QUILLEN STRATIFICATION FOR

THE STABLE MODULE CATEGORY

Wayne W. Wheeler^{∗}

Abstract. One of the major developments in the cohomology of finite groups over the past few decades has been the theory of varieties for modules, and the Quillen Stratification Theorem is a fundamental result in this area. If G is a finite group and k is an algebraically closed field of characteristic p > 0, then this result shows that the prime ideal spectrum V_{G}(k) of the cohomology ring H^{∗}(G, k) is a disjoint union of locally closed subsets V_{G,E}^{+} corresponding to conjugacy classes of elementary abelian p-subgroups E of G. The purpose of this paper is to prove an analogous result for the stable category kG-\underline{Mod} of kG-modules modulo projectives. The categorical equivalent of a stratification is the concept of a recollement of categories, and we show that one can use this concept to decompose the category kG-\underline{Mod} into certain quotient categories C_{G,E}^{+} corresponding to conjugacy classes of elementary abelian p-subgroups E of G. Moreover, the categories C_{G,E}^{+} are determined by the local structure of the group G.

1. Introduction

One of the most useful results in the cohomology of finite groups is the Quillen Stratification Theorem [12], which describes the structure of the prime ideal spectrum of the cohomology ring. If G is a finite group and k is an algebraically closed field of characteristic p, write V_{G}(k) for the prime ideal spectrum of H^{∗}(G, k). For any elementary abelian p-subgroup E of G the restriction map res_{G,E} : H^{∗}(G, k) → H^{∗}(E, k) induces a map res_{G,E}^{∗} : V_{E}(k) → V_{G}(k), and we set V_{G,E} = res_{G,E}^{∗}(V_{E}(k)) and

V_{G,E}^{−} = ⋃_{E′<E} res_{G,E′}^{∗}(V_{E′}(k)).

If V_{G,E}^{+} = V_{G,E} − V_{G,E}^{−}, then Quillen’s result states that the set V_{G,E}^{+} is locally closed; moreover, if E is a set of representatives for the conjugacy classes of elementary abelian p-subgroups of G, then the prime ideal spectrum V_{G}(k) is the disjoint union of the locally closed subsets V_{G,E}^{+} for E ∈ E.

In this paper we will prove an analogue of the Quillen Stratification Theorem for the stable category kG-\underline{Mod} of left kG-modules modulo projectives. As Cline, Parshall, and Scott have pointed out in [7], the idea of a recollement is the appropriate categorical

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C20, 20J06.

∗Partially supported by the NSF.

concept needed to replace that of a stratification of a topological space. This idea was first introduced by Beilinson, Bernstein, and Deligne in their work on singular spaces [2]. A recollement allows one to decompose a category into other categories in much the same way that one can use a short exact sequence to decompose a module into a submodule and a quotient module.

Theorem 3.4 below shows that if E is a set of representatives for the conjugacy classes of nontrivial elementary abelian p-subgroups of G, then the stable module category kG- \underline{Mod} decomposes into a collection of certain categories C_{G,E}^{+} for E ∈ E; stated differently, the category kG-\underline{Mod} can be obtained from the categories C_{G,E}^{+} by iterated recollement. Moreover, Theorem 4.6 shows that C_{G,E}^{+} is equivalent to C_{N_{G}(E),E}^{+}, so it is locally determined. As one might guess from the analogy with decompositions of modules, the categories C_{G,E}^{+} are quotients of subcategories of kG-\underline{Mod}.

It is interesting to note that the main results of this paper all seem to require dealing with infinitely generated modules. The ideas make heavy use of the recent work of Rickard on idempotent modules [13] and of Benson, Carlson, and Rickard on varieties for infinitely generated modules [5, 6]. In particular, idempotent modules are used to define the functors occurring in the recollements that decompose kG-\underline{Mod} into other subcategories. Because idempotent modules are usually infinitely generated, it seems doubtful that one could obtain similar results for the stable category kG-\underline{mod} of finitely generated modules.

2. Idempotent modules and recollement

It has long been recognized that the stable category of finitely generated modules provides an appropriate context for studying cohomology theory, but only in the past few years have people realized that infinitely generated modules have advantages that their finitely generated counterparts lack. In particular, Rickard has shown that the stable module category contains certain infinitely generated modules M that are idempotent in the sense that M ⊗ M ≅ M in the stable category (see [13]). In this section we reinterpret the work of Rickard to show that one can obtain stratifications of the stable module category from certain subcategories of the finitely generated stable category.

Throughout this paper G will denote a finite group, and k will be an algebraically closed field whose characteristic p divides |G|. We write kG-Mod for the category of all left kG-modules and kG-mod for the full subcategory of all finitely generated modules. Similarly, kG-\underline{Mod} and kG-\underline{mod} denote the corresponding stable categories. Explicitly, the objects of kG-\underline{Mod} are the same as those of kG-Mod. If M and N are kG-modules, let PHom_{kG}(M, N) denote the k-subspace of Hom_{kG}(M, N) consisting of those maps that factor through a projective kG-module. Then the morphisms from M to N in the stable category kG-\underline{Mod} are defined by setting

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/ PHom_{kG}(M, N),

and kG-\underline{mod} is the full subcategory of kG-\underline{Mod} consisting of all finitely generated modules. If f : M → N is a kG-homomorphism, then we sometimes also write f for the corresponding map in kG-\underline{Mod} if no confusion is likely to arise. The categories kG-\underline{Mod} and kG-\underline{mod} are triangulated (Theorem I.2.6 of [10]), and the translation functor is given by Ω^{−1}.

Recall that if C is a triangulated category and D is a triangulated subcategory of C, then D is said to be a thick subcategory if it is closed under taking direct summands of objects. In this case a localization procedure is used to define the quotient C/D, which is again a triangulated category. A good discussion of localization can be found in [9].

Now suppose that C is any triangulated subcategory of kG-\underline{Mod}, and let C^{⊕} denote the smallest full triangulated subcategory of kG-\underline{Mod} that contains C and is closed under arbitrary direct sums. In [13] a module M is said to be C-local if \underline{Hom}_{kG}(C, M) = 0 for all C in C. In this paper we write C^{⊥} for the subcategory of all C-local objects. It is easy to see that C^{⊥} is always a thick subcategory of kG-\underline{Mod}, and C^{⊥} = (C^{⊕})^{⊥} by Lemma 5.2 of [13].

Definition 2.1. Let C be a thick subcategory of kG-\underline{mod} such that if M is in C and N is in kG-\underline{mod}, then M ⊗ N is in C. Then C will be called an ideal subcategory of kG-\underline{mod}.

