LOCAL SUBGROUPS AND THE STABLE CATEGORY

WAYNE W. WHEELER

Abstract. If G is a finite group and k is an algebraically closed field of characteristic p > 0, then this paper uses the local subgroup structure of G to define a category 𝔏(G, k) that is equivalent to the stable category of all left kG-modules modulo projectives. A subcategory of 𝔏(G, k) equivalent to the stable category of finitely generated kG-modules is also identified. The definition of 𝔏(G, k) depends largely but not exclusively upon local data; one condition on the objects involves compatibility with respect to conjugations by arbitrary group elements rather than just elements of p-local subgroups.

1. Introduction

One of the main themes of modular representation theory, going back many decades to the fundamental work of R. Brauer, is the idea that the representations of a finite group are closely related to those of its local subgroups. To some extent this paper is intended to provide a general explanation for why this idea has proven to be so fruitful over such a long period. In particular, the work presented here shows that it is possible to use the local subgroups of a finite group to construct a category equivalent to the stable category.

Let G be a finite group, let k be an algebraically closed field of characteristic p, and let P(G) be the collection of all p-subgroups of G. The stable category kG-\underline{Mod} is obtained by factoring out the projective modules from the category of all left kG-modules. Section 3 defines a category 𝔏(G, k) in which the objects are essentially certain collections of modules. In particular, an object L in 𝔏(G, k) determines a module L(P) in kN_{G}(P)-\underline{Mod} for each P ∈ P(G). Each module L(P) must satisfy a condition on its variety, and the family of all modules determined by L must be compatible under conjugation and restriction. In the compatibility conditions conjugations by arbitrary group elements are allowed, so the definition of 𝔏(G, k) does not depend solely on the local structure of G. The main result of this paper is that 𝔏(G, k) is equivalent to kG-\underline{Mod}. If Q is a Sylow p-subgroup of G, let 𝔩(G, k) be the full subcategory of 𝔏(G, k) consisting of the objects L such that L(Q) is stably isomorphic to a finitely generated module. Then 𝔩(G, k) is equivalent to the full subcategory kG-\underline{mod} of finitely generated modules in kG-\underline{Mod}.

The work presented here makes extensive use of Rickard’s work on idempotent modules [3] as well as Benson, Carlson, and Rickard’s theory of varieties for infinitely generated modules [1].

2 WAYNE W. WHEELER

These topics are reviewed in Section 2. The third section defines the category 𝔏(G, k) and a canonical functor 𝔉 : kG-\underline{Mod} → 𝔏(G, k). It is possible to define the tensor product of a kG-module M and an object L of 𝔏(G, k), and the result is another object of 𝔏(G, k). This idea is considered in Section 4. Finally, Section 5 is devoted to proving that the canonical functor 𝔉 : kG-\underline{Mod} → 𝔏(G, k) is an equivalence of categories.

2. Preliminary results

Throughout this paper G denotes a finite group, and k is an algebraically closed field of prime characteristic p. If g ∈ G and H is a subgroup of G, let ^{g}H = gHg^{−1}. For any kH-module M set M↑^{G} = kG ⊗_{kH} M; if M is a kG-module, then M↓_{H} denotes the restriction of M to H. It will often be convenient simply to write M↓ when the subgroup H can be inferred from the context. We write ε : k_{H}^{↑G} → k for the augmentation map given by

ε( ∑_{g∈G/H} g ⊗ x_{g}) = ∑_{g} x_{g}.

Let kG-Mod denote the category of all left kG-modules, and let kG-mod be the full subcategory of finitely generated kG-modules. If M and M′ are kG-modules, let PHom_{kG}(M, M′) denote the k-subspace of Hom_{kG}(M, M′) consisting of those maps that factor through a projective kG-module. The stable category kG-\underline{Mod} has the same objects as kG-Mod, but the morphisms from M to M′ in kG-\underline{Mod} are defined by setting

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

The full subcategory of kG-\underline{Mod} consisting of finitely generated kG-modules is denoted kG- \underline{mod}. It is well known that the categories kG-\underline{Mod} and kG-\underline{mod} are triangulated (Theorem I.2.6 of [2]), and the translation functor is given by Ω^{−1}. For convenience we often identify Ω^{−1} with the isomorphic functor Ω^{−1}k ⊗ −.

If γ : M → M′ is a kG-homomorphism, then we normally also write γ for the corresponding map in kG-\underline{Mod}. In fact, we will generally only be concerned with maps in the stable category. In a few cases homomorphisms are defined in the module category, but even then it is always the image in kG-\underline{Mod} that is of interest.

Recall that if T is a triangulated category and C is a full triangulated subcategory of T , then C is said to be a thick subcategory if it is closed under taking direct summands of objects. Now suppose that C is a thick subcategory of kG-\underline{mod}. As in [3], we say that C is a tensor-ideal subcategory of kG-\underline{mod} if M ⊗ M′ is in C whenever M is in C and M′ is in kG-\underline{mod}. Let C^{⊕} denote the smallest full triangulated subcategory of kG-\underline{Mod} that contains C and is closed

LOCAL SUBGROUPS AND THE STABLE CATEGORY 3

under arbitrary direct sums. A module M is said to be C-local if \underline{Hom}_{kG}(C, M) = 0 for all C in C.

The following proposition summarizes the fundamental facts about idempotent modules that will be needed in the following sections. Proofs can be found in [3].

Proposition 2.1. Let C be a tensor-ideal subcategory of kG-\underline{mod}. For any object M in kG- \underline{Mod} there is a triangle

\xymatrix{e_{C}(M) \ar[r] & M \ar[r] & f_{C}(M) \ar[r] & Ω^{−1}(e_{C}(M))}

in kG-\underline{Mod} such that e_{C}(M) is in C^{⊕} and f_{C}(M) is C-local, 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 a C-local object. 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, e_{C}(k) ⊗ M ≅ e_{C}(M) and f_{C}(k) ⊗ M ≅ f_{C}(M) in kG-\underline{Mod} for any module M.

In the following work it will often be necessary to consider the universal map η : e_{C}(k) → k described in Proposition 2.1. By abuse of notation we usually use the same symbol η to denote this map for any thick subcategory C of kG-\underline{mod} and for any finite group G.

Benson, Carlson, and Rickard have used idempotent modules to develop a theory of varieties for arbitrary kG-modules, and we give a brief review of this theory. Although it is common to consider the maximal ideal spectrum of the cohomology ring H^{∗}(G, k), we will use the space Proj H^{∗}(G, k) of all homogeneous prime ideals that do not contain the ideal ⨁_{n=1}^{∞} H^{n}(G, k). If I is a homogeneous ideal in H^{∗}(G, k), we set V̄_{G}(I) = {𝔭 ∈ Proj H^{∗}(G, k) | I ⊆ 𝔭}. If M is a finitely generated kG-module, let J(M) be the annihilator of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k), and let V̄_{G}(M) = V̄_{G}(J(M)). Then V̄_{G}(M) is closed in Proj H^{∗}(G, k), and V̄_{G}(k) = Proj H^{∗}(G, k).

If M is infinitely generated, however, then the definition of V̄_{G}(M) is more complicated. 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 tensor-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 f_{C′(𝔭)} ⊗ e_{C(𝔭)} ⊗ M is not projective.

Although the set V̄_{G}(M) is not necessarily closed in V̄_{G}(k) if M is not finitely generated, these sets do retain the most important properties of varieties for finitely generated modules. In particular, the following results hold for any kG-modules M and M′:

(1) M is projective if and only if V̄_{G}(M) = ∅;

(2) V̄_{G}(M ⊕ M′) = V̄_{G}(M) ∪ V̄_{G}(M′);

4 WAYNE W. WHEELER

(3) V̄_{G}(M ⊗ M′) = V̄_{G}(M) ∩ V̄_{G}(M′). The reader should be warned, however, that the definition of V̄_{G}(M) given here is not used in [1]. In that paper the authors define the so-called variety V_{G}(M) of a module M to be a collection of homogeneous irreducible subvarieties of the maximal ideal spectrum of H^{∗}(G, k). The results of [1] can be translated into the notation used here 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 a closed subset of V̄_{G}(k), and 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. For simplicity we write e_{V} for e_{C(V)} and f_{V} for f_{C(V)}. If H is a subgroup of G, let res_{G,H}^{∗} : V̄_{H}(k) → V̄_{G}(k) denote the map induced by the restriction homomorphism res_{G,H} : H^{∗}(G, k) → H^{∗}(H, k).

Proposition 2.2. Let V and W be closed subsets of V̄_{G}(k), and let H be a subgroup of G. Then

(1) e_{V} ⊗ e_{W} ≅ e_{V ∩W} and f_{V} ⊗ f_{W} ≅ f_{V ∪W} in kG-\underline{Mod};

(2) e_{V} ↓_{H} ≅ e_{(res_{G,H}^{∗})^{−1}(V)} and f_{V} ↓_{H} ≅ f_{(res_{G,H}^{∗})^{−1}(V)};

(3) V̄_{G}(e_{V}) = V and V̄_{G}(f_{V}) = V̄_{G}(k) − V.

Proof. The first two statements are proven in [3]; the third statement is Proposition 3.1 of [4].

Proposition 2.3. Suppose that H is a subgroup of G.

(1) If M is a kG-module, then V̄_{H}(M↓_{H}) = (res_{G,H}^{∗})^{−1}(V̄_{G}(M)).

(2) If M is a kH-module, then V̄_{G}(M↑^{G}) = res_{G,H}^{∗}(V̄_{H}(M)).

Proof. See Propositions 4.1 and 4.2 of [4].

