VARIETIES AND LOCALIZATIONS OF

MODULE CATEGORIES

Jon F. Carlson^{∗}  Wayne W. Wheeler
Department of Mathematics School of Mathematical Sciences
University of Georgia  Australian National University
Athens, Georgia 30602  Canberra ACT 0200
USA  Australia
(email: jfc@sloth.math.uga.edu) (email: wheeler@pell.anu.edu.au)

1. Introduction

In [3] the authors in collaboration with Peter Donovan employed the quotient category construction as a technique for studying the modular representation theory of finite groups. Quotient categories can be used to filter the category of all modules over group algebras, and they provide a whole new viewpoint on the module theory in general. In addition to giving some fresh insights into the cohomology of finite groups, the study has also pointed the way to some new results on the induction functor from certain subgroups containing centralizers of elementary abelian p-subgroups [2].

Let G be a finite group, and suppose that k is an algebraically closed field of characteristic p > 0. The primary emphasis of [3] was on the stable category \overline{mod}-kG of all kG-modules modulo projectives, filtered by subcategories determined by complexity. The complexity of a module M is the polynomial rate of growth of a minimal projective resolution of M. It is also the dimension of the support variety V_{G}(M) in the maximal ideal spectrum V_{G}(k) of H^{∗}(G, k). Hence it is never larger than the dimension of V_{G}(k), which is equal to the p-rank of G. Let M_{c} be the full subcategory of \overline{mod}-kG consisting of all modules of complexity at most c, and let Q_{c} = M_{c}/M_{c−1}. Then Q_{c} has the same objects as M_{c}, but the morphisms are obtained by a process that turns out to be a homogeneous localization of the cohomology (See Theorem 3.5 of [3]). The connection with the cohomology allows for the decompositions of objects and morphisms according to the components of the varieties of the modules. Specifically, if r = dim V_{G}(k), then End_{Q_{r}}(k) is a direct sum of local rings that are in one-to-one correspondence with the r-dimensional components of V_{G}(k). A similar decomposition holds for the endomorphism ring End_{Q_{c}}(M) of any module M in Q_{c}.

A main purpose of this paper is to deal with the case of a single component in V_{G}(k). That is, we fix a component V and let M be the full subcategory of \overline{mod}-kG consisting of modules whose varieties do not contain V. In the quotient category Q = \overline{mod}-kG/M, the endomorphism ring End_{Q}(k) of the trivial module is a commutative local ring obtained

Typeset by AMS-TEX

∗Partially supported by a grant from NSF.

by homogeneous localization of H^{∗}(G, k). Some of the basic properties of the ring are set out in Section 4. We show, for example, that it contains a subfield over which the set of morphisms between any two modules in Q is a finite-dimensional vector space. In Section 5 we look at equivalences of quotient categories. In particular, the component V corresponds to the conjugacy class of a maximal elementary abelian p-subgroup E, and under proper circumstances there is an equivalence between the quotient category for G and that for the normalizer of E.

A second purpose of this paper is to lay the groundwork for some future investigations. A major open question concerns the structure of endomorphism rings of modules in the complexity quotient categories M_{c}/M_{d} when the difference c − d is greater than one. The decompositions of objects according to the structure of the varieties can still be obtained, and it should be the case that the sets of morphisms between modules are finitely generated over End_{M_{r}/M_{d}}(k). But in spite of some recent progress in [5], the structure of End_{M_{r}/M_{d}}(k) is still a mystery.

As motivation for the studying the quotient categories arising from a single component of V_{G}(k), we present the following.

Theorem 1.1. Let M be a kG-module of complexity c > 0. Suppose that V_{G}(M) = V_{1} ∪ ⋯ ∪ V_{t} ∪ W where V_{1}, …, V_{t} are the components of dimension c and W is the union of the components of smaller dimension. For 1 ≤ i ≤ t let N_{i} be the full subcategory of M_{c} consisting of all modules L such that V_{G}(L) ∩ V_{i} < V_{i}. Then

Hom_{M_{c}/M_{c−1}}(M, M) ≅ ∐_{i=1}^{t} Hom_{M_{c}/N_{i}}(M, M).