The following proposition summarizes the fundamental facts about idempotent modules from [13].

Proposition 2.2. Let C be an ideal subcategory of kG-\underline{mod}. For any object M in kG- \underline{Mod} there is a triangle

e_{C}(M) ⟶ M ⟶ f_{C}(M) ⟶ Ω^{−1}(e_{C}(M))

in kG-\underline{Mod} such that e_{C}(M) is in C^{⊕} and f_{C}(M) is in C^{⊥}, and such a triangle is unique up to isomorphism. The morphism e_{C}(M) → M is the universal map in kG-\underline{Mod} from an object of C^{⊕} to M, and M → f_{C}(M) is the universal map from M to an object of C^{⊥}. The modules e_{C}(k) and f_{C}(k) are idempotent in the sense that e_{C}(k) ⊗ e_{C}(k) ≅ e_{C}(k) and f_{C}(k)⊗f_{C}(k) ≅ f_{C}(k) in kG-\underline{Mod}. Moreover, M ⊗e_{C}(k) ≅ e_{C}(M) and M ⊗f_{C}(k) ≅ f_{C}(M) in kG-\underline{Mod} for any module M.

If C is an ideal subcategory of kG-\underline{mod}, then for simplicity we write e for e_{C}(k) and f for f_{C}(k) throughout the remainder of this section.

Proposition 2.3. Let C be an ideal subcategory of kG-\underline{mod}. If M is any object of kG- \underline{Mod}, then Hom_{k}(f, M) lies in C^{⊥}. Moreover, Hom_{k}(f, −) : kG-\underline{Mod} → C^{⊥} is a right adjoint of the inclusion functor ι : C^{⊥} → kG-\underline{Mod}, and − ⊗ f : kG-\underline{Mod} → C^{⊥} is a left adjoint of the inclusion functor.

Proof. Let M be an object of kG-\underline{Mod}, and let C be an object of C. Then Proposition 2.2 implies that C ⊗ f ≅ f_{C}(C) ≅ 0 in kG-\underline{Mod}, so

\underline{Hom}_{kG}(C, Hom_{k}(f, M)) ≅ \underline{Hom}_{kG}(C ⊗ f, M) = 0.

Thus Hom_{k}(f, M) lies in C^{⊥}. Moreover, if D is an object of C^{⊥}, then D ⊗ f is stably isomorphic to D so that there is a natural isomorphism

Hom_{C^{⊥}}(D, Hom_{k}(f, M)) = \underline{Hom}_{kG}(D, Hom_{k}(f, M))
≅ \underline{Hom}_{kG}(D ⊗ f, M)
≅ \underline{Hom}_{kG}(ιD, M).

Hence Hom_{k}(f, −) is a right adjoint of ι. Finally, if M is any kG-module and D is an object of C^{⊥}, then any stable map from M to D factors uniquely through f_{C}(M) ≅ M ⊗f by Proposition 2.2. Thus \underline{Hom}_{kG}(M, D) ≅ \underline{Hom}_{kG}(M ⊗ f, D), and there is a natural isomorphism

Hom_{C^{⊥}}(M ⊗ f, D) = \underline{Hom}_{kG}(M ⊗ f, D) ≅ \underline{Hom}_{kG}(M, ιD).

This completes the proof. □

To state the next result, we recall that if i_{∗} : D′ → D is any functor, then the strict image of D′ under i_{∗} is the full subcategory of D consisting of all objects isomorphic to i_{∗}D′ for some object D′ in D′. Proposition 2.3 shows that the inclusion functor ι : C^{⊥} → kG-\underline{Mod} satisfies the assumptions of the following proposition.

Proposition 2.4. Let D and D′ be triangulated categories, and suppose that i_{∗} : D′ → D is an exact functor that is a full embedding. Assume that i_{∗} has both a left adjoint i^{∗} and a right adjoint i^{!}, and let E be the strict image of D′ under i_{∗}. Then E is a thick subcategory of D, and the quotient morphism j^{∗} : D → D/E has both a left adjoint j_{!} and a right adjoint j_{∗}. Moreover, j_{!} and j_{∗} are full embeddings.

Proof. Adjoints of exact functors between triangulated categories are exact by Lemma 3.9 of [11]. The proposition is simply a restatement of Theorem 2.1 of [7] obtained by using this fact. □

When the conditions of Proposition 2.4 hold, we say that D admits a recollement (with respect to D′ and D/E). This situation is sometimes summarized by the diagram

[figure: D′; D; D/E]

in which each functor is the left adjoint of the one occurring above it. When (X, O_{X}) is a ringed space, F ⊆ X is a closed subspace, and U = X − F, a recollement naturally arises relating certain derived categories of O_{X}-modules, O_{F} -modules, and O_{U}-modules (see Section 1.4 of [2]).

In the situation of interest to us C is an ideal subcategory of kG-\underline{mod}, and the resulting recollement takes the form

[figure: C^{⊥}; kG-\underline{Mod}; kG-\underline{Mod}/C^{⊥}.]

The functor − ⊗ e : kG-\underline{Mod} → C^{⊕} vanishes on C^{⊥}, so it induces a functor − ⊗ e : kG-\underline{Mod}/C^{⊥} → C^{⊕} of triangulated categories.

Proposition 2.5. Let C be an ideal subcategory of kG-\underline{mod}. Then the composition C^{⊕} ↪ kG-\underline{Mod} → kG-\underline{Mod}/C^{⊥} is an equivalence of triangulated categories with a quasi-inverse given by − ⊗ e : kG-\underline{Mod}/C^{⊥} → C^{⊕}.

Proof. The composition C^{⊕} ↪ kG-\underline{Mod} → kG-\underline{Mod}/C^{⊥} → C^{⊕} is just the functor − ⊗ e : C^{⊕} → C^{⊕}. Similarly, the composition kG-\underline{Mod}/C^{⊥} → C^{⊕} ↪ kG-\underline{Mod} → kG-\underline{Mod}/C^{⊥} is the functor induced by − ⊗ e. It is easy to check that in both cases these functors are naturally isomorphic to the identity. □

It follows from Proposition 2.5 that if C is an ideal subcategory of kG-\underline{mod}, then there is a commutative diagram

[figure: kG-\underline{Mod}; −⊗e; kG-\underline{Mod}/C^{⊥}; −⊗e; C^{⊕},]

and the functor − ⊗ e : kG-\underline{Mod}/C^{⊥} → C^{⊕} is an equivalence. Using arguments similar to those given in the proof of Proposition 2.3, one can easily check that the inclusion functor ι : C^{⊕} → kG-\underline{Mod} is a left adjoint of − ⊗ e : kG-\underline{Mod} → C^{⊕}, and Hom_{k}(e, −) : C^{⊕} → kG-\underline{Mod} is a right adjoint. Thus we obtain the following result.