Now suppose that P is an arbitrary p-subgroup of G, and define V_{G,P} = res_{G,P}^{∗} (V̄_{P} (k)). We shall write e_{G,P} for e_{V_{G,P}} and f_{G,P} for f_{V_{G,P}}. If E is an elementary abelian p-subgroup of G, set

V_{G,E}^{−} = ⋃_{E_{0}<E} V_{G,E_{0}}

and V_{G,E}^{+} = V_{G,E} − V_{G,E}^{−}. Then the module λ_{G,E} = f_{V_{G,E}^{−}} ⊗ e_{V_{G,E}} satisfies V̄_{G}(λ_{G,E}) = V_{G,E}^{+}.

Proposition 2.4. Let E be an elementary abelian p-subgroup of G, and set N = N_{G}(E). Then λ_{N,E}^{↑G} is stably isomorphic to λ_{G,E}. Moreover, the map 1 ⊗ εη↑^{G} : λ_{G,E} ⊗ e_{N,E}^{↑G} → λ_{G,E} is a stable isomorphism.

Proof. The first statement is Proposition 4.5 of [4]. The proof of this proposition also shows that the second statement is true, although it is not explicitly stated in [4].

LOCAL SUBGROUPS AND THE STABLE CATEGORY 5

Lemma 2.5. Let V, W, and X be closed subsets of V̄_{G}(k), and assume that W ∩ X ⊆ V. Then there are canonical stable isomorphisms (f_{V} ⊗ e_{W}) ⊕ (f_{V} ⊗ e_{X}) → f_{V} ⊗ e_{W ∪X} and f_{V} ⊗ e_{W} → f_{V ∪X} ⊗ e_{W}.

Proof. See Lemma 3.2 of [4].

Let r be the p-rank of G. For 0 ≤ s ≤ r let

V_{s} = ⋃_{E} V_{G,E},

where the union is taken over all elementary abelian p-subgroups E of G with rank E ≤ s. Set e_{s} = e_{V_{s}} and f_{s} = f_{V_{s}}.

The proof of the following result is similar to that of Proposition 4.3 of [4].

Proposition 2.6. Let 0 ≤ s ≤ r − 1, and let E_{1}, …,E_{n} be a set of representatives for the conjugacy classes of elementary abelian p-subgroups of rank s + 1 in G. Then

f_{s} ⊗ e_{s+1} ≅ ⨁_{i=1}^{n} λ_{G,E_{i}}.

Proof. For 1 ≤ i ≤ n set V_{s+1}^{(i)} = V_{G,E_{i}}, and let V_{s+1}^{(n+1)} be the union of all components of V_{s+1} of dimension at most s. Then V_{s+1} = V_{s+1}^{(1)} ∪ ⋯ ∪ V_{s+1}^{(n+1)}, and we prove by induction on t that f_{s} ⊗ e_{V_{s+1}^{(1)}∪···∪V_{s+1}^{(t)}} ≅ ⨁_{i=1}^{t} f_{s} ⊗ e_{V_{s+1}^{(i)}} for 1 ≤ t ≤ n + 1. There is nothing to prove if t = 1, so assume that 1 < t ≤ n + 1. Because (V_{s+1}^{(1)} ∪ ⋯ ∪ V_{s+1}^{(t−1)}) ∩ V_{s+1}^{(t)} ⊆ V_{s}, Lemma 2.5 and the inductive assumption give

f_{s} ⊗ e_{V_{s+1}^{(1)}∪···∪V_{s+1}^{(t)}} ≅ (f_{s} ⊗ e_{V_{s+1}^{(1)}∪···∪V_{s+1}^{(t−1)}}) ⊕ (f_{s} ⊗ e_{V_{s+1}^{(t)}}) ≅ ⨁_{i=1}^{t} f_{s} ⊗ e_{V_{s+1}^{(i)}},

as desired. Moreover, V_{s+1}^{(n+1)} ⊆ V_{s} so that f_{s} ⊗ e_{V_{s+1}^{(n+1)}} ≅ 0 in kG-\underline{Mod}. Hence

f_{s} ⊗ e_{s+1} ≅ ⨁_{i=1}^{n+1} f_{s} ⊗ e_{V_{s+1}^{(i)}} ≅ ⨁_{i=1}^{n} f_{s} ⊗ e_{V_{G,E_{i}}} .

Fix i with 1 ≤ i ≤ n, and let V′_{i} be the union of all components of V_{s} that are not contained in V_{G,E_{i}}^{−}. Then V_{G,E_{i}} ∩ V′_{i} ⊆ V_{G,E_{i}}^{−}, so Lemma 2.5 implies that

f_{s} ⊗ e_{V_{G,E_{i}}} = f_{V_{G,E_{i}}^{−}∪V′_{i}} ⊗ e_{V_{G,E_{i}}} ≅ f_{V_{G,E_{i}}^{−}} ⊗ e_{V_{G,E_{i}}} = λ_{G,E_{i}}.

Thus f_{s} ⊗ e_{s+1} ≅ ⨁_{i=1}^{n} λ_{G,E_{i}}, as desired.

Proposition 2.7. Let 0 ≤ s ≤ r − 1, and let E_{1}, …, E_{n} be a set of representatives for the conjugacy classes of elementary abelian p-subgroups of rank s + 1 in G. Set N_{j} = N_{G}(E_{j}) for all j, and let η_{j} : e_{N_{j},E_{j}} → k be the universal map described in Proposition 2.1. Then for 1 ≤ j ≤ n there are maps α_{j} : e_{N_{j},E_{j}}^{↑G} → e_{s+1} such that ηα_{j} = εη_{j}↑^{G} and 1 ⊗ (⨁_{j=1}^{n} α_{j}) : f_{s} ⊗ (⨁_{j=1}^{n} e_{N_{j},E_{j}}^{↑G}) → f_{s} ⊗ e_{s+1} is a stable isomorphism.

Proof. For each j there is a commutative diagram

\xymatrix{e_{s+1} ⊗ e_{N_{j},E_{j}}^{↑G} \ar[r]^{1⊗εη_{j}↑^{G}} \ar[d]_{η⊗1} & e_{s+1} ⊗ k \ar[d]^{η⊗1} \\ k ⊗ e_{N_{j},E_{j}}^{↑G} \ar[r]_{1⊗εη_{j}↑^{G}} & k ⊗ k,}

and V̄_{G}(e_{N_{j},E_{j}}^{↑G}) = V_{G,E_{j}} ⊆ V_{s+1} so that η ⊗ 1 : e_{s+1} ⊗ e_{N_{j},E_{j}}^{↑G} → e_{N_{j},E_{j}}^{↑G} is a stable isomorphism for all j. Set α_{j} = (1 ⊗ εη_{j}↑^{G})(η ⊗ 1)^{−1}. Then the above diagram shows that ηα_{j} = εη_{j}↑^{G} for all j.

To prove that 1 ⊗ (⨁_{j=1}^{n} α_{j}) : f_{s} ⊗ (⨁_{j=1}^{n} e_{N_{j},E_{j}}^{↑G}) → f_{s} ⊗ e_{s+1} is a stable isomorphism, it suffices to prove that 1 ⊗ 1 ⊗ (⨁_{j=1}^{n} εη_{j}↑^{G}) : f_{s} ⊗ e_{s+1} ⊗ (⨁_{j=1}^{n} e_{N_{j},E_{j}}^{↑G}) → f_{s} ⊗ e_{s+1} is a stable isomorphism. Now f_{s} ⊗e_{s+1} ≅ ⨁_{i=1}^{n} λ_{G,E_{i}}, and V̄_{G}(λ_{G,E_{i}} ⊗e_{N_{j},E_{j}}^{↑G}) = V_{G,E_{i}}^{+} ∩V_{G,E_{j}} = ∅ if i ≠ j. Thus it is only necessary to show that ⨁_{j=1}^{n}(1 ⊗ εη_{j}↑^{G}) : ⨁_{j=1}^{n}(λ_{G,E_{j}} ⊗ e_{N_{j},E_{j}}^{↑G}) → ⨁_{j=1}^{n} λ_{G,E_{j}} is a stable isomorphism. But this is an immediate consequence of Proposition 2.4.

3. The category of local modules

The main purpose of this section is to define the category 𝔏(G, k) of G-local modules and a canonical functor 𝔉 : kG-\underline{Mod} → 𝔏(G, k) that will be studied in the following sections. It will be useful to begin by fixing some notation. Let P(G) be the collection of all p-subgroups of G. If P ∈ P(G), then throughout the remainder of the paper we usually write N for N_{G}(P). Similar notation will be used for normalizers of other p-subgroups. For example, if P_{0}, P_{1} ∈ P(G), then we write N_{0} for N_{G}(P_{0}) and N_{1} for N_{G}(P_{1}).