In other words, the endomorphisms in M_{c}/M_{c−1} are determined by the one-component quotients. Furthermore, if M and N are objects in M_{c} and if their varieties do not share an irreducible component of dimension c, then Hom_{M_{c}/M_{c−1}}(M, N) = 0 (See Lemma 3.7 of [3]). Theorem 1.1 is proven in Section 3, along with some other results on decompositions.

Although it is known that any object in one of the quotient categories is a finite direct sum of indecomposable objects, the decompositions of the endomorphism rings do not translate directly into decompositions of modules because there is no Krull-Schmidt Theorem. Instead, adjustments must be made in order to get the direct sum splittings that are compatible with the structure of the varieties. In [2] the adjustment was to take a multiple of the module. In Section 3 we use a different technique that also works with some success in the quotient categories M_{c}/M_{d} (See [5]). To this point most of the applications of the quotient construction have involved the splittings, as in [4].

2. Notation and Preliminaries

In this section we fix the notation that will be used throughout the rest of the paper, and we recall a number of results from [3] that will be useful.

Let G be a finite group, and let p be a prime divisor of |G|. Fix an algebraically closed field k of characteristic p. If H is any subgroup of G, then the trivial kH-module will also be denoted by k, or sometimes by k_{H} if we wish to emphasize the subgroup. If M is a kG-module, then M_{↓H} denotes the restriction of M to H; if N is a kH-module, then N^{↑G} denotes the induced module. When we wish to think of restriction and induction as functors, we write them as Res_{H}^{G} and Ind_{H}^{G}. The restriction map between the cohomology rings will be denoted by res_{G,H} : H^{∗}(G, k) → H^{∗}(H, k).

The cohomology ring H^{∗}(G, k) is a finitely generated, graded k-algebra, and if ζ, η ∈ H^{∗}(G, k) are homogeneous, then ζη = (−1)^{(deg ζ)(deg η)}ηζ. In particular, H^{∗}(G, k) is commutative if p = 2. If p > 2, then all elements of odd degree are nilpotent, so they lie in the radical Rad H^{∗}(G, k). Thus H^{∗}(G, k)/Rad H^{∗}(G, k) is always a commutative ring, and the maximal ideal spectrum V_{G}(k) of H^{∗}(G, k) is a homogeneous affine variety. If ζ is any element of H^{∗}(G, k), then V_{G}(ζ) denotes the subvariety consisting of all maximal ideals containing ζ. If M is a kG-module, then Ext_{kG}^{∗}(M, M) ≅ H^{∗}(G, Hom_{k}(M, M)) is a finitely generated module over H^{∗}(G, k). Let J(M) denote the annihilator of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k), and let V_{G}(M) ⊆ V_{G}(k) denote the subvariety consisting of all maximal ideals containing J(M). The complexity of the module M is the dimension of V_{G}(M). If H is a subgroup of G, then the restriction map res_{G,H} : H^{∗}(G, k) → H^{∗}(H, k) induces a map res_{G,H}^{∗} : V_{H}(k) → V_{G}(k). For more details on varieties and cohomology rings see [1] or [6].

Let mod-kG denote the category of all finitely generated right kG-modules. If M and N are kG-modules, let PHom_{kG}(M, N) ⊆ Hom_{kG}(M, N) denote the subspace of all kGhomomorphisms that factor through a projective module. The stable category \overline{mod}-kG has the same objects as mod-kG, but morphisms are defined by setting \overline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N). The stable category is triangulated by Theorem I.2.6 of [8], and the translation functor is given by Ω^{−1}.

If ζ ∈ H^{n}(G, k), then we write ζ̂: Ω^{n}(k) → k for the morphism in \overline{mod}-kG representing ζ. Since there are no projective homomorphisms from Ω^{n}(k) to k, we can also regard ζ̂ as a homomorphism in mod-kG. If M is a kG-module, then we will also write ζ_{M} for ζ ∪ Id_{M} ∈ Ext_{kG}^{n}(M, M), and we let ζ̂_{M} : Ω^{n}(M) → M be the morphism in \overline{mod}-kG representing ζ_{M}. If ζ = 0, set L_{ζ} = Ω^{n}(k) ⊕ Ω(k); if ζ ≠ 0, set L_{ζ} = Ker ζ̂. Then L_{ζ} can also be described as the module with no projective summands such that \overline{mod}-kG has a triangle of the form

L_{ζ} ⟶ Ω^{n}(k) ⟶^{ζ̂} k ⟶ Ω^{−1}(L_{ζ}).