Theorem 2.6. Let C be an ideal subcategory of kG-\underline{mod}. Then there is a recollement

[figure: C^{⊥}; kG-\underline{Mod}; C^{⊕}.]

The inclusion functor C^{⊥} ↪ kG-\underline{Mod} has a left adjoint given by −⊗f and a right adjoint given by Hom_{k}(f, −); the functor − ⊗ e : kG-\underline{Mod} → C^{⊕} has a left adjoint given by the inclusion and a right adjoint given by Hom_{k}(e, −).

The recollement of Theorem 2.6 identifies C^{⊕} as the quotient kG-\underline{Mod}/C^{⊥}. On the other hand, C^{⊕} is also a thick subcategory of kG-\underline{Mod}, and the next result shows that this recollement has the property that kG-\underline{Mod}/C^{⊕} is also equivalent to C^{⊥}. The proof is straightforward and is left to the reader.

Proposition 2.7. Let C be an ideal subcategory of kG-\underline{mod}. Then the composition C^{⊥} ↪ kG-\underline{Mod} → kG-\underline{Mod}/C^{⊕} is an equivalence of triangulated categories with a quasi-inverse given by − ⊗ f : kG-\underline{Mod}/C^{⊕} → C^{⊥}.

3. Quillen stratification for the stable category

In this section we apply the results of Section 2 to obtain an analogue of the Quillen Stratification Theorem for kG-\underline{Mod}. The strata that correspond to the locally closed subspaces in Quillen’s Theorem are certain categories corresponding to the elementary abelian p-subgroups of G. These categories are shown to be determined by the local structure of G in the next section.

We begin with a brief review of the theory of varieties for modules. Although it is common to deal exclusively with subvarieties of the maximal ideal spectrum of H^{∗}(G, k), we will instead use subschemes of the prime ideal spectrum. First assume that M is a finitely generated kG-module, and let J(M) denote the annihilator of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k). The collection of all prime ideals in H^{∗}(G, k) that contain J(M) is a closed subscheme V_{G}(M) of the prime ideal spectrum V_{G}(k) of H^{∗}(G, k). In the situation to be studied here, however, it is most natural to deal with homogeneous prime ideals. We will therefore write V̄_{G}(M) for the collection of all essential homogeneous prime ideals of H^{∗}(G, k) containing J(M). Then V̄_{G}(M) is a projective subscheme of V̄_{G}(k) = Proj H^{∗}(G, k).

For infinitely generated modules, however, this definition of V̄_{G}(M) does not seem to be the appropriate one to use. Recent work of Benson, Carlson, and Rickard [5, 6] develops a theory of varieties for infinitely generated modules, and this theory is closely related to idempotent modules. For 𝔭 ∈ V̄_{G}(k) let C(𝔭) denote the full subcategory of kG- \underline{mod} consisting of all finitely generated kG-modules M such that V̄_{G}(M) ⊆ V̄_{G}(𝔭), and let C′(𝔭) denote the full subcategory of C(𝔭) consisting of all finitely generated modules M such that 𝔭 ∉ V̄_{G}(M). Then C(𝔭) and C′(𝔭) are ideal subcategories of kG-\underline{mod}. If M is an arbitrary kG-module, then V̄_{G}(M) can be defined as the collection of all primes 𝔭 ∈ V̄_{G}(k) such that M ⊗e_{C(𝔭)} ⊗f_{C′(𝔭)} is not projective. Note that V̄_{G}(e_{C(𝔭)} ⊗f_{C′(𝔭)}) = {𝔭} by Lemma 10.4 of [6].

Although the set V̄_{G}(M) is not necessarily closed in V̄_{G}(k) if M is not finitely generated, these sets do enjoy many of the same properties that hold in the case of finitely generated modules. For example, M is projective if and only if V̄_{G}(M) = ∅. Moreover, if M and N are any kG-modules, then V̄_{G}(M ⊕ N) = V̄_{G}(M) ∪ V̄_{G}(N) and V̄_{G}(M ⊗ N) = V̄_{G}(M) ∩ V̄_{G}(N). The reader should observe, however, that the definition of V̄_{G}(M) given here is not used in [6]. In that paper the so-called variety V_{G}(M) of a module is defined to be a collection of homogeneous irreducible subvarieties of the maximal ideal spectrum of H^{∗}(G, k). The connection between V_{G}(M) and V̄_{G}(M) is given by observing that V ∈ V_{G}(M) if and only if the generic point of V lies in V̄_{G}(M).

Now let V be any subset of V̄_{G}(k) that is closed under specialization; in other words, if 𝔭 ∈ V and 𝔮 ∈ V̄_{G}(k) with 𝔮 ⊇ 𝔭, then 𝔮 ∈ V. Let C(V) denote the full subcategory of kG-\underline{mod} consisting of all finitely generated kG-modules M such that V̄_{G}(M) ⊆ V. Then C(V) is an ideal subcategory of kG-\underline{mod}, and for simplicity we will write e_{V} for e_{C(V)} and f_{V} for f_{C(V)}. If W ⊆ V̄_{G}(k) is also closed under specialization, then the idempotent modules have the property that e_{V} ⊗ e_{W} ≅ e_{V ∩W} and f_{V} ⊗ f_{W} ≅ f_{V ∪W} in kG-\underline{Mod}. This result is proven in [13] under the assumption that V and W are closed subsets of V̄_{G}(k). Like many of the results of [13], however, essentially the same proof gives the result for subsets of V̄_{G}(k) closed under specialization. We will frequently need results in this greater generality, and we will usually use them without further comment.

The first proposition of this section is well known to experts in the area.

Proposition 3.1. Let V be a subset of V̄_{G}(k) that is closed under specialization. Then V̄_{G}(e_{V}) = V and V̄_{G}(f_{V}) = V̄_{G}(k) − V.