Suppose that for every P ∈ P(G) we have a module L(P) in kN-\underline{Mod} such that V̄_{N}(L(P)) ⊆ V_{N,P}. Assume in addition that whenever P_{1} ⊆ P_{2} in P(G), there is a homomorphism ϕ_{P_{1},P_{2}} : L(P_{1})↓_{N_{1}∩N_{2}} → L(P_{2})↓_{N_{1}∩N_{2}} such that the diagram

[figure: L(P_{1})↓_{N_{1}∩N_{2}}; ϕ_{P_{1},P_{2}}; ψ_{P_{1},P_{2}}; L(P_{2})↓_{N_{1}∩N_{2}}; e_{N_{1}∩N_{2},P_{1}} ⊗ L(P_{2})↓_{N_{1}∩N_{2}}; η⊗1; L(P_{2})↓_{N_{1}∩N_{2}}]

commutes for some stable isomorphism ψ_{P_{1},P_{2}}. Assume that ϕ_{P,P} = 1_{L(P)} for all P ∈ P(G) and that ϕ_{P_{2},P_{3}} ∘ ϕ_{P_{1},P_{2}} = ϕ_{P_{1},P_{3}} in k[N_{1} ∩ N_{2} ∩ N_{3}]-\underline{Mod} whenever P_{1} ⊆ P_{2} ⊆ P_{3} in P(G). For

LOCAL SUBGROUPS AND THE STABLE CATEGORY 7

each g ∈ G assume that there is a stable isomorphism c_{g}(P) : g ⊗ L(P) → L(^{g}P) such that the diagram

\xymatrix{g ⊗ h ⊗ L(P) \ar[r]^{≅} \ar[d]_{g⊗c_{h}(P)} & gh ⊗ L(P) \ar[d]^{c_{gh}(P)} \\ g ⊗ L(^{h}P) \ar[r]_{c_{g}(^{h}P)} & L(^{gh}P)}

commutes for all g, h ∈ G and all P ∈ P(G); assume in addition that if g ∈ N, then c_{g}(P) : g ⊗ L(P) → L(P) is the map given by g ⊗ x ↦ gx. Finally, suppose that if P_{1} ⊆ P_{2} and g ∈ G, then there is a commutative diagram

\xymatrix{(g ⊗ L(P_{1}))↓_{^{g}(N_{1}∩N_{2})} \ar[r]^{c_{g}(P_{1})} \ar[d]_{g⊗ϕ_{P_{1},P_{2}}} & L(^{g}P_{1})↓_{^{g}(N_{1}∩N_{2})} \ar[d]^{ϕ_{^{g}P_{1},^{g}P_{2}}} \\ (g ⊗ L(P_{2}))↓_{^{g}(N_{1}∩N_{2})} \ar[r]_{c_{g}(P_{2})} & L(^{g}P_{2})↓_{^{g}(N_{1}∩N_{2})}.}

Then we say that (L, ϕ, c) is a G-local module (over k). We usually abbreviate the notation by writing L for the G-local module (L, ϕ, c). For simplicity we sometimes also write ϕ instead of ϕ_{P_{1},P_{2}} when the subgroups P_{1} and P_{2} can be determined from the context.

If L = (L, ϕ, c) and L′ = (L′, ϕ′, c′) are G-local modules, let ξ(P) : L(P) → L′(P) be a map in kN-\underline{Mod} for all P ∈ P(G). Assume that there is a commutative diagram

\xymatrix{g ⊗ L(P) \ar[r]^{c_{g}(P)} \ar[d]_{g⊗ξ(P)} & L(^{g}P) \ar[d]^{ξ(^{g}P)} \\ g ⊗ L′(P) \ar[r]_{c′_{g}(P)} & L′(^{g}P)}

for all P ∈ P(G) and g ∈ G; assume in addition that if P_{1}, P_{2} ∈ P(G) and P_{1} ⊆ P_{2}, then there is a commutative diagram

\xymatrix{L(P_{1})↓_{N_{1}∩N_{2}} \ar[r]^{ϕ_{P_{1},P_{2}}} \ar[d]_{ξ(P_{1})} & L(P_{2})↓_{N_{1}∩N_{2}} \ar[d]^{ξ(P_{2})} \\ L′(P_{1})↓_{N_{1}∩N_{2}} \ar[r]_{ϕ′_{P_{1},P_{2}}} & L′(P_{2})↓_{N_{1}∩N_{2}}.}

Under these circumstances we say that the sequence of maps ξ = {ξ(P)}_{P ∈P(G)} is a G-local homomorphism.

Let 𝔏(G, k) denote the category in which the objects are the G-local modules and the morphisms are the G-local homomorphisms. Our next objective is to define a canonical functor

8 WAYNE W. WHEELER

𝔉 : kG-\underline{Mod} → 𝔏(G, k). For any kG-module M set (𝔉M)(P) = e_{N,P} ⊗ M↓_{N} for all P ∈ P(G). The map ϕ_{P_{1},P_{2}} : (𝔉M)(P_{1}) → (𝔉M)(P_{2}) is defined to be the composition

e_{N_{1},P_{1}}↓_{N_{1}∩N_{2}} ⊗ M↓_{N_{1}∩N_{2}} ⟶^{(1⊗η⊗1)^{−1}} e_{N_{1},P_{1}}↓ ⊗ e_{N_{2},P_{2}}↓ ⊗ M↓ ⟶^{η⊗1⊗1} e_{N_{2},P_{2}}↓_{N_{1}∩N_{2}} ⊗ M↓_{N_{1}∩N_{2}}

whenever P_{1} ⊆ P_{2}. For each P ∈ P(G) and each g ∈ G there is a stable isomorphism

g ⊗ e_{N,P} \underrightarrow{((g⊗1)⊗η)^{−1}} (g ⊗ e_{N,P}) ⊗ e_{^{g}N,^{g}P} ⟶^{(g⊗η)⊗1} e_{^{g}N,^{g}P} .

Combining this map with the isomorphism g ⊗M↓_{N} → M↓_{^{g}N} given by g ⊗m ↦ gm, we obtain a stable isomorphism c_{g}(P) by taking the composition

g ⊗ (e_{N,P} ⊗ M↓_{N}) ≅ (g ⊗ e_{N,P}) ⊗ (g ⊗ M↓_{N}) ≅ e_{^{g}N,^{g}P} ⊗ M↓_{^{g}N}.

If γ : M → M′ is a map in kG-\underline{Mod}, then 𝔉γ is the G-local homomorphism satisfying (𝔉γ)(P) = 1 ⊗ γ : e_{N,P} ⊗ M↓_{N} → e_{N,P} ⊗ M′↓_{N} for all P ∈ P(G).

We have now defined a canonical functor 𝔉 : kG-\underline{Mod} → 𝔏(G, k). It will sometimes be useful to consider the analogous functor kH-\underline{Mod} → 𝔏(H, k) for some subgroup H of G. By abuse of notation we use the same symbol 𝔉 to denote this functor for any subgroup of G.

The following proposition characterizes the isomorphisms in 𝔏(G, k). The proof is straightforward and is left to the reader.

Proposition 3.1. Let ξ : L → L′ be a G-local homomorphism. Then ξ is an isomorphism in 𝔏(G, k) if and only if ξ(P) : L(P) → L′(P) is an isomorphism for all P ∈ P(G).

4. Tensor products

If one thinks of a G-local module essentially as a kG-module, then one would expect to be able to define the tensor product of two G-local modules. Such a definition is indeed possible, but for our purposes it will be more useful to consider the tensor product of a kG-module and a G-local module. This construction has the advantage of being slightly easier to define and to use. The main result of this section is the existence of a certain natural isomorphism relating tensor products and the functor 𝔉. This isomorphism is needed in the next section to show that 𝔉 is an equivalence.

Let M be a kG-module, and let L = (L, ϕ, c) be a G-local module. For any P ∈ P(G) set (M ⊗L)(P) = M↓_{N} ⊗L(P). If P_{1}, P_{2} ∈ P(G) with P_{1} ⊆ P_{2}, then there is a stable isomorphism 1⊗ϕ_{P_{1},P_{2}} : M↓_{N_{1}∩N_{2}} ⊗L(P_{1})↓_{N_{1}∩N_{2}} → M↓_{N_{1}∩N_{2}} ⊗L(P_{2})↓_{N_{1}∩N_{2}}. For any g ∈ G and P ∈ P(G) let m_{g}(P) : g ⊗ M↓_{N} → M↓_{^{g}N} be the isomorphism given by g ⊗ x ↦ gx. Then there is a stable isomorphism given by the composition

g ⊗ (M↓_{N} ⊗ L(P)) ≅ (g ⊗ M↓_{N}) ⊗ (g ⊗ L(P)) ⟶^{m_{g}(P)⊗c_{g}(P)} M↓_{^{g}N} ⊗ L(^{g}P).

LOCAL SUBGROUPS AND THE STABLE CATEGORY 9

With these definitions it is easy to see that M ⊗ L = (M ⊗ L, 1 ⊗ ϕ, m ⊗ c) is again a G-local module. In order to provide a connection between this definition and the functor 𝔉, we begin with the following lemma.

Lemma 4.1. Let L be a G-local module, and let P_{1}, P_{2} ∈ P(G). Then there is a stable isomorphism θ_{L}(P_{1}, P_{2}) of k[N_{1} ∩ N_{2}]-modules given by the composition

e_{N_{1},P_{1}}↓_{N_{1}∩N_{2}} ⊗ L(P_{2})↓ ⟶_{≅}^{(1⊗η⊗ϕ)^{−1}} e_{N_{1},P_{1}}↓ ⊗ e_{N_{2},P_{2}}↓ ⊗ L(P_{1} ∩ P_{2})↓ ⟶_{≅}^{η⊗1⊗ϕ} e_{N_{2},P_{2}}↓ ⊗ L(P_{1})↓.

Moreover, the isomorphism θ_{L}(P_{1}, P_{2}) is natural in L. For each g ∈ G there is a commutative diagram in k[^{g}(N_{1} ∩ N_{2})]-\underline{Mod} of the form

[figure: g ⊗ (e_{N_{1},P_{1}}↓ ⊗ L(P_{2})↓); ≅; g⊗θ_{L}(P_{1},P_{2}); e_{^{g}N_{1},^{g}P_{1}}↓ ⊗ (g ⊗ L(P_{2}))↓; 1⊗c_{g}(P_{2}); e_{^{g}N_{1},^{g}P_{1}}↓ ⊗ L(^{g}P_{2})↓; θ_{L}(^{g}P_{1},^{g}P_{2}); g ⊗ (e_{N_{2},P_{2}}↓ ⊗ L(P_{1})↓); ≅; e_{^{g}N_{2},^{g}P_{2}}↓ ⊗ (g ⊗ L(P_{1}))↓; 1⊗c_{g}(P_{1}); e_{^{g}N_{2},^{g}P_{2}}↓ ⊗ L(^{g}P_{1})↓.]