Let E be a maximal elementary abelian p-subgroup of G. Then V_{E} = res_{G,E}^{∗}(V_{E}(k)) is a component of the variety V_{G}(k). Indeed, the Dimension Theorem of Quillen says that V_{G}(k) = ∪V_{E}, where the union is over a complete set of representatives of the conjugacy classes of maximal elementary abelian p-subgroups of G. Moreover, if M is any kG-module with V_{E} ⊆ V_{G}(M), then V_{E}(k) = V_{E}(M_{↓E}). If E ⊆ H ⊆ G, let N (H, E) denote the full subcategory of \overline{mod}-kH consisting of all modules M such that res_{H,E}^{∗}(V_{E}(k)) ⊈ V_{H}(M), and let Q(H, E) denote the quotient \overline{mod}-kH/N (H, E). Explicitly, the objects in Q(H, E) are the same as those of \overline{mod} − kH. A morphism from M to M′ is represented by a triple (s, N, f), where s : N → M and f : N → M′ are morphisms in \overline{mod}-kH, and \overline{mod}-kH has a triangle of the form

N ⟶^{s} M ⟶ Z ⟶ Ω^{−1}(N)

with Z in N (H, E). A morphism s : N → M embeds in such a triangle if and only if the triple (1, N, s) : N → M represents an isomorphism in Q(H, E). In this case we will simply say that s is a Q(H, E)-isomorphism, even though, strictly speaking, s is not even a morphism in Q(H, E). Instead of writing the morphism (s, N, f) as a triple, we will sometimes express it as a diagram

M ⟵^{s} N ⟶^{f} M′.

Another triple (s′, N′, f′) represents the same morphism from M to M′ if and only if there exist an object N_{1} and maps t : N_{1} → N and t′ : N_{1} → N′ in \overline{mod}-kH such that t and t′ are Q(H, E)-isomorphisms with st = s′t′ and ft = f′t′. It is often useful to think of the morphism (s, N, f) : M → M′ as f ∘ s^{−1}, even though s may not be invertible in \overline{mod}-kH. For a more detailed description of localization and quotient categories the reader is referred to [7] or [9].

Now let c be a nonnegative integer that is at most the p-rank of G, and let M_{c} be the full subcategory of \overline{mod}-kG consisting of modules having complexity at most c. For c ≥ 1, set Q_{c} = M_{c}/M_{c−1}. The categories Q_{c} admit a description that is entirely analogous to that given above for Q(G, E). Because N (G, E) and M_{c−1} are thick subcategories of \overline{mod}-kG, the quotient categories Q(G, E) and Q_{c} inherit a triangulated structure from \overline{mod}-kG. The categories Q_{c} were studied in [3], and further results are given in the next section of this paper. We begin by describing the results about Q_{c} from [3] that we will need here, along with some analogous results for the quotients Q(G, E).

Let M_{1} and M_{2} be objects of Q_{c}, and set

S_{1} = {ζ ∪ Id_{M_{1}} | ζ ∈ H^{∗}(G, k) is homogeneous and dim (V_{G}(ζ) ∩ V_{G}(M_{1})) < c}.

If ζ_{M_{1}} = ζ ∪ Id_{M_{1}} ∈ S_{1} has degree n, then ζ̂_{M_{1}} : Ω^{n}(M_{1}) → M_{1} is a Q_{c}-isomorphism. Moreover, the set S_{1} lies in the center of the graded ring Êxt_{kG}^{∗}(M_{1}, M_{1}), and E^{∗} = Êxt_{kG}^{∗}(M_{1}, M_{2}) is a graded module for Êxt_{kG}^{∗}(M_{1}, M_{1}), so we can consider the localization E^{∗}S_{1}^{−1}. If M_{3} is another object of Q_{c}, then there is a well-defined composition

∘ : Êxt_{kG}^{∗}(M_{2}, M_{3})S_{2}^{−1} ⊗ Êxt_{kG}^{∗}(M_{1}, M_{2})S_{1}^{−1} → Êxt_{kG}^{∗}(M_{1}, M_{3})S_{1}^{−1}.