Proof. If 𝔭 ∈ V, then e_{V} ⊗ e_{C(𝔭)} ≅ e_{C(𝔭)} in kG-\underline{Mod}, so e_{V} ⊗ e_{C(𝔭)} ⊗ f_{C′(𝔭)} ≅ e_{C(𝔭)} ⊗ f_{C′(𝔭)} is not projective. Hence 𝔭 ∈ V̄_{G}(e_{V}). If 𝔭 ∉ V, set V′ = V̄_{G}(𝔭) − {𝔭}. Then e_{V} ⊗ e_{C(𝔭)} ≅ e_{V ∩V′}, so e_{V} ⊗ e_{C(𝔭)} ⊗ f_{C′(𝔭)} ≅ e_{V ∩V′} ⊗ f_{C′(𝔭)} ≅ e_{V} ⊗ e_{V′} ⊗ f_{C′(𝔭)} = e_{V} ⊗ e_{C′(𝔭)} ⊗ f_{C′(𝔭)}. But e_{C′(𝔭)} ⊗ f_{C′(𝔭)} lies in C′(𝔭)^{⊕} ∩ C′(𝔭)^{⊥} by Proposition 2.2, so e_{C′(𝔭)} ⊗ f_{C′(𝔭)} is projective and hence 𝔭 ∉ V̄_{G}(e_{V}).

To prove that V̄_{G}(f_{V}) = V̄_{G}(k) − V, note that the module e_{V} ⊗ f_{V} is projective so that V̄_{G}(e_{V} ⊗ f_{V}) = ∅ and hence V̄_{G}(f_{V}) ⊆ V̄_{G}(k) − V. But because there is a triangle

e_{V} ⟶ k ⟶ f_{V} ⟶ Ω^{−1}e_{V}

in kG-\underline{Mod}, it follows that V̄_{G}(k) ⊆ V̄_{G}(e_{V}) ∪ V̄_{G}(f_{V}). Thus V̄_{G}(k) − V ⊆ V̄_{G}(f_{V}), as desired. □

Lemma 3.2. Let V, W, and X be subsets of V̄_{G}(k) that are closed under specialization, and assume that V ∩ W ⊆ X. Then there are canonical stable isomorphisms (e_{V} ⊗ f_{X}) ⊕ (e_{W} ⊗ f_{X}) → e_{V ∪W} ⊗ f_{X} and e_{V} ⊗ f_{X} → e_{V} ⊗ f_{W ∪X}.

Proof. Theorem 8.1 of [13] proves the existence of a so-called Mayer–Vietoris triangle

e_{V ∩W} ⟶ e_{V} ⊕ e_{W} ⟶ e_{V ∪W} ⟶ Ω^{−1}e_{V ∩W}

in kG-\underline{Mod}. Because V ∩ W ⊆ X, it follows that V̄_{G}(e_{V ∩W} ⊗ f_{X}) = ∅ so that e_{V ∩W} ⊗ f_{X} is projective. Hence the map (e_{V} ⊗ f_{X}) ⊕ (e_{W} ⊗ f_{X}) → e_{V ∪W} ⊗ f_{X} is a canonical stable isomorphism.

To prove the second isomorphism, consider the triangle

e_{W} ⟶ k ⟶ f_{W} ⟶ Ω^{−1}e_{W} .

We have already seen that e_{W} ⊗ e_{V} ⊗ f_{X} ≅ e_{V ∩W} ⊗ f_{X} is projective, so it follows that there are canonical stable isomorphisms

e_{V} ⊗ f_{X} ≅ k ⊗ e_{V} ⊗ f_{X} ⟶ f_{W} ⊗ e_{V} ⊗ f_{X} ≅ e_{V} ⊗ f_{W ∪X}.

This completes the proof. □

Proposition 3.3. Let V and W be subsets of V̄_{G}(k) that are closed under specialization. Then there is a recollement

[figure: C(V ∪ W)^{⊥}; C(W)^{⊥}; C(V)^{⊕}/C(V ∩ W)^{⊕}.]

The inclusion functor C(V ∪ W)^{⊥} ↪ C(W)^{⊥} has a left adjoint given by − ⊗ f_{V ∪W} and a right adjoint given by Hom_{k}(f_{V ∪W} , −). If λ = e_{V} ⊗ f_{W}, then the functor C(W)^{⊥} → C(V)^{⊕}/C(V ∩ W)^{⊕} is the composition

C(W)^{⊥} ⟶^{−⊗λ} C(V)^{⊕} ∩ C(W)^{⊥} ↪ C(V)^{⊕} → C(V)^{⊕}/C(V ∩ W)^{⊕}.

It has a left adjoint induced by the functor − ⊗ λ : C(V)^{⊕} → C(W)^{⊥} and a right adjoint induced by the functor Hom_{k}(λ, − ⊗ λ) : C(V)^{⊕} → C(W)^{⊥}.

Proof. We begin by verifying that the functors occurring in the statement of the proposition do indeed map into the appropriate subcategories of kG-\underline{Mod}. By Proposition 2.3 we know that − ⊗ f_{V ∪W} and Hom_{k}(f_{V ∪W} , −) both map kG-\underline{Mod} into C(V ∪ W)^{⊥}, and the functor − ⊗ λ maps kG-\underline{Mod} into C(V)^{⊕} ∩ C(W)^{⊥} by Proposition 2.2. Note that λ is stably isomorphic to e_{V} ⊗ f_{V ∩W} by Lemma 3.2 so that − ⊗ λ also induces a functor from C(V)^{⊕}/C(V ∩W)^{⊕} into C(W)^{⊥}. Finally, we show that Hom_{k}(λ, −⊗λ) maps kG-\underline{Mod} into C(W)^{⊥}. Let M be a kG-module, and let N be an object of C(W) so that N ≅ N ⊗ e_{W} in kG-\underline{Mod}. Then

\underline{Hom}_{kG}(N, Hom_{k}(λ, M ⊗ λ)) ≅ \underline{Hom}_{kG}(N ⊗ λ, M ⊗ λ).

But N ⊗ λ ≅ N ⊗ e_{W} ⊗ e_{V} ⊗ f_{W} ≅ 0 in kG-\underline{Mod} because V̄_{G}(e_{W} ⊗ f_{W}) = ∅. Hence Hom_{k}(λ, M ⊗ λ) is an object of C(W)^{⊥}, as desired.

By Theorem 2.6 there is a recollement

[figure: C(V)^{⊥}; kG-\underline{Mod}; C(V)^{⊕}.]

The essential idea of the proof is to show that one can restrict to the subcategory C(W)^{⊥} and modify the functors appropriately to obtain another recollement

[figure: C(V)^{⊥} ∩ C(W)^{⊥}; C(W)^{⊥}; C(V)^{⊕} ∩ C(W)^{⊥}.]