If P_{1} ⊆ P_{2} and P ∈ P(G), then there is also a commutative diagram in k[N_{1} ∩ N_{2} ∩ N]-\underline{Mod} given by

\xymatrix{e_{N_{1},P_{1}}↓ ⊗ e_{N_{2},P_{2}}↓ ⊗ L(P)↓ \ar[rr]^{≅} \ar[d]_{1⊗θ_{L}(P_{2},P)} &  & e_{N_{2},P_{2}}↓ ⊗ e_{N_{1},P_{1}}↓ ⊗ L(P)↓ \ar[d]^{1⊗θ_{L}(P_{1},P)} \\ e_{N_{1},P_{1}}↓ ⊗ e_{N,P} ↓ ⊗ L(P_{2})↓ \ar@{-->}[dr]_{η⊗1⊗1} &  & e_{N_{2},P_{2}}↓ ⊗ e_{N,P} ↓ ⊗ L(P_{1})↓ \ar@{-->}[dl]^{η⊗1⊗ϕ} \\  & e_{N,P} ↓ ⊗ L(P_{2})↓.}

Proof. Set P_{3} = P_{1} ∩ P_{2}. Because N_{1} ∩ N_{2} ⊆ N_{3}, we can restrict the maps ϕ_{P_{3},P_{i}} to N_{1} ∩ N_{2} for i = 1, 2, and we claim that the resulting maps 1 ⊗ η ⊗ ϕ_{P_{3},P_{2}}↓ and η ⊗ 1 ⊗ ϕ_{P_{3},P_{1}}↓ occurring in the definition of θ_{L}(P_{1}, P_{2}) are isomorphisms. There is a triangle of k[N_{1} ∩ N_{2}]-modules of the form

\xymatrix{e_{N_{2}∩N_{3},P_{3}}↓ ⊗ L(P_{2})↓ \ar[r]^{η⊗1} & L(P_{2})↓ \ar[r] & f_{N_{2}∩N_{3},P_{3}}↓ ⊗ L(P_{2})↓ \ar[r] & Ω^{−1}k ⊗ e_{N_{2}∩N_{3},P_{3}}↓ ⊗ L(P_{2})↓.}

Because V̄_{N_{1}∩N_{2}}(e_{N_{1},P_{1}}↓ ⊗ f_{N_{2}∩N_{3},P_{3}}↓ ⊗ L(P_{2})↓) = ∅, the map η ⊗ 1 occurring in this triangle becomes an isomorphism upon tensoring with e_{N_{1},P_{1}}↓. The commutativity of the diagram

[figure: e_{N_{1},P_{1}}↓_{N_{1}∩N_{2}} ⊗ e_{N_{2},P_{2}}↓_{N_{1}∩N_{2}} ⊗ L(P_{3})↓_{N_{1}∩N_{2}}; 1⊗η⊗ϕ_{P_{3},P_{2}}↓; 1⊗1⊗ψ_{P_{3},P_{2}}; ≅; e_{N_{1},P_{1}}↓_{N_{1}∩N_{2}} ⊗ L(P_{2})↓_{N_{1}∩N_{2}}; e_{N_{1},P_{1}}↓ ⊗ e_{N_{2},P_{2}}↓ ⊗ e_{N_{2}∩N_{3},P_{3}}↓ ⊗ L(P_{2})↓; 1⊗η⊗η⊗1; 1⊗η⊗1⊗1; ≅; e_{N_{1},P_{1}}↓_{N_{1}∩N_{2}} ⊗ L(P_{2})↓_{N_{1}∩N_{2}}; e_{N_{1},P_{1}}↓_{N_{1}∩N_{2}} ⊗ e_{N_{2}∩N_{3},P_{3}}↓_{N_{1}∩N_{2}} ⊗ L(P_{2})↓_{N_{1}∩N_{2}}; ≅; 1⊗η⊗1; e_{N_{1},P_{1}}↓_{N_{1}∩N_{2}} ⊗ L(P_{2})↓_{N_{1}∩N_{2}}]

shows that 1 ⊗ η ⊗ ϕ_{P_{3},P_{2}}↓ is an isomorphism. Similarly, η ⊗ 1 ⊗ ϕ_{P_{3},P_{1}}↓ is an isomorphism, and so is θ_{L}(P_{1}, P_{2}). It is straightforward to check that this isomorphism is natural in L.

Now suppose in addition that g ∈ G. Observe that there is a commutative diagram in k[^{g}(N_{1} ∩ N_{2})]-\underline{Mod} of the form

\xymatrix{g ⊗ (e_{N_{1},P_{1}}↓ ⊗ e_{N_{2},P_{2}}↓ ⊗ L(P_{3})↓) \ar[r]^{g⊗(1⊗η⊗ϕ)} \ar[d]_{≅} & g ⊗ (e_{N_{1},P_{1}}↓ ⊗ L(P_{2})↓) \ar[d]^{≅} \\ e_{^{g}N_{1},^{g}P_{1}}↓ ⊗ e_{^{g}N_{2},^{g}P_{2}}↓ ⊗ (g ⊗ L(P_{3}))↓ \ar[d]_{1⊗1⊗c_{g}(P_{3})} \ar[r]^{1⊗η⊗(g⊗ϕ)} & e_{^{g}N_{1},^{g}P_{1}}↓ ⊗ (g ⊗ L(P_{2}))↓ \ar[d]^{1⊗c_{g}(P_{2})} \\ e_{^{g}N_{1},^{g}P_{1}}↓ ⊗ e_{^{g}N_{2},^{g}P_{2}}↓ ⊗ L(^{g}P_{3})↓ \ar[r]_{1⊗η⊗ϕ} & e_{^{g}N_{1},^{g}P_{1}}↓ ⊗ L(^{g}P_{2})↓.}

Combining this diagram with the analogous diagram in which P_{1} and P_{2} are interchanged, we conclude that the first diagram given in the statement of the lemma commutes.

Finally, if P_{1} ⊆ P_{2} and P ∈ P(G), then it is straightforward to verify that the last diagram given in the statement of the lemma commutes.

Proposition 4.2. Let L be a G-local module, and let P ∈ P(G). Then there is an isomorphism 𝔉(L(P)↑^{G}) ≅ e_{N,P}^{↑G} ⊗ L, and this isomorphism is natural in L.

LOCAL SUBGROUPS AND THE STABLE CATEGORY 11

Proof. Let P_{0} ∈ P(G). By Lemma 4.1 there are isomorphisms

e_{N_{0},P_{0}} ⊗ L(P)↑^{G}↓_{N_{0}} ≅ ⨁_{g∈N_{0}\G/N} e_{N_{0},P_{0}} ⊗ (g ⊗ L(P))↓_{^{g}N∩N_{0}}↑^{N_{0}}

≅ ⨁_{g}(e_{N_{0},P_{0}}↓_{^{g}N∩N_{0}} ⊗ L(^{g}P)↓_{^{g}N∩N_{0}})↑^{N_{0}}

≅ ⨁_{g}(e_{^{g}N,^{g}P} ↓_{^{g}N∩N_{0}} ⊗ L(P_{0})↓_{^{g}N∩N_{0}})↑^{N_{0}}

≅ ⨁_{g}(g ⊗ e_{N,P})↓_{^{g}N∩N_{0}}↑^{N_{0}} ⊗ L(P_{0})

≅ e_{N,P}^{↑G} ↓_{N_{0}} ⊗ L(P_{0}),

and it is straightforward to check that these isomorphisms are natural in L. Let Φ(P_{0}) : e_{N_{0},P_{0}} ⊗ L(P)↑^{G}↓_{N_{0}} → e_{N,P}^{↑G} ↓_{N_{0}} ⊗ L(P_{0}) be the above isomorphism. Then it is only necessary to show that Φ : 𝔉(L(P)↑^{G}) → e_{N,P}^{↑G} ⊗ L is a G-local homomorphism.

Suppose that P_{1}, P_{2} ∈ P(G) with P_{1} ⊆ P_{2}. We wish to show that there is a commutative diagram

\xymatrix{e_{N_{1},P_{1}}↓_{N_{1}∩N_{2}} ⊗ L(P)↑^{G}↓_{N_{1}∩N_{2}} \ar[d]_{Φ(P_{1})} & e_{N_{1},P_{1}}↓ ⊗ e_{N_{2},P_{2}}↓ ⊗ L(P)↑^{G}↓ \ar[l]^{≅}_{1⊗η⊗1} \ar[r]^{η⊗1⊗1} & e_{N_{2},P_{2}}↓ ⊗ L(P)↑^{G}↓ \ar[d]^{Φ(P_{2})} \\ e_{N,P}^{↑G} ↓_{N_{1}∩N_{2}} ⊗ L(P_{1})↓_{N_{1}∩N_{2}} \ar[rr]_{1⊗ϕ_{P_{1},P_{2}}} &  & e_{N,P}^{↑G} ↓ ⊗ L(P_{2})↓.}

If g ∈ G, then Lemma 4.1 shows that there is a commutative diagram in k[^{g}N ∩ N_{1} ∩ N_{2}]-\underline{Mod} given by

\xymatrix{e_{N_{1},P_{1}}↓ ⊗ e_{N_{2},P_{2}}↓ ⊗ L(^{g}P)↓ \ar[rr]^{≅} \ar[d]_{1⊗θ_{L}(P_{2},^{g}P)} &  & e_{N_{2},P_{2}}↓ ⊗ e_{N_{1},P_{1}}↓ ⊗ L(^{g}P)↓ \ar[d]^{1⊗θ_{L}(P_{1},^{g}P)} \\ e_{N_{1},P_{1}}↓ ⊗ e_{^{g}N,^{g}P} ↓ ⊗ L(P_{2})↓ \ar[dr]_{η⊗1⊗1} &  & e_{N_{2},P_{2}}↓ ⊗ e_{^{g}N,^{g}P} ↓ ⊗ L(P_{1})↓ \ar[dl]^{η⊗1⊗ϕ} \\  & e_{^{g}N,^{g}P} ↓ ⊗ L(P_{2})↓.}