Suppose that f/ζ_{M_{1}} lies in (E^{∗}S_{1}^{−1})^{0}, the elements of degree zero in E^{∗}S_{1}^{−1}. Then deg f = deg ζ_{M_{1}}, and f can be regarded as a map Ω^{n}(M_{1}) → M_{2} for some n.

Theorem 2.1. The map Φ : (E^{∗}S_{1}^{−1})^{0} → Hom_{Q_{c}}(M_{1}, M_{2}) defined by setting Φ(f/ζ_{M_{1}}) = (ζ̂_{M_{1}}, Ω^{n}(M_{1}), f) is a vector space isomorphism that commutes with compositions.

Proof. See Theorem 3.5 of [3]. □

A similar result holds for the category Q(G, E). Indeed, suppose that M_{1} and M_{2} are objects of Q(G, E), and set

S_{1} = {ζ_{M_{1}}|ζ ∈ H^{∗}(G, k) is homogeneous and V_{E} ⊈ V_{G}(ζ) ∩ V_{G}(M_{1})}.

If E^{∗} = Êxt_{kG}^{∗}(M_{1}, M_{2}), then again an element of (E^{∗}S_{1}^{−1})^{0} can be regarded as a pair of maps f : Ω^{n}(M_{1}) → M_{2} and ζ̂_{M_{1}} : Ω^{n}(M_{1}) → M_{1}. The same techniques used to prove Theorem 2.1 give the following result.

Theorem 2.2. The map Ψ : (E^{∗}S_{1}^{−1})^{0} → Hom_{Q(G,E)}(M_{1}, M_{2}) defined by setting Ψ(f/ζ_{M_{1}}) = (ζ̂_{M_{1}}, Ω^{n}(M_{1}), f) is a vector space isomorphism that commutes with compositions.

Using the isomorphisms of Theorems 2.1 and 2.2, we will sometimes identify Hom_{Q_{c}}(M_{1}, M_{2}) or Hom_{Q(G,E)}(M_{1}, M_{2}) with the elements of degree zero in the appropriate localization of Êxt_{kG}^{∗}(M_{1}, M_{2}).

Let ζ_{1}, …, ζ_{r} be homogeneous elements of H^{∗}(G, k) such that k[ζ_{1}, …, ζ_{r}] is a polynomial subring of H^{∗}(G, k). If H^{∗}(G, k) is finitely generated as a module over k[ζ_{1}, …, ζ_{r}], then ζ_{1}, …, ζ_{r} is called a homogeneous set of parameters for H^{∗}(G, k). This is equivalent to the condition that dim V_{G}(k) = r and V_{G}(ζ_{1}) ∩ ⋯ ∩ V_{G}(ζ_{r}) = {0}. The cohomology ring H^{∗}(G, k) always has a homogeneous set of parameters by the Noether Normalization Lemma.

The next proposition is the analogue for Q(G, E) of Proposition 3.6 of [3]. Although the proof given in [3] also works here with straightforward modifications, the proof given below is somewhat simpler.

Proposition 2.3. For some integer n > 0 there is an element ζ ∈ H^{n}(G, k) such that ζ̂_{M} : Ω^{n}(M) → M is a Q(G, E)-isomorphism for every object M.

Proof. Let ζ_{1}, …, ζ_{r} be a homogeneous set of parameters for H^{∗}(G, k). Since V_{G}(ζ_{1}) ∩ ⋯ ∩ V_{G}(ζ_{r}) = {0}, there is an i such that V_{E} ⊈ V_{G}(ζ_{i}). Set ζ = ζ_{i} and n = deg ζ_{i}. Then for any object M of Q(G, E) it follows that V_{E} ⊈ V_{G}(ζ) ∩ V_{G}(M), and ζ̂_{M} is a Q(G, E)-isomorphism, as desired. □