Specifically, the inclusion functor C(V)^{⊥} ∩ C(W)^{⊥} ↪ C(W)^{⊥} has a left adjoint given by − ⊗ f_{V} ⊗ f_{W} and a right adjoint given by Hom_{k}(f_{V} ⊗ f_{W} , −); the functor − ⊗ λ : C(W)^{⊥} → C(V)^{⊕} ∩ C(W)^{⊥} has a left adjoint given by the inclusion and a right adjoint given by Hom_{k}(λ, −). Using the properties of idempotent modules and the fact that \underline{Hom}_{kG}(L, Hom_{k}(M, N)) is naturally isomorphic to \underline{Hom}_{kG}(L ⊗ M, N), one can easily check that these functors satisfy the stated adjointness conditions and map into the appropriate subcategories of kG-\underline{Mod} to give a recollement. The details are straightforward and are left to the reader.

A module M lies in C(V)^{⊥} ∩C(W)^{⊥} if and only if V̄_{G}(M)∩V = ∅ and V̄_{G}(M)∩W = ∅, and this condition holds precisely when M lies in C(V ∪ W)^{⊥}. Hence C(V)^{⊥} ∩ C(W)^{⊥} = C(V ∪ W)^{⊥}. Moreover, it is easy to check that the composition of functors

C(V)^{⊕} ∩ C(W)^{⊥} ↪ C(V)^{⊕} → C(V)^{⊕}/C(V ∩ W)^{⊕}

is an equivalence of triangulated categories with a quasi-inverse induced by − ⊗ λ : C(V)^{⊕} → C(V)^{⊕} ∩ C(W)^{⊥}. We can therefore replace C(V)^{⊕} ∩ C(W)^{⊥} by the category C(V)^{⊕}/C(V ∩ W)^{⊕}, and the result is a recollement satisfying the required conditions. This completes the proof. □

It is also possible to give a direct proof of Proposition 3.3 that does not appeal to Theorem 2.6. One checks that the inclusion functor C(V ∪ W)^{⊥} → C(W)^{⊥} has a left adjoint given by −⊗f_{V ∪W} and a right adjoint given by Hom_{k}(f_{V ∪W} , −). Proposition 2.4 then implies that there is a recollement

[figure: C(V ∪ W)^{⊥}; C(W)^{⊥}; C(W)^{⊥}/C(V ∪ W)^{⊥}.]

It is then necessary to show that the category C(W)^{⊥}/C(V ∪ W)^{⊥} is equivalent to C(V)^{⊕}/C(V ∩ W)^{⊕} and that the remaining functors in the resulting recollement are those stated in Proposition 3.3.

If E is an arbitrary collection of elementary abelian p-subgroups of G, set

V_{G,E} = ⋃_{E∈E} res_{G,E}^{∗}(V̄_{E}(k))

and C_{G,E} = C(V_{G,E})^{⊕}. When E is an elementary abelian p-subgroup of G, we also write V_{G,E} for V_{G,{E}} and C_{G,E} for C_{G,{E}}. Set

V_{G,E}^{−} = ⋃_{E′<E} res_{G,E′}^{∗}(V̄_{E′}(k))

and V_{G,E}^{+} = V_{G,E} − V_{G,E}^{−}. Finally, define the categories C_{G,E}^{−} and C_{G,E}^{+} by setting C_{G,E}^{−} = C(V_{G,E}^{−})^{⊕} and C_{G,E}^{+} = C_{G,E}/C_{G,E}^{−}. Then the following categorical analogue of the Quillen Stratification Theorem is an immediate consequence of Proposition 3.3.

Theorem 3.4. Let E_{1}, …, E_{n} be a complete set of representatives for the conjugacy classes of nontrivial elementary abelian p-subgroups of G, and assume that the indices are chosen so that ∪_{j=1}^{i}V_{G,E_{i}}^{+} is closed in V̄_{G}(k) for 1 ≤ i ≤ n. Let E_{0} = ∅ and E_{i} = {E_{1}, …, E_{i}}. Then for 1 ≤ i ≤ n there is a recollement

[figure: C_{G,E_{i}}^{⊥}; C_{G,E_{i−1}}^{⊥}; C_{G,E_{i}}^{+}.]

In particular, C_{G,E_{0}}^{⊥} = kG-\underline{Mod} can be obtained from the categories C_{G,E_{i}}^{+} by iterated recollement.

For any collection E of elementary abelian p-subgroups of G we write e_{G,E} for e_{C(V_{G,E})} = e_{V_{G,E}} and f_{G,E} for f_{C(V_{G,E})} = f_{V_{G,E}}. If E is an elementary abelian p-subgroup of G, then we also write e_{G,E} for e_{G,{E}} and f_{G,E} for f_{G,{E}}. With this notation we can now prove a result that is related to the Avrunin–Scott Stratification Theorem [1] for the variety V̄_{G}(M) of a finitely generated module M.

Proposition 3.5. Let M be a kG-module that is not projective, and let E_{1}, …, E_{n} be a complete set of representatives for the conjugacy classes of nontrivial elementary abelian p-subgroups E of G such that V̄_{G}(M) ∩ V_{G,E}^{+} ≠ ∅. Assume that the indices are chosen so that

⋃_{j=1}^{i}(V̄_{G}(M) ∩ V_{G,E_{j}}^{+})

is closed in V̄_{G}(M) for 1 ≤ i ≤ n. Then there exist a projective module P and a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M ⊕ P

such that V̄_{G}(M_{i}/M_{i−1}) = V̄_{G}(M) ∩ V_{G,E_{i}}^{+} for 1 ≤ i ≤ n.

Proof. Let E_{i} = {E_{1}, …, E_{i}} for 1 ≤ i ≤ n. Because

⋃_{j=1}^{n−1}(V̄_{G}(M) ∩ V_{G,E_{j}}^{+})

is closed in V̄_{G}(M), the set V̄_{G}(M) is the disjoint union of the subsets V̄_{G}(M) ∩ V_{G,E_{n−1}} and V̄_{G}(M) ∩ V_{G,E_{n}}^{+}. By induction on n it suffices to show that there exist a projective module P_{n} and a short exact sequence

0 → M_{n−1} → M ⊕ P_{n} → M′_{n} → 0

such that V̄_{G}(M_{n−1}) = V̄_{G}(M) ∩ V_{G,E_{n−1}} and V̄_{G}(M′_{n}) = V̄_{G}(M) ∩ V_{G,E_{n}}^{+}. Proposition 2.2 implies that there is a distinguished triangle

M ⊗ e_{G,E_{n−1}} → M → M ⊗ f_{G,E_{n−1}} → Ω^{−1}(M ⊗ e_{G,E_{n−1}}),

so it is enough to show that V̄_{G}(M ⊗e_{G,E_{n−1}}) = V̄_{G}(M)∩V_{G,E_{n−1}} and V̄_{G}(M ⊗f_{G,E_{n−1}}) = V̄_{G}(M) ∩ V_{G,E_{n}}^{+}. Both of these equalities follow easily from Proposition 3.1. □

4. Quillen stratification and local subgroups

In the last section we saw that the Quillen Stratification Theorem has an analogue for the stable module category kG-\underline{Mod}. In particular, the stable category can be obtained by iterated recollement from certain quotient categories corresponding to elementary abelian p-subgroups. In this section we show that these quotient categories are locally determined. The equivalence of categories obtained here is similar to equivalences obtained by Benson [3, 4], and the techniques are based upon those used in [3].