After inducing from^{g}N ∩N_{1}∩N_{2} to N_{1}∩N_{2} and taking the direct sum over g ∈ (N_{1}∩N_{2})\G/N, some tedious verifications show that there is a commutative diagram

\xymatrix{e_{N_{2},P_{2}}↓ ⊗ e_{N_{1},P_{1}}↓ ⊗ L(P)↑^{G}↓_{N_{1}∩N_{2}} \ar[d]_{≅} \ar[r]^{η⊗Φ(P_{1})} & e_{N,P}^{↑G} ↓_{N_{1}∩N_{2}} ⊗ L(P_{1})↓_{N_{1}∩N_{2}} \ar[d]^{1⊗ϕ_{P_{1},P_{2}}} \\ e_{N_{1},P_{1}}↓ ⊗ e_{N_{2},P_{2}}↓ ⊗ L(P)↑^{G}↓_{N_{1}∩N_{2}} \ar[r]_{η⊗Φ(P_{2})} & e_{N,P}^{↑G} ↓_{N_{1}∩N_{2}} ⊗ L(P_{2})↓_{N_{1}∩N_{2}}.}

12 WAYNE W. WHEELER

It is easy to see that this is equivalent to the commutativity of the desired diagram.

Finally, suppose that g ∈ G and that P_{0} ∈ P(G). We must show that there is a commutative diagram in k[^{g}N_{0}]-\underline{Mod} of the form

[figure: g ⊗ (e_{N_{0},P_{0}} ⊗ L(P)↑^{G}↓); g⊗Φ(P_{0}); ≅; e_{^{g}N_{0},^{g}P_{0}} ⊗ (g ⊗ L(P)↑^{G}↓); e_{^{g}N_{0},^{g}P_{0}} ⊗ L(P)↑^{G}↓; Φ(^{g}P_{0}); g ⊗ (e_{N,P}^{↑G} ↓_{N_{0}} ⊗ L(P_{0})); ≅; e_{N,P}^{↑G} ↓_{^{g}N_{0}} ⊗ (g ⊗ L(P_{0})); e_{N,P}^{↑G} ↓_{^{g}N_{0}} ⊗ L(^{g}P_{0}).]

Let T be a set of representatives for the (N_{0}, N) double cosets in G. It follows from Lemma 4.1 that there is a commutative diagram

\xymatrix{g ⊗ (e_{N_{0},P_{0}} ⊗ L(P)↑^{G}↓_{N_{0}}) \ar[r]^{≅} \ar[d] & e_{^{g}N_{0},^{g}P_{0}} ⊗ L(P)↑^{G}↓_{^{g}N_{0}} \ar[d] \\ ⨁_{h∈T} (g ⊗ (e_{N_{0},P_{0}}↓ ⊗ (h ⊗ L(P))↓_{^{h}N∩N_{0}}↑^{N_{0}})) \ar[r]^{≅} \ar[d] & ⨁_{h} e_{^{g}N_{0},^{g}P_{0}} ⊗ (gh ⊗ L(P))↓_{^{gh}N∩^{g}N_{0}}↑^{^{g}N_{0}} \ar[d] \\ ⨁_{h∈T} (g ⊗ (e_{N_{0},P_{0}}↓ ⊗ L(^{h}P)↓_{^{h}N∩N_{0}})↑^{N_{0}}) \ar[r]^{≅} \ar[d] & ⨁_{h∈T} (e_{^{g}N_{0},^{g}P_{0}}↓ ⊗ L(^{gh}P)↓)↑^{^{g}N_{0}} \ar[d] \\ ⨁_{h∈T} (g ⊗ (e_{^{h}N,^{h}P} ↓ ⊗ L(P_{0})↓)↑^{N_{0}}) \ar[r]^{≅} \ar[d] & ⨁_{h∈T} (e_{^{gh}N,^{gh}P} ↓ ⊗ L(^{g}P_{0})↓)↑^{^{g}N_{0}} \ar[d] \\ g ⊗ (e_{N,P}^{↑G} ↓_{N_{0}} ⊗ L(P_{0})) \ar[r]^{≅} & e_{N,P}^{↑G} ↓_{^{g}N_{0}} ⊗ L(^{g}P_{0}),}

and this completes the proof.

It is possible to define not only the tensor product but also operations such as induction and restriction of G-local modules. In the case of restriction, for example, let L be a G-local module, and let H ⊆ G. Define an H-local module L↓_{H} by setting

L↓_{H}(P) = L(P)↓_{H∩N}

for every p-subgroup P ⊆ H. Note that V̄_{N}(L(P)) ⊆ V_{N,P}, so

V̄_{H∩N}(L↓_{H}(P)) = (res_{N,H∩N}^{∗})^{−1}(V̄_{N}(L(P))) ⊆ (res_{N,H∩N}^{∗})^{−1}(V_{N,P}) = V_{H∩N,P} .

If P_{1} and P_{2} are p-subgroups of H with P_{1} ⊆ P_{2}, then the map ϕ_{P_{1},P_{2}} : L(P_{1})↓_{N_{1}∩N_{2}} → L(P_{2})↓_{N_{1}∩N_{2}} gives a commutative diagram

[figure: L(P_{1})↓_{N_{1}∩N_{2}∩H}; ϕ_{P_{1},P_{2}}↓; ψ_{P_{1},P_{2}}↓; L(P_{2})↓_{N_{1}∩N_{2}∩H}; e_{N_{1}∩N_{2},P_{1}}↓ ⊗ L(P_{2})↓; η⊗1; L(P_{2})↓_{N_{1}∩N_{2}∩H}.]

Moreover, (res_{N_{1}∩N_{2},N_{1}∩N_{2}∩H}^{∗})^{−1}(V_{N_{1}∩N_{2},P_{1}}) = V_{N_{1}∩N_{2}∩H,P_{1}} so that e_{N_{1}∩N_{2},P_{1}}↓_{N_{1}∩N_{2}∩H} ≅ e_{N_{1}∩N_{2}∩H,P_{1}}. Finally, if h ∈ H, then c_{h}(P) : h ⊗ L(P) → L(^{h}P) is a stable isomorphism, so c_{h}(P)↓_{^{h}N∩H} : h ⊗ L↓_{H}(P) → L↓_{H}(^{h}P) is also a stable isomorphism. It is easy to check that the resulting object L↓_{H} = (L↓_{H}, ϕ↓_{H}, c↓_{H}) is an H-local module. If ξ : L → L′ is a G-local homomorphism, set ξ↓_{H}(P) = ξ(P)↓_{N∩H} : L(P)↓_{N∩H} → L′(P)↓_{N∩H} for every p-subgroup P of H. Then ξ↓_{H} : L↓_{H} → L′↓_{H} is an H-local homomorphism, so there is a restriction functor res_{G,H} : 𝔏(G, k) → 𝔏(H, k).

Using arguments similar to those given in the proof of Proposition 4.2, one can show that for any P ∈ P(G) there is a natural isomorphism 𝔉(L(P)) ≅ e_{N,P} ⊗ L↓_{N} in 𝔏(N, k). It is much easier, however, to deduce the existence of this isomorphism from Theorem 5.9, so we postpone any further discussion of the restriction functor to the next section.

5. An equivalence of categories

The current section is devoted to showing that the canonical functor 𝔉 : kG-\underline{Mod} → 𝔏(G, k) is an equivalence of categories. Constructing an explicit adjoint seems to be rather difficult, so we prove that 𝔉 is an equivalence by showing that it is essentially surjective, full, and faithful. As a first step toward proving faithfulness, we begin with a result showing that any G-local homomorphism is uniquely determined by its value on a Sylow p-subgroup.

Proposition 5.1. Let L and L′ be G-local modules, and let ξ : L → L′ be a G-local homomorphism such that ξ(P) = 0 for some Sylow p-subgroup P of G. Then ξ = 0.

Proof. Let Q be a p-subgroup of G, and let R be a Sylow p-subgroup of N_{G}(Q). Then there is an element g ∈ G with ^{g}R ⊆ P. The commutative diagram

\xymatrix{g ⊗ L(Q) \ar[r]^{c_{g}(Q)} \ar[d]_{g⊗ξ(Q)} & L(^{g}Q) \ar[d]^{ξ(^{g}Q)} \\ g ⊗ L′(Q) \ar[r]_{c′_{g}(Q)} & L′(^{g}Q)}

14 WAYNE W. WHEELER

shows that ξ(Q) = 0 if and only if ξ(^{g}Q) = 0. In order to show that ξ(Q) = 0, then, we may replace Q by ^{g}Q and may therefore assume that g = 1 and that R ⊆ P.