If t is the smallest positive integer such that Ω^{t}(M) ≅ M in Q(G, E) for every object M, then we call t the period of the category Q(G, E). This periodicity has one consequence that will be important to us later. Let M_{1} and M_{2} be objects of Q(G, E), and consider the situation in Theorem 2.2. Since the degree of every element of S_{1} is a multiple of t, we see that every element of (E^{∗}S_{1}^{−1})^{0} is a quotient f/ζ_{M_{1}}, and deg f and deg ζ_{M_{1}} are both divisible by t. Thus instead of considering the elements of degree zero in the localization of Êxt_{kG}^{∗}(M_{1}, M_{2}), we can consider the elements of degree zero in the localization of ⨁_{n=0}^{∞} Êxt_{kG}^{nt}(M_{1}, M_{2}) regarded as a module for ⨁_{n=0}^{∞} Êxt_{kG}^{nt}(M_{1}, M_{1}). The result is still isomorphic to Hom_{Q(G,E)}(M_{1}, M_{2}).

Finally, we end this section with an observation on direct sum decompositions in Q(G, E). Note that the following proposition does not provide a Krull–Schmidt Theorem in Q(G, E) because it includes no uniqueness assertion.

Proposition 2.4. Any object in Q(G, E) is a finite direct sum of indecomposable objects.

Sketch of proof. This result was proven for Q_{c} in Corollary 2.3 of [3] by constructing a function η_{G} from objects of Q_{c} to nonnegative integers. The function η_{G} is constant on isomorphism classes and additive on direct sums, and η_{G}(M) = 0 if and only if M ≅ 0 in Q_{c}. Although η_{G} is not constant on isomorphism classes of Q(G, E), we get such a function η′_{G} by setting η′_{G}(M) = η_{E}(M_{↓E}). The argument of Corollary 2.3 of [3] then gives the desired result. □

3. Direct Sum Decompositions

The main purpose of this section is to prove a result on direct sum decompositions in the category Q_{c}. For any object M the decomposition obtained will depend upon the cdimensional components of V_{G}(M). We begin by proving Theorem 1.1, which shows that endomorphism rings in Q_{c} have such a decomposition.

Proof of Theorem 1.1. We are given a kG-module M ∈ M_{c} such that V_{G}(M) = V_{1} ∪ ⋯ ∪ V_{t} ∪ W, where V_{1}, …, V_{t} are the components of dimension c and W is the union of all components of smaller dimension. For 1 ≤ i ≤ t set

S_{i} = {ζ_{M} | ζ ∈ H^{∗}(G, k) is homogeneous and V_{i} ⊈ V_{G}(ζ)}.

As in Theorems 2.1 and 2.2, it follows that

Hom_{M_{c}/N_{i}}(M, M) ≅ (Êxt_{kG}^{∗}(M, M) · S_{i}^{−1})^{0}.

The proof now follows the steps of Section 4 of [3].

Choose homogeneous elements ζ^{(1)}, …, ζ^{(t)} such that W ⊆ V_{G}(ζ^{(i)}) for all i and V_{i} ⊆ V_{G}(ζ^{(j)}) if and only if i ≠ j. Then for i ≠ j it follows that V_{G}(M) ⊆ V_{G}(ζ^{(i)}) ∪ V_{G}(ζ^{(j)}) = V_{G}(ζ^{(i)}ζ^{(j)}), so some power of ζ^{(i)}ζ^{(j)} annihilates Ext_{kG}^{∗}(M, M). Thus if we replace each ζ^{(i)} by a suitable power, we may assume that the following conditions are satisfied:

(a) ζ_{M}^{(i)}ζ_{M}^{(j)} = (ζ^{(i)}ζ^{(j)})_{M} = 0 for i ≠ j;

(b) ζ_{M}^{(i)} ∈ S_{j} if and only if i = j; and

(c) deg ζ^{(1)} = ⋯ = deg ζ^{(t)}.