Suppose that 𝔭 ∈ V̄_{G}(k), and set V = V̄_{G}(𝔭). Then there is an elementary abelian p-subgroup Ẽ of G and a prime ideal 𝔭̃ ∈ V̄Ẽ(k) such that 𝔭 = res_{G,Ẽ}^{∗}(𝔭̃). If Ẽ is minimal

with respect to this property, then the pair (Ẽ, 𝔭̃) is unique up to conjugacy; moreover, if Ṽ = V̄Ẽ(𝔭̃), then V = res_{G,Ẽ}^{∗}(Ṽ). As in [8], we say that Ẽ is a vertex for V or for 𝔭,

that 𝔭̃ is a source for 𝔭, and that Ṽ is a source for V.

We begin this section with two propositions describing the sets V̄_{G}(M) for induced and restricted modules. Both results are already known even for infinitely generated modules, but they do not seem to have been stated in a convenient form.

Proposition 4.1. Suppose that H is a subgroup of G and M is a kG-module. Then V̄_{H}(M↓_{H}) = (res_{G,H}^{∗})^{−1}(V̄_{G}(M)).

Proof. Let 𝔮 ∈ V̄_{H}(k), and set 𝔭 = res_{G,H}^{∗}(𝔮). Let Ẽ be a vertex of 𝔮, and let 𝔮̃ ∈ V̄Ẽ(k) be a prime ideal with res_{H,Ẽ}^{∗}(𝔮̃) = 𝔮. Then Theorems 10.6 and 10.7 of [6] show that 𝔭 ∈ V̄_{G}(M) if and only if 𝔮̃ ∈ V̄Ẽ(M↓Ẽ). Similarly, 𝔮 ∈ V̄_{H}(M↓_{H}) if and only if 𝔮̃ ∈ V̄Ẽ(M↓Ẽ). Thus V̄_{H}(M↓_{H}) = (res_{G,H}^{∗})^{−1}(V̄_{G}(M)), as desired. □

Proposition 4.2. Suppose that H is a subgroup of G and M is a kH-module. Then V̄_{G}(M↑^{G}) = res_{G,H}^{∗}(V̄_{H}(M)).

Proof. Because M is a summand of M↑^{G}↓_{H}, we see that V̄_{H}(M) ⊆ V̄_{H}(M↑^{G}↓_{H}) = (res_{G,H}^{∗})^{−1}(V̄_{G}(M↑^{G})) by Proposition 4.1, so res_{G,H}^{∗}(V̄_{H}(M)) ⊆ V̄_{G}(M↑^{G}). On the other hand, if 𝔭 ∈ V̄_{G}(M↑^{G}), then (M ⊗ e_{C(𝔭)}↓_{H} ⊗ f_{C′(𝔭)}↓_{H})↑^{G} ≅ M↑^{G} ⊗ e_{C(𝔭)} ⊗ f_{C′(𝔭)} is not projective, so neither is M⊗e_{C(𝔭)}↓_{H}⊗f_{C′(𝔭)}↓_{H}. Thus V̄_{H}(M)∩V̄_{H}(e_{C(𝔭)}↓_{H}⊗f_{C′(𝔭)}↓_{H}) ≠ ∅. But V̄_{G}(e_{C(𝔭)} ⊗ f_{C′(𝔭)}) = {𝔭}, so Proposition 4.1 implies that V̄_{H}(e_{C(𝔭)}↓_{H} ⊗ f_{C′(𝔭)}↓_{H}) = (res_{G,H}^{∗})^{−1}(𝔭). It follows that there is an element 𝔮 ∈ V̄_{H}(M) with res_{G,H}^{∗}(𝔮) = 𝔭, and 𝔭 ∈ res_{G,H}^{∗}(V̄_{H}(M)). Thus V̄_{G}(M↑^{G}) = res_{G,H}^{∗}(V̄_{H}(M)), and this completes the proof. □

Throughout the remainder of this section let E be a fixed elementary abelian psubgroup of G, and set λ_{G,E} = e_{V_{G,E}} ⊗ f_{V_{G,E}^{−}}. Note that V̄_{G}(λ_{G,E}) = V_{G,E}^{+} and that λ_{G,E} depends only upon the conjugacy class of the subgroup E. Let N = N_{G}(E), and let V_{N,E} = V_{1}, …, V_{n} be all of the distinct components of (res_{G,N}^{∗})^{−1}(V_{G,E}) such that res_{G,N}^{∗}(V_{i}) = V_{G,E}. Then for 1 ≤ i ≤ n there is an elementary abelian p-subgroup E_{i} such

that V_{i} = res_{N,E_{i}}^{∗}(V̄_{E_{i}}(k)). Set V_{i}^{−} = V_{i} ∩ (res_{G,N}^{∗})^{−1}(V_{G,E}^{−}). If i ≠ j, then the vertices E_{i} and E_{j} are G-conjugates of E that are not N-conjugate, so V_{i} ∩V_{j} ⊆ (res_{G,N}^{∗})^{−1}(V_{G,E}^{−}) and hence V_{i} ∩ V_{j} = V_{i}^{−} ∩ V_{j}^{−}.

Proposition 4.3. The restriction λ_{G,E}↓_{N} is stably isomorphic to λ_{N,E_{1}} ⊕ ⋯ ⊕ λ_{N,E_{n}}.