The commutative diagram

[figure: L(Q)↓_{N_{G}(Q)∩N_{G}(P)}; ϕ_{Q,P}; ξ(Q)↓; L(P)↓_{N_{G}(Q)∩N_{G}(P)}; ξ(P)↓=0; L′(Q)↓_{N_{G}(Q)∩N_{G}(P)}; ϕ′_{Q,P}; ψ_{Q,P}; L′(P)↓_{N_{G}(Q)∩N_{G}(P)}; e_{N_{G}(Q)∩N_{G}(P),Q} ⊗ L′(P)↓; η⊗1; L′(P)↓_{N_{G}(Q)∩N_{G}(P)}]

shows that (η ⊗ 1) ∘ ψ_{Q,P} ∘ ξ(Q)↓ = 0. But V̄_{N_{G}(Q)∩N_{G}(P)}(L(Q)↓) ⊆ V_{N_{G}(Q)∩N_{G}(P),Q}, so the map (η ⊗ 1) ∘ ψ_{Q,P} ∘ ξ(Q)↓ factors uniquely through η ⊗ 1 and hence ψ_{Q,P} ∘ ξ(Q)↓ = 0. Because ψ_{Q,P} is a stable isomorphism, it follows that ξ(Q)↓_{N_{G}(Q)∩N_{G}(P)} = 0 and hence ξ(Q)↓_{R} = 0. Thus

ξ(Q) = \frac{1}{|N_{G}(Q) : R|} Tr_{R}^{N_{G}(Q)}(ξ(Q)↓_{R}) = 0,

as desired.

Let P be a Sylow p-subgroup of G. Set N = N_{G}(P), and let g ∈ G. If Q is a Sylow p-subgroup of ^{g}N ∩ N, then Q ⊆ ^{g}P ∩ P ⊆ ^{g}N ∩ N and hence Q = ^{g}P ∩ P. It follows that the map η : e_{^{g}N∩N,^{g}P ∩P} → k is a stable isomorphism and hence so are

ϕ_{^{g}P ∩P,^{g}P} : L(^{g}P ∩ P)↓_{^{g}N∩N} → L(^{g}P)↓_{^{g}N∩N}

and

ϕ_{^{g}P ∩P,P} : L(^{g}P ∩ P)↓_{^{g}N∩N} → L(P)↓_{^{g}N∩N}.

Definition 5.2. Let L and L′ be G-local modules, let P be a Sylow p-subgroup of G, and set N = N_{G}(P). Suppose that γ : L(P) → L′(P) is a kN-homomorphism. We say that γ is G-stable if there is a commutative diagram in k[^{g}N ∩ N]-\underline{Mod} of the form

\xymatrix{(g ⊗ L(P))↓ \ar[d]_{g⊗γ} \ar[r]^{c_{g}(P)} & L(^{g}P)↓ \ar[r]^{ϕ^{−1}} & L(^{g}P ∩ P)↓ \ar[r]^{ϕ} & L(P)↓ \ar[d]^{γ} \\ (g ⊗ L′(P))↓ \ar[r]_{c′_{g}(P)} & L′(^{g}P)↓ \ar[r]_{(ϕ′)^{−1}} & L′(^{g}P ∩ P)↓ \ar[r]_{ϕ′} & L′(P)↓}

for all g ∈ G.

The following result is easy to verify, and the proof is left to the reader.

LOCAL SUBGROUPS AND THE STABLE CATEGORY 15

Proposition 5.3. Let L and L′ be G-local modules, let P be a Sylow p-subgroup of G, and set N = N_{G}(P). Suppose that γ : L(P) → L′(P) is a kN-homomorphism. If there is a G-local homomorphism ξ : L → L′ with ξ(P) = γ, then γ is G-stable.

Let L and L′ be G-local modules, and let P be a Sylow p-subgroup of G. Set N = N_{G}(P). We will define a homomorphism

T = T_{P}^{L,L′} : \underline{Hom}_{kN}(L(P), L′(P)) → Hom_{𝔏(G,k)}(L, L′).

Because P is a Sylow p-subgroup of G, we have e_{N,P} ≅ k_{N}. The maps ι : k_{G} → k_{N}^{↑G} and ε : k_{N}^{↑G} → k_{G} given by ι(x) = ∑_{g∈G/N} g ⊗ x and ε(∑_{g∈G/N} g ⊗ x_{g}) = ∑_{g} x_{g} satisfy ει = 1_{k} because |G : N| ≡ 1 (mod p). If γ : L(P) → L′(P) is a kN-homomorphism, let T γ : L → L′ be the G-local homomorphism given by the composition

\xymatrix{L \ar[r]^{ι⊗1} & k_{N}^{↑G} ⊗ L ≅ e_{N,P}^{↑G} ⊗ L ≅ 𝔉(L(P)↑^{G}) \ar[r]^{𝔉(γ↑^{G})} & 𝔉(L′(P)↑^{G}) ≅ e_{N,P}^{↑G} ⊗ L′ ≅ k_{N}^{↑G} ⊗ L′ \ar[r]^{ε⊗1} & L′.}

Proposition 5.4. Let L and L′ be G-local modules, and let P be a Sylow p-subgroup of G. Set N = N_{G}(P), and let γ : L(P) → L′(P) be a kN-homomorphism.

(1) If γ is G-stable, then T γ : L → L′ is a G-local homomorphism with (T γ)(P) = γ.

(2) If L = 𝔉M and L′ = 𝔉M′ for some kG-modules M and M′, then T γ = 𝔉(Tr_{N}^{G} γ).

Proof. To compute (T γ)(P), we begin by observing that the definition of the isomorphism in Proposition 4.2 shows that the composition of the isomorphisms

L(P)↑^{G}↓_{N} ≅ e_{N,P} ⊗ L(P)↑^{G}↓_{N} = 𝔉(L(P)↑^{G})(P) ≅ e_{N,P}^{↑G} ↓_{N} ⊗ L(P) ≅ k_{N}^{↑G}↓_{N} ⊗ L(P)

is given by

L(P)↑^{G}↓_{N} ≅ ⨁_{g∈N\G/N}(g ⊗ L(P))↓_{^{g}N∩N}↑^{N}

≅ ⨁_{g} L(^{g}P)↓_{^{g}N∩N}↑^{N}

≅ ⨁_{g} L(^{g}P ∩ P)↓_{^{g}N∩N}↑^{N}

≅ ⨁_{g} L(P)↓_{^{g}N∩N}↑^{N}

≅ ⨁_{g} k_{^{g}N∩N}^{↑G} ⊗ L(P)

≅ k_{N}^{↑G}↓_{N} ⊗ L(P).

If γ is G-stable, then there is a commutative diagram

[figure: L(P); γ; ≅; k ⊗ L(P); 1⊗γ; ι⊗1; k_{N}^{↑G}↓_{N} ⊗ L(P); 1⊗γ; ≅; L(P)↑^{G}↓_{N}; γ↑^{G}↓_{N}; L′(P); ≅; k ⊗ L′(P); k_{N}^{↑G}↓_{N} ⊗ L′(P); ε⊗1; ≅; L′(P)↑^{G}↓_{N}.]

Hence (T γ)(P) = γ, and (1) holds.

Now suppose that L = 𝔉M and L′ = 𝔉M′. Identifying e_{N,P} with k_{N}, we may assume that L(P) = M↓_{N} and L′(P) = M′↓_{N}. Then the isomorphism

k_{N}^{↑G}↓_{N} ⊗ L(P) = k_{N}^{↑G}↓_{N} ⊗ M↓_{N} ≅ M↓_{N}↑^{G}↓_{N} = L(P)↑^{G}↓_{N}

is given by (g ⊗ 1) ⊗ m ↦ g ⊗ g^{−1}m, and similarly for L′ instead of L. It follows that the composition

\xymatrix{M↓_{N} \ar[r]^{ι⊗1} & k_{N}^{↑G}↓_{N} ⊗ M↓_{N} ≅ M↓_{N}↑^{G}↓_{N} \ar[r]^{γ↑^{G}↓_{N}} & M′↓_{N}↑^{G}↓_{N} ≅ k_{N}^{↑G}↓_{N} ⊗ M′↓_{N} \ar[r]^{ε⊗1} & M′↓_{N}}

is equal to Tr_{N}^{G} γ. Thus T γ : 𝔉M → 𝔉M′ and 𝔉(Tr_{N}^{G} γ) : 𝔉M → 𝔉M′ are G-local homomorphisms with (T γ)(P) = 𝔉(Tr_{N}^{G} γ)(P). Proposition 5.1 now implies that T γ = 𝔉(Tr_{N}^{G} γ), as desired.

Corollary 5.5. The functor 𝔉 : kG-\underline{Mod} → 𝔏(G, k) is full.

Proof. Let M and M′ be kG-modules, and let ξ : 𝔉M → 𝔉M′ be a G-local homomorphism. Let P be a Sylow p-subgroup of G, and set N = N_{G}(P). Then ξ(P) is G-stable, so T ξ(P) : 𝔉M → 𝔉M′ satisfies (T ξ(P))(P) = ξ(P). Hence ξ = T ξ(P) = 𝔉(Tr_{N}^{G} ξ(P)), and 𝔉 is full.

As another consequence of Proposition 5.4, we present a result that strengthens the statement of Proposition 3.1.

Corollary 5.6. Let ξ : L → L′ be a G-local homomorphism, and let P be a Sylow p-subgroup of G. Then ξ is an isomorphism if and only if ξ(P) is an isomorphism.

Proof. It is clear that if ξ is an isomorphism, then ξ(P) is an isomorphism. To prove the converse, suppose that ξ(P) is an isomorphism. Because ξ(P) is G-stable, the map ξ(P)^{−1} : L′(P) → L(P) is also G-stable. By Proposition 5.4 there is a G-local homomorphism ξ′ : L′ → L with ξ′(P) = ξ(P)^{−1}. Then (ξ′ξ)(P) = 1_{L(P)} and (ξξ′)(P) = 1_{L′(P)}, so ξ′ξ = 1_{L} and ξξ′ = 1_{L′} by Proposition 5.1. Thus ξ is an isomorphism.

The following technical lemma is needed to show that the functor 𝔉 is essentially surjective.