Let ζ = ζ^{(1)}+⋯+ζ^{(t)}. For each i there is a maximal ideal m_{i} ∈ V_{i} such that ζ^{(j)} ∈ m_{i} if and only if j ≠ i. Hence ζ ∉ m_{i}, and V_{G}(ζ)∩V_{i} ≠ V_{i} for all i so that dim(V_{G}(ζ)∩V_{G}(M)) < c. If n = deg ζ, then it follows that the map ζ̂⊗1_{M} : Ω^{n}(k)⊗M → M is a Q_{c}-isomorphism representing ζ_{M}. Thus e_{i} = ζ_{M}^{(i)}/ζ_{M} defines an element of Hom_{Q_{c}}(M, M) for 1 ≤ i ≤ t, and 1 = e_{1} + ⋯ + e_{t}. Moreover, the elements e_{1}, …, e_{t} are orthogonal idempotents of the center of Hom_{Q_{c}}(M, M) so that

Hom_{Q_{c}}(M, M) = ∐_{i=1}^{t} Hom_{Q_{c}}(M, M) · e_{i}.

It remains only to show that Hom_{Q_{c}}(M, M)e_{i} ≅ Hom_{M_{c}/N_{i}}(M, M). The multiplicative set

S = {η_{M} | η ∈ H^{∗}(G, k) is homogeneous and dim(V_{G}(η) ∩ V_{G}(M)) < c}

is a subset of S_{i} for 1 ≤ i ≤ t, so there is a natural ring homomorphism

Ψ_{i} : Hom_{Q_{c}}(M, M) ≅ (Êxt_{kG}^{∗}(M, M)S^{−1})^{0} → (Êxt_{kG}^{∗}(M, M)S_{i}^{−1})^{0} ≅ Hom_{M_{c}/N_{i}}(M, M) such that Ψ_{i}(e_{i}) is an isomorphism. If j ≠ i, then 0 = Ψ_{i}(e_{i}e_{j}) = Ψ_{i}(e_{i})Ψ_{i}(e_{j}), so it follows that Ψ_{i}(e_{j}) = 0.

Next we show that Ψ_{i} gives an isomorphism of Hom_{Q_{c}}(M, M)e_{i} onto Hom_{M_{c}/N_{i}}(M, M). Without loss of generality it suffices to consider the case i = 1. Let β ∈ H^{∗}(G, k) be a homogeneous element such that V_{1} ⊈ V_{G}(β), and let α/β_{M} ∈ Hom_{M_{c}/N_{1}}(M, M). Let s and u be positive integers such that deg(β^{s}) = s deg β = un, and set η = β^{s}ζ^{(1)} + (ζ^{(2)})^{u+1} + ⋯ + (ζ^{(t)})^{u+1}. Then it is easy to check that η_{M} ∈ S, and

\frac{α}{β_{M}} = \frac{αβ_{M}^{s−1}(ζ_{M}^{(1)})^{2}}{β_{M}^{s}(ζ_{M}^{(1)})^{2}} = \frac{αβ_{M}^{s−1}(ζ_{M}^{(1)})^{2}}{η_{M}ζ_{M}^{(1)}} = \frac{αβ_{M}^{s−1}ζ_{M}^{(1)}}{η_{M}}

in Hom_{M_{c}/N_{1}}(M, M). It follows that Ψ_{1} maps αβ_{M}^{s−1}ζ_{M}^{(1)}/η_{M} to α/β_{M}, so Ψ_{1} is surjective.

Finally, we show that Ψ_{1} is a monomorphism on Hom_{Q_{c}}(M, M)e_{1}. Suppose that α/β_{M} ∈ Hom_{Q_{c}}(M, M)e_{1} is in the kernel of Ψ_{1}. Then α/β_{M} = 0 in Hom_{M_{c}/N_{1}}(M, M), so there is an element γ_{M} ∈ S_{1} such that αγ_{M} = 0 in \overline{mod}-kG. Let s and u be positive integers such that deg(γ^{s}) = s deg γ = un, and set η = γ^{s}ζ^{(1)} + (ζ^{(2)})^{u+1} + ⋯ + (ζ^{(t)})^{u+1}. Then η_{M} ∈ S, and hence in Hom_{Q_{c}}(M, M)e_{1} we have

\frac{α}{β_{M}} = \frac{αη_{M}ζ_{M}^{(1)}}{β_{M}η_{M}ζ_{M}} = \frac{αγ_{M}^{s}(ζ_{M}^{(1)})^{2}}{β_{M}η_{M}ζ_{M}} = 0.

