_{ρ} 𝔐′, N^{C′_{G}})
≅ Hom_{kB_{b}^{c}}(𝔐′, (N^{C′_{G}})^{Z})
≅ Hom_{kB_{b}^{c}}(𝔐′, N^{C_{G}}).
□

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

Hom_{L_{P}} (P_{γ}, Q_{δ}) = Hom_{G∝P}(P_{γ}, Q_{δ})/C_{G}(P).

Then some results in last section can be obtained accordingly.

If 𝔐 = κ_{M} for some M ∈ kG-mod, we denote by M^{C_{G}} the kL_{P}-module RK_{ρ}κ_{M}. Since a morphism (g,^{g}P ≤ Q) provides a group homomorphism P → Q and thus induces an injection c_{g^{−1}} : C_{G}(Q) → C_{G}(P), we obtain an injection M^{C_{G}(P)} → M^{C_{G}(Q)}. This leads to the kL_{P}-action on M^{C_{G}}.

Combining the Frobenius Reciprocity (2’) and Lemma 5.1 (2), we obtain the following isomorphism.

Proposition 5.8. Let 𝔐 ∈ kL_{P}-mod and N ∈ kG-mod. Then

Hom_{kG}(\underrightarrow{lim}_{P}𝔐, N) ≅ Hom_{kL_{P}} (𝔐, N^{C_{G}}),

where N^{C_{G}} is as above.

5.4. Orbit categories of fusion systems. This method also works for the orbit category of a fusion system [9]. Since we are not going to recall the definition of an abstract fusion system, here we only deal with a special case. Suppose B_{b} is a Brauer category and F_{b} is a fusion system as introduced after Corollary 5.5. One may continue to define the orbit category O_{F_{b}} as a quotient category with

Hom_{O_{F_{b}}} (P, Q) = Q\Hom_{F_{b}}(P, Q).

Then we obtain an extension sequence

S ↪ F_{b} ↠ O_{F_{b}},

where S is given by S(Q, e_{Q}) = Q for any object (Q, e_{Q}) ∈ Ob F_{b}. The above constructions still work if we replace F_{b} by F_{b}^{c}. Lemma 5.1 (1) leads to the following statements.

Proposition 5.9. If 𝔐 ∈ kF_{b}-mod and 𝔑 ∈ kO_{F_{b}}-mod, then

Hom_{kF_{b}}(𝔐, Res_{ρ}𝔑) ≅ Hom_{kO_{F_{b}}} (𝔐_{S}, 𝔑).

Given 𝔐 ∈ kF_{b}^{c}-mod, we get Hom_{kG}(\underrightarrow{lim}_{P}𝔐, k) ≅ Hom_{kO_{F^{c}_{b}}} (𝔐_{S}, \underline{k}).

Proof. The first isomorphism comes directly from Lemma 5.1 (1). Replacing F_{b} and O_{F_{b}} by F_{b}^{c} and O_{F_{b}^{c}}, respectively, we obtain Hom_{kF_{b}^{c}}(𝔐, Res_{ρ}𝔑) ≅ Hom_{kO_{F^{c}_{b}}} (𝔐_{S}, 𝔑). Then we apply Corollary 5.5 to it and get

Hom_{kG}(\underrightarrow{lim}_{P}𝔐, k) ≅ Hom_{kB_{b}^{c}}(𝔐, \underline{k})
≅ Hom_{kF_{b}^{c}}(𝔐, \underline{k})
≅ Hom_{kO_{F^{c}_{b}}} (𝔐_{S}, \underline{k}).
□

References

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

[2] D. Benson, Representations and Cohomology II, Cambridge Studies in Adv. Math. 31 (Cambridge University Press 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: xu@mat.uab.cat