LOCAL SUBGROUPS AND THE STABLE CATEGORY 17

Lemma 5.7. Let ξ_{1} : L_{1} → L′_{1} and ξ_{2} : L_{2} → L′_{2} be G-local homomorphisms, and let P be a Sylow p-subgroup of G. Set N = N_{G}(P). Suppose that γ : L_{1}(P) → L_{2}(P) and γ′ : L′_{1}(P) → L′_{2}(P) are kN-homomorphisms such that there is a commutative diagram

\xymatrix{L_{1}(P) \ar[d]_{γ} \ar[r]^{ξ_{1}(P)} & L′_{1}(P) \ar[d]^{γ′} \\ L_{2}(P) \ar[r]_{ξ_{2}(P)} & L′_{2}(P).}

Then (T γ′) ∘ ξ_{1} = ξ_{2} ∘ (T γ).

Proof. The result follows immediately from the commutativity of the diagram

[figure: L_{1}; ξ_{1}; ι⊗1; k_{N}^{↑G} ⊗ L_{1}; 1⊗ξ_{1}; ≅; 𝔉(L_{1}(P)↑^{G}); 𝔉(ξ_{1}(P)↑^{G}); 𝔉(γ↑^{G}); 𝔉(L_{2}(P)↑^{G}); 𝔉(ξ_{2}(P)↑^{G}); ≅; k_{N}^{↑G} ⊗ L_{2}; 1⊗ξ_{2}; ε⊗1; L_{2}; ξ_{2}; L′_{1}; ι⊗1; k_{N}^{↑G} ⊗ L′_{1}; ≅; 𝔉(L′_{1}(P)↑^{G}); 𝔉(γ′↑^{G}); 𝔉(L′_{2}(P)↑^{G}); ≅; k_{N}^{↑G} ⊗ L′_{2}; ε⊗1; L′_{2}.]

Let r denote the p-rank of G, and suppose that s is an integer with 0 ≤ s ≤ r. Let kG- \underline{Mod}(s) denote the full subcategory of kG-\underline{Mod} consisting of all modules M with e_{s} ⊗ M ≅ M, and let 𝔏_{s}(G, k) be the full subcategory of 𝔏(G, k) consisting of all objects L such that e_{s} ⊗ L ≅ L. It is easy to check that if M is any kG-module, then M is in kG-\underline{Mod}(s) if and only if 𝔉M is in 𝔏_{s}(G, k). In particular, the restriction of 𝔉 to kG-\underline{Mod}(s) defines a functor 𝔉_{s} : kG-\underline{Mod}(s) → 𝔏_{s}(G, k).

Lemma 5.8. Let 0 ≤ s ≤ r − 1, and assume that the functor 𝔉_{s} : kG-\underline{Mod}(s) → 𝔏_{s}(G, k) is essentially surjective. Then 𝔉_{s+1} is also essentially surjective.

Proof. Let L be an object of 𝔏_{s+1}(G, k). Then e_{s} ⊗ L is an object of 𝔏_{s}(G, k), so there is a module M in kG-\underline{Mod}(s) such that 𝔉M ≅ e_{s} ⊗ L. Let ξ : 𝔉M → L denote the composition

𝔉M ≅ e_{s} ⊗ L ⟶^{η⊗1} L.

Then 1 ⊗ ξ : e_{s} ⊗ 𝔉M → e_{s} ⊗ L is an isomorphism.

Let E_{1}, …, E_{n} be representatives for the conjugacy classes of elementary abelian p-subgroups of rank s + 1 in G. Fix i with 1 ≤ i ≤ n, and consider the map β_{E_{i}} = β_{s} ⊗ 1 occurring in the triangle

\xymatrix{e_{s}↓_{N_{i}} ⊗ L(E_{i}) \ar[r] & L(E_{i}) \ar[r] & f_{s}↓_{N_{i}} ⊗ L(E_{i}) \ar[r]^{β_{s}⊗1} & Ω^{−1}k ⊗ e_{s}↓_{N_{i}} ⊗ L(E_{i}).}

Using the isomorphisms

\underline{Hom}_{kN_{i}}(f_{s}↓_{N_{i}} ⊗ L(E_{i}), Ω^{−1}k ⊗ e_{s}↓_{N_{i}} ⊗ L(E_{i})) ≅ \underline{Hom}_{kN_{i}}(f_{s}↓_{N_{i}} ⊗ L(E_{i}), Ω^{−1}k ⊗ (𝔉M)(E_{i}))
= \underline{Hom}_{kN_{i}}(f_{s}↓ ⊗ L(E_{i}), Ω^{−1}k ⊗ e_{N_{i},E_{i}} ⊗ M↓)
≅ \underline{Hom}_{kN_{i}}(f_{s}↓_{N_{i}} ⊗ L(E_{i}), Ω^{−1}k ⊗ M↓_{N_{i}})
≅ \underline{Hom}_{kG}(f_{s} ⊗ L(E_{i})↑^{G}, Ω^{−1}k ⊗ M),

we obtain a map β′_{E_{i}} : f_{s} ⊗ L(E_{i})↑^{G} → Ω^{−1}k ⊗ M corresponding to β_{E_{i}}. Set M_{s+1} = ⨁_{i=1}^{n} L(E_{i})↑^{G}. Taking the direct sum of the maps β′_{E_{i}} for 1 ≤ i ≤ n gives a map β : f_{s}⊗M_{s+1} → Ω^{−1}k ⊗ M. Let M′ be the kG-module defined by the triangle

\xymatrix{M \ar[r]^{μ} & M′ \ar[r] & f_{s} ⊗ M_{s+1} \ar[r]^{β} & Ω^{−1}k ⊗ M.}

Then V̄_{G}(M′) ⊆ V̄_{G}(M) ∪ V̄_{G}(f_{s} ⊗ M_{s+1}) ⊆ V_{s+1}, so M′ is an object of kG-\underline{Mod}(s + 1). We will show that 𝔉M′ ≅ L.

Observe that for 1 ≤ i ≤ n the map β′_{E_{i}} : f_{s} ⊗ L(E_{i})↑^{G} → Ω^{−1}k ⊗ M factors as

f_{s} ⊗ L(E_{i})↑^{G} ⟶^{β_{s}⊗1} Ω^{−1}k ⊗ e_{s} ⊗ L(E_{i})↑^{G} ≅ Ω^{−1}k ⊗ (e_{s}↓_{N_{i}} ⊗ L(E_{i}))↑^{G}

≅ Ω^{−1}k ⊗ (e_{N_{i},E_{i}} ⊗ M↓_{N_{i}})↑^{G} ≅ Ω^{−1}k ⊗ e_{N_{i},E_{i}}^{↑G} ⊗ M ⟶^{1⊗εη_{i}↑^{G}⊗1} Ω^{−1}k ⊗ M.

Let P be a Sylow p-subgroup of G, and set N = N_{G}(P). Then the map (𝔉β′_{E_{i}})(P) in kN-\underline{Mod} is given by the composition

e_{N,P} ⊗ f_{s}↓_{N} ⊗ L(E_{i})↑^{G}↓_{N} → e_{N,P} ⊗ Ω^{−1}k ⊗ e_{N_{i},E_{i}}^{↑G}↓ ⊗ M↓ ⟶^{1⊗1⊗εη_{i}↑^{G}⊗1} e_{N,P} ⊗ Ω^{−1}k ⊗ M↓_{N}.

After a permutation of the tensor factors, the first map in this composition is just the composition around the diagram

[figure: f_{s}↓_{N} ⊗ e_{N,P} ⊗ L(E_{i})↑^{G}↓_{N}; ≅; β_{s}⊗1⊗1; f_{s}↓_{N} ⊗ e_{N_{i},E_{i}}^{↑G}↓_{N} ⊗ L(P); β_{s}⊗1⊗1; Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ e_{N,P} ⊗ L(E_{i})↑^{G}↓_{N}; ≅; (1⊗1⊗1⊗ξ(E_{i})↑↓)^{−1}; Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ e_{N_{i},E_{i}}^{↑G}↓_{N} ⊗ L(P); (1⊗1⊗1⊗ξ(P))^{−1}; Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ e_{N,P} ⊗ (e_{N_{i},E_{i}} ⊗ M↓_{N_{i}})↑^{G}↓_{N}; ≅; 1⊗η⊗1⊗1; ≅; Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ e_{N_{i},E_{i}}^{↑G}↓_{N} ⊗ e_{N,P} ⊗ M↓_{N}; ≅; 1⊗η⊗1⊗1⊗1; Ω^{−1}k ⊗ e_{N,P} ⊗ (e_{N_{i},E_{i}} ⊗ M↓_{N_{i}})↑^{G}↓_{N}; ≅; Ω^{−1}k ⊗ e_{N_{i},E_{i}}^{↑G}↓_{N} ⊗ e_{N,P} ⊗ M↓_{N},]

LOCAL SUBGROUPS AND THE STABLE CATEGORY 19

in which the middle square commutes by Proposition 4.2. It follows, therefore, that the triangle in which the third map is

\xymatrix{e_{N,P} ⊗ f_{s}↓_{N} ⊗ M_{s+1}↓_{N} \ar[r]^{1⊗β} & e_{N,P} ⊗ Ω^{−1}k ⊗ M↓_{N} ≅ Ω^{−1}k ⊗ e_{N,P} ⊗ M↓_{N}}

is isomorphic to the triangle in which the third map is given by the composition

f_{s}↓_{N} ⊗ (⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G}↓_{N}) ⊗ L(P) ⟶^{β_{s}⊗1⊗1} Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ (⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G}↓_{N}) ⊗ L(P)

\underrightarrow{(1⊗1⊗1⊗ξ(P))^{−1}} Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ (⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G}↓_{N}) ⊗ e_{N,P} ⊗ M↓_{N}

\underrightarrow{1⊗1⊗(⨁ εη_{i}↑↓)⊗1⊗1} Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ e_{N,P} ⊗ M↓_{N}.