Proof. Note that e_{G,E}↓_{N} = e_{(res_{G,N}^{∗})^{−1}(V_{G,E})} and f_{V_{G,E}^{−}}↓_{N} = f_{(res_{G,N}^{∗})^{−1}(V_{G,E}^{−})} by Proposition 7.1 of [13]. We will use these facts to prove by induction that e_{V_{1}∪···∪V_{s}} ⊗ f_{V_{G,E}^{−}}↓_{N} ≅ (e_{V_{1}} ⊗ f_{V_{G,E}^{−}}↓_{N}) ⊕ ⋯ ⊕ (e_{V_{s}} ⊗ f_{V_{G,E}^{−}}↓_{N}) in kG-\underline{Mod} for all s with 1 ≤ s ≤ n. There is nothing to prove when s = 1, so assume that 1 < s ≤ n. Set V = V_{1} ∪ ⋯ ∪ V_{s−1} so that V ∩ V_{s} ⊆ (res_{G,N}^{∗})^{−1}(V_{G,E}^{−}). Lemma 3.2 implies that there is a stable isomorphism e_{V ∪V_{s}} ⊗ f_{V_{G,E}^{−}}↓_{N} ≅ (e_{V} ⊗ f_{V_{G,E}^{−}}↓_{N}) ⊕ (e_{V_{s}} ⊗ f_{V_{G,E}^{−}}↓_{N}) and hence e_{V_{1}∪···∪V_{s}} ⊗ f_{V_{G,E}^{−}}↓_{N} ≅ (e_{V_{1}} ⊗ f_{V_{G,E}^{−}}↓_{N}) ⊕ ⋯ ⊕ (e_{V_{s}} ⊗ f_{V_{G,E}^{−}}↓_{N}) by induction.

Now let V′ denote the union of all components V′_{1} of (res_{G,N}^{∗})^{−1}(V_{G,E}) such that res_{G,N}^{∗}(V′_{1}) ≠ V_{G,E}. If 𝔭′ ∈ V′, then any vertex E′ of 𝔭′ must also be a vertex of 𝔭 = res_{G,N}^{∗}(𝔭′), so E′ is conjugate to a subgroup of E. But E′ is not conjugate to E by the definition of V′, so 𝔭′ ∈ (res_{G,N}^{∗})^{−1}(V_{G,E}^{−}). Then Lemma 3.2 implies that

e_{V_{G,E}} ↓_{N} ⊗ f_{V_{G,E}^{−}}↓_{N} = e_{V_{1}∪···∪V_{n}∪V′} ⊗ f_{V_{G,E}^{−}}↓_{N}
≅ (e_{V_{1}∪···∪V_{n}} ⊗ f_{V_{G,E}^{−}}↓_{N}) ⊕ (e_{V′} ⊗ f_{V_{G,E}^{−}}↓_{N})
≅ (e_{V_{1}} ⊗ f_{V_{G,E}^{−}}↓_{N}) ⊕ ⋯ ⊕ (e_{V_{n}} ⊗ f_{V_{G,E}^{−}}↓_{N}).

Fix i with 1 ≤ i ≤ n, and set W_{i} = V_{1}^{−} ∪ ⋯ ∪ V_{i−1}^{−} ∪ V_{i+1}^{−} ∪ ⋯ ∪ V_{n}^{−} ∪ V′. Then (res_{G,N}^{∗})^{−1}(V_{G,E}^{−}) = W_{i} ∪ V_{i}^{−} and V_{i} ∩ W_{i} ⊆ V_{i}^{−}, so it follows from Lemma 3.2 that there are stable isomorphisms e_{V_{i}} ⊗ f_{V_{G,E}^{−}}↓_{N} = e_{V_{i}} ⊗ f_{W_{i}∪V_{i}^{−}} ≅ e_{V_{i}} ⊗ f_{V_{i}^{−}} = λ_{N,E_{i}}. Hence λ_{G,E}↓_{N} = e_{V_{G,E}} ↓_{N} ⊗ f_{V_{G,E}^{−}}↓_{N} ≅ λ_{N,E_{1}} ⊕ ⋯ ⊕ λ_{N,E_{n}}, as desired. □

Lemma 4.4. Let M be a kN-module such that V̄_{N}(M) ⊆ V_{N,E}, and let g ∈ G−N. Then V̄_{N}((g ⊗ M)↓_{gNg^{−1}∩N}↑^{N}) ∩ V_{N,E}^{+} = ∅.

Proof. Let 𝔭 ∈ V_{N,E}^{+}. Then there is a prime ideal 𝔭_{0} ∈ V̄_{E}(k) with vertex E such that res_{N,E}^{∗}(𝔭_{0}) = 𝔭. Proposition 4.1 shows that if M′ is any kN-module, then 𝔭 ∈ V̄_{N}(M′) if and only if 𝔭_{0} ∈ V̄_{E}(M′↓_{E}), so by the Mackey Decomposition Theorem it is enough to show that 𝔭_{0} ∉ V̄_{E}((g ⊗ M)↓_{gNg^{−1}∩E}↑^{E}) for all g ∈ G − N. Because E is the vertex of 𝔭_{0}, we know by Proposition 4.2 that 𝔭_{0} ∉ V̄_{E}((g ⊗ M)↓_{gNg^{−1}∩E}↑^{E}) if E ⊈ gNg^{−1}. Thus we may assume that E ⊆ gNg^{−1}, and we wish to show that 𝔭_{0} ∉ V̄_{E}((g ⊗ M)↓_{E}).

Because V̄_{N}(M) ⊆ V_{N,E}, we have M ≅ M ⊗ e_{N,E} in kN-\underline{Mod}. Hence

(g ⊗ M)↓_{E} ≅ (g ⊗ (M ⊗ e_{N,E}))↓_{E} ≅ (g ⊗ M)↓_{E} ⊗ (g ⊗ e_{N,E})↓_{E},

and V̄_{E}((g ⊗ M)↓_{E}) ⊆ V̄_{E}((g ⊗ e_{N,E})↓_{E}) = V̄_{E}(e_{gNg^{−1},gEg^{−1}}↓_{E}). If g ∉ N, then E is not conjugate to gEg^{−1} in gNg^{−1} so that the prime ideal 𝔭_{1} = res_{gNg^{−1},E}^{∗}(𝔭_{0}) does not have a vertex contained in gEg^{−1}. Thus 𝔭_{1} ∉ V̄_{gNg^{−1}}(e_{gNg^{−1},gEg^{−1}}), so 𝔭_{0} ∉ V̄_{E}(e_{gNg^{−1},gEg^{−1}}↓_{E}) and hence 𝔭_{0} ∉ V̄_{E}((g ⊗ M)↓_{E}). This completes the proof. □

12 WAYNE W. WHEELER

Proposition 4.5. The module λ_{N,E}^{↑G} is stably isomorphic to λ_{G,E}. Proof. Let α : e_{N,E} → k denote the universal map in kN-\underline{Mod} from an object of C_{N,E} to k, as in Proposition 2.2. Let ε : k_{N}^{↑G} → k denote the augmentation map, and set ϕ = ε ∘ α↑^{G}. Suppose that D is a set of representatives for the (N, N) double cosets contained in G − N, and set

M_{0} = ⨁_{g∈D}(g ⊗ e_{N,E})↓_{gNg^{−1}∩N}↑^{N}

and

M_{1} = ⨁_{g∈D}(g ⊗ k_{N})↓_{gNg^{−1}∩N}↑^{N}.