Let us call this composition β̃. For 1 ≤ i ≤ n let α_{i} : e_{N_{i},E_{i}}^{↑G} → e_{s+1} be the map defined in Proposition 2.7, and let α : ⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G} → e_{s+1} be the direct sum of the α_{i}. Then ηα = ⨁_{i=1}^{n} εη_{i}↑^{G}, and one can check that there is a commutative diagram

\xymatrix{f_{s}↓_{N} ⊗ (⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G}↓_{N}) ⊗ L(P) \ar[d]_{1⊗α⊗1} \ar[r]^{β̃} & Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ e_{N,P} ⊗ M↓_{N} \ar[d]^{1⊗1⊗ξ(P)} \\ f_{s}↓_{N} ⊗ e_{s+1}↓_{N} ⊗ L(P) \ar[r]_{β_{s}⊗η⊗1} & Ω^{−1}k ⊗ e_{s}↓_{N} ⊗ L(P).}

Now 1 ⊗ α : f_{s} ⊗ (⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G}) → f_{s} ⊗ e_{s+1} is a stable isomorphism by Proposition 2.7, and 1 ⊗ ξ(P) : e_{s}↓_{N} ⊗ e_{N,P} ⊗ M↓_{N} → e_{s}↓_{N} ⊗ L(P) is also an isomorphism. It follows that there are isomorphisms of triangles

[figure: e_{N,P} ⊗M↓_{N}; ≅; (η⊗1⊗1)^{−1}; 1⊗μ; e_{N,P} ⊗M′↓_{N}; f_{s}↓_{N} ⊗e_{N,P} ⊗M_{s+1}↓_{N}; ≅; γ; Ω^{−1}k⊗e_{N,P} ⊗M↓_{N}; (1⊗η⊗1⊗1)^{−1}; e_{s}↓⊗e_{N,P} ⊗M↓; 1⊗ξ(P); η⊗1⊗μ; e_{N,P} ⊗M′↓_{N}; ζ; f_{s}↓⊗(⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G}↓)⊗L(P); β̃; 1⊗α⊗1; Ω^{−1}k⊗e_{s}↓⊗e_{N,P} ⊗M↓; 1⊗1⊗ξ(P); e_{s}↓_{N} ⊗L(P); η⊗1; L(P); f_{s}↓_{N} ⊗e_{s+1}↓_{N} ⊗L(P); β_{s}⊗η⊗1; Ω^{−1}k⊗e_{s}↓_{N} ⊗L(P)]

for some ζ : e_{N,P} ⊗ M′↓_{N} → L(P).

The first and last triangles in this commutative diagram correspond to G-local homomorphisms

\xymatrix{𝔉M \ar[r]^{𝔉μ} & 𝔉M′ \ar[r] & f_{s} ⊗ 𝔉M_{s+1} \ar[r] & Ω^{−1}k ⊗ 𝔉M}

and

\xymatrix{e_{s} ⊗ L \ar[r]^{η⊗1} & L \ar[r] & f_{s} ⊗ e_{s+1} ⊗ L \ar[r] & Ω^{−1}k ⊗ e_{s} ⊗ L.}

20 WAYNE W. WHEELER

Because there are G-local homomorphisms

𝔉M ⟶^{(η⊗1)^{−1}} e_{s} ⊗ 𝔉M ⟶^{1⊗ξ} e_{s} ⊗ L

and

\xymatrix{f_{s} ⊗ 𝔉M_{s+1} ≅ f_{s} ⊗ ⨁_{i=1}^{n} 𝔉(L(E_{i})↑^{G}) ≅ f_{s} ⊗ (⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G}) ⊗ L \ar[r]^{1⊗α⊗1} & f_{s} ⊗ e_{s+1} ⊗ L,}

Proposition 5.3 shows that the compositions

e_{N,P} ⊗ M↓_{N} ⟶^{(η⊗1⊗1)^{−1}} e_{s}↓_{N} ⊗ e_{N,P} ⊗ M↓_{N} ⟶^{1⊗ξ(P)} e_{s}↓_{N} ⊗ L(P)

and

\xymatrix{f_{s}↓_{N} ⊗ e_{N,P} ⊗ M_{s+1}↓_{N} \ar[r]^{γ}_{≅} & f_{s}↓ ⊗ (⨁_{i=1}^{n} e_{N_{i},E_{i}}^{↑G}↓) ⊗ L(P) \ar[r]^{1⊗α⊗1} & f_{s}↓_{N} ⊗ e_{s+1}↓_{N} ⊗ L(P)}

are both G-stable. Proposition 5.4 and Lemma 5.7 imply that there is a commutative diagram of triangles

[figure: e_{N,P} ⊗M↓_{N}; 1⊗μ; (1⊗ξ(P))∘(η⊗1⊗1)^{−1}; e_{N,P} ⊗M′↓_{N}; (T ζ)(P); f_{s}↓_{N} ⊗e_{N,P} ⊗M_{s+1}↓_{N}; (1⊗α⊗1)γ; Ω^{−1}k⊗e_{N,P} ⊗M↓_{N}; e_{s}↓_{N} ⊗L(P); η⊗1; L(P); f_{s}↓_{N} ⊗e_{s+1}↓_{N} ⊗L(P); Ω^{−1}k⊗e_{s}↓_{N} ⊗L(P).]

Hence (T ζ)(P) : e_{N,P} ⊗ M′↓_{N} → L(P) is an isomorphism. Corollary 5.6 implies that T ζ : 𝔉M′ → L is an isomorphism in 𝔏_{s+1}(G, k), and this completes the proof.

We can now prove the main result of the paper.

Theorem 5.9. The functor 𝔉 : kG-\underline{Mod} → 𝔏(G, k) is an equivalence of categories.

Proof. The functor 𝔉_{0} : kG-\underline{Mod}(0) → 𝔏_{0}(G, k) is trivially an equivalence because all objects of both kG-\underline{Mod}(0) and 𝔏_{0}(G, k) are isomorphic to zero. For 1 ≤ s ≤ r Lemma 5.8 implies by induction that 𝔉_{s} is essentially surjective. In particular, 𝔉 = 𝔉_{r} is essentially surjective.

Now suppose that γ : M → M′ is a kG-homomorphism such that 𝔉γ = 0. Let P be a Sylow psubgroup of G, and set N = N_{G}(P). Identifying e_{N,P} with k_{N}, we see that 0 = (𝔉γ)(P) = γ↓_{N} and hence γ = 0. Thus 𝔉 is faithful. Since 𝔉 is full by Corollary 5.5, it follows that 𝔉 is an equivalence, as desired.

Assume that P is a Sylow p-subgroup of G, and set N = N_{G}(P). Let 𝔩(G, k) be the full subcategory of 𝔏(G, k) consisting of the objects L such that L(P) is stably isomorphic to a finitely generated kN-module. If M is a finitely generated kG-module, then 𝔉M is an object of 𝔩(G, k), so the restriction of 𝔉 defines a functor 𝔣 : kG-\underline{mod} → 𝔩(G, k).

Corollary 5.10. The functor 𝔣 : kG-\underline{mod} → 𝔩(G, k) is an equivalence of categories.

LOCAL SUBGROUPS AND THE STABLE CATEGORY 21

Proof. Theorem 5.9 implies that 𝔣 is full and faithful, so it is only necessary to show that 𝔣 is essentially surjective. Let L be an object of 𝔩(G, k). Then there is a kG-module M with 𝔉M ≅ L in 𝔏(G, k), and we may assume without loss of generality that M has no projective summands. The definition of 𝔩(G, k) shows that M↓_{N} ≅ M_{0} ⊕ Q in kN-Mod for some finitely generated module M_{0} and some projective module Q. But M is a summand of M↓_{N}↑^{G} ≅ M_{0}↑^{G} ⊕ Q↑^{G}, and M has no projective summands. Thus M is a summand of M_{0}↑^{G} so that M is finitely generated. It follows that 𝔣M ≅ L in 𝔩(G, k), as desired.

We now prove the result on restrictions of G-local modules stated in Section 4.

Corollary 5.11. Let L be an object of 𝔏(G, k), and let P ∈ P(G). Then there is an isomorphism 𝔉(L(P)) ≅ e_{N,P} ⊗ L↓_{N} in 𝔏(N, k), and the isomorphism is natural in L.

Proof. Let M be a kG-module such that there is an isomorphism ξ : L → 𝔉M in 𝔏(G, k). Then we obtain isomorphisms

\xymatrix{𝔉(L(P)) \ar[r]^{𝔉(ξ(P))} & 𝔉(e_{N,P} ⊗ M↓_{N}) ≅ e_{N,P} ⊗ 𝔉(M↓_{N}) ≅ e_{N,P} ⊗ (𝔉M)↓_{N} ≅ e_{N,P} ⊗ L↓_{N}}

in 𝔏(N, k). It is easy to check that the resulting isomorphism 𝔉(L(P)) ≅ e_{N,P} ⊗ L↓_{N} is natural.

If B is a block of kG, then it seems reasonable to expect that there is a category that is equivalent to the stable category B-\underline{Mod} and has a definition similar to that of 𝔏(G, k). Unfortunately, the definition of 𝔏(G, k) does not seem to be compatible with the block structure of kG. It is not clear, therefore, whether an appropriate construction can be obtained simply by modifying the techniques used here; significantly different ideas may be necessary.

References

[1] D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Cambridge Philos. Soc. 120 (1996), 597–615.

[2] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, Cambridge Univ. Press, Cambridge, 1988.

[3] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. (2) 56 (1997), 149–170.

[4] W. W. Wheeler, Quillen stratification for the stable module category, Quart. J. Math. Oxford (2), 50 (1999), 355–369. Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121, USA

   E-mail address: wheeler@member.ams.org