Then ϕ↓_{N} is the composition

(1 ⊗ e_{N,E}) ⊕ M_{0} ⟶^{α↑^{G}↓_{N}} (1 ⊗ k_{N}) ⊕ M_{1} ⟶^{ε↓_{N}} k,

and ϕ↓_{N} is just α on the summand 1 ⊗ e_{N,E} ≅ e_{N,E}. Moreover, V̄_{N}(M_{0}) ∩ V_{N,E}^{+} = ∅ by Lemma 4.4, so it follows that M_{0} ⊗ λ_{N,E} is projective. Consider the triangle

e_{N,E}^{↑G} ⟶^{ϕ} k_{G} ⟶ F ⟶ Ω^{−1}(e_{N,E}^{↑G}).

Then there is an isomorphism of triangles

e_{N,E}^{↑G}↓_{N} ⊗ λ_{N,E} ⟶^{ϕ⊗1} k_{N} ⊗ λ_{N,E} ⟶ F ↓_{N} ⊗ λ_{N,E} ⟶ Ω^{−1}(e_{N,E}^{↑G}↓_{N} ⊗ λ_{N,E})
↓ ∥ ↓ ↓
e_{N,E} ⊗ λ_{N,E} ⟶^{α⊗1} k_{N} ⊗ λ_{N,E} ⟶ f_{N,E} ⊗ λ_{N,E} ⟶ Ω^{−1}(e_{N,E} ⊗ λ_{N,E}).

The module f_{N,E} ⊗ λ_{N,E} = f_{N,E} ⊗ e_{N,E} ⊗ f_{V_{N,E}^{−}} is projective, so F ↓_{N} ⊗ λ_{N,E} is also projective. Let 𝔭 ∈ V_{G,E}^{+}, and let 𝔭_{1} ∈ V_{N,E}^{+} be a prime ideal with res_{G,N}^{∗}(𝔭_{1}) = 𝔭. Because V̄_{N}(F ↓_{N})∩V_{N,E}^{+} = V̄_{N}(F ↓_{N} ⊗λ_{N,E}) = ∅, we know that 𝔭_{1} ∉ V̄_{N}(F ↓_{N}) and hence 𝔭 ∉ V̄_{G}(F) by Proposition 4.1. Thus F ⊗ λ_{G,E} is projective, and ϕ ⊗ 1 : e_{N,E}^{↑G} ⊗ λ_{G,E} → k ⊗ λ_{G,E} is a stable isomorphism. By Proposition 4.3 it follows that

λ_{G,E} ≅ e_{N,E}^{↑G} ⊗ λ_{G,E} ≅ (e_{N,E} ⊗ λ_{G,E}↓_{N})↑^{G} ≅ ⨁_{i=1}^{n}(e_{N,E} ⊗ λ_{N,E_{i}})↑^{G}.

Now λ_{N,E_{i}} = e_{V_{i}} ⊗ f_{V_{i}^{−}}, and for i > 1 we have V_{i} ≠ V_{N,E} so that

V̄_{G}(e_{N,E} ⊗ e_{V_{i}} ⊗ f_{V_{i}^{−}}) = V̄_{G}(e_{V_{N,E}∩V_{i}}) ∩ V̄_{G}(f_{V_{i}^{−}}) = ∅

because V_{N,E} ∩ V_{i} ⊆ V_{i}^{−}. Hence λ_{G,E} ≅ (e_{N,E} ⊗ λ_{N,E})↑^{G} ≅ λ_{N,E}^{↑G}, as desired. □

We are now ready to prove the main result of this section, which gives a local description of the quotient category C_{G,E}^{+} appearing in Theorem 3.4.

Theorem 4.6. The functors (− ⊗ λ_{N,E}) ∘ res_{G,N} : C_{G,E}^{+} → C_{N,E}^{+} and ind_{N,G} : C_{N,E}^{+} → C_{G,E}^{+} are equivalences of categories that are quasi-inverses.

Proof. If M is a kG-module with V̄_{G}(M) ⊆ V_{G,E}^{−}, then

V̄_{G}(M↓_{N} ⊗ λ_{N,E}) ⊆ (res_{G,N}^{∗})^{−1}(V_{G,E}^{−}) ∩ V_{N,E}^{+} = ∅.

Thus the functor (− ⊗ λ_{N,E}) ∘ res_{G,N} : C_{G,E} → C_{N,E} vanishes on objects in C_{G,E}^{−}, so it induces a functor C_{G,E}^{+} → C_{N,E}^{+}. Similarly, if M is a kN-module with V̄_{N}(M) ⊆ V_{N,E}^{−}, then V̄_{G}(M↑^{G}) ⊆ res_{G,N}^{∗}(V_{N,E}^{−}) = V_{G,E}^{−}. Thus ind_{N,G} : C_{N,E} → C_{G,E} induces a functor C_{N,E}^{+} → C_{G,E}^{+}.

Now let M be an object of C_{N,E}, and let D be a set of representatives for the (N, N) double cosets contained in G − N. By the Mackey Decomposition Theorem we have

M↑^{G}↓_{N} = (1 ⊗ M) ⊕ ⨁_{g∈D}(g ⊗ M)↓_{gNg^{−1}∩N}↑^{N},

and Lemma 4.4 implies that V̄_{N}((g ⊗ M)↓_{gNg^{−1}∩N}↑^{N}) ∩ V_{N,E}^{+} = ∅ for all g ∈ D. Hence M↑^{G}↓_{N} ⊗λ_{N,E} is naturally isomorphic to M⊗λ_{N,E} in kN-\underline{Mod}. But M⊗λ_{N,E} is naturally isomorphic to M in C_{N,E}^{+}, so the composition of functors (− ⊗ λ_{N,E}) ∘ res_{G,N} ∘ ind_{N,G} : C_{N,E}^{+} → C_{N,E}^{+} is isomorphic to the identity.

Finally, let M be an object of C_{G,E}. Then Proposition 4.5 implies that in the quotient C_{G,E}^{+} there are natural isomorphisms

(M↓_{N} ⊗ λ_{N,E})↑^{G} ≅ M ⊗ λ_{N,E}^{↑G} ≅ M ⊗ λ_{G,E} ≅ M,

and this completes the proof. □

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Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA

  E-mail address: www@sloth.math.uga.edu