This completes the proof. □

Thus we have a decomposition of the endomorphism rings corresponding to the components of the variety of the module. In a standard module category we would expect a decomposition of the module itself, but that may not happen in Q_{c} because there is no Krull-Schmidt Theorem. In particular, M may be indecomposable in Q_{c} even though its endomorphism ring decomposes. On the other hand, as in [2], a multiple of M may decompose into a sum of modules each of which has a variety with a single c-dimensional component. In what follows we offer a different sort of decomposition. First we need a lemma.

Lemma 3.1. Let ζ ∈ H^{m}(G, k) and γ ∈ H^{n}(G, k). Then there exist a projective module P and a short exact sequence of the form

0 ⟶ L_{ζ} ⊗ L_{γ} ⟶ Ω^{n}(L_{ζ}) ⊕ Ω^{m}(L_{γ}) ⊕ P ⟶ L_{ζγ} ⟶ 0.

Remark. It is possible that the projective modules arising in the statement of Lemma 3.1 and in the proof below might be zero.

Proof. First suppose that γ = 0. Then L_{γ} = Ω^{n}(k) ⊕ Ω(k) and L_{ζγ} = Ω^{m+n}(k) ⊕ Ω(k). It follows from the definition of L_{ζ} that there is a short exact sequence

0 ⟶ Ω(L_{ζ}) ⟶ Ω^{m+1}(k) ⊕ P_{1} ⟶ Ω(k) ⟶ 0

for some projective module P_{1}. Thus there is also a short exact sequence

0 ⟶ Ω^{n}(L_{ζ}) ⊕ Ω(L_{ζ}) ⟶ Ω^{n}(L_{ζ}) ⊕ Ω^{m+n}(k) ⊕ Ω^{m+1}(k) ⊕ P_{1}
⟶ Ω^{m+n}(k) ⊕ Ω(k) ⟶ 0.

Since L_{ζ}⊗L_{γ} ≅ Ω^{n}(L_{ζ})⊕Ω(L_{ζ})⊕P_{2} for some projective module P_{2} and Ω^{n}(L_{ζ})⊕Ω^{m}(L_{γ}) ≅ Ω^{n}(L_{ζ}) ⊕ Ω^{m+n}(k) ⊕ Ω^{m+1}(k), the desired result follows in this case. Similarly, the result follows if ζ = 0.

Now assume that ζ ≠ 0 and γ ≠ 0. Because L_{ζ} = Ker ζ̂ and L_{γ} = Ker γ̂, we see that there are chain complexes

C^{•}(ζ) : ⋯ ⟶ 0 ⟶ L_{ζ} ⟶ Ω^{m}(k) ⟶ 0 ⟶ ⋯

and

C^{•}(γ) : ⋯ ⟶ 0 ⟶ L_{γ} ⟶ Ω^{n}(k) ⟶ 0 ⟶ ⋯

with C^{0}(ζ) = Ω^{m}(k) and C^{0}(γ) = Ω^{n}(k); moreover, these complexes are exact except in degree zero, and H^{0}(C^{•}(ζ)) = H^{0}(C^{•}(γ)) = k. It follows from the Künneth Theorem that the tensor product C^{•}(ζ) ⊗ C^{•}(γ) is a complex that is also exact everywhere except in degree zero, and H^{0}(C^{•}(ζ) ⊗ C^{•}(γ)) = k. Hence there exist projective modules P_{3} and P_{4} and an exact sequence

0 ⟶ L_{ζ} ⊗ L_{γ} ⟶ Ω^{m}(L_{γ}) ⊕ Ω^{n}(L_{ζ}) ⊕ P_{3}
⟶ Ω^{m+n}(k) ⊕ P_{4} ⟶ k ⟶ 0.

Since the map Ω^{m+n}(k) ⊕ P_{4} → k represents ζγ, its kernel is L_{ζγ} ⊕ P_{5} for some projective module P_{5}, and we get an exact sequence

0 ⟶ L_{ζ} ⊗ L_{γ} ⟶ Ω^{m}(L_{γ}) ⊕ Ω^{n}(L_{ζ}) ⊕ P ⟶ L_{ζγ} ⟶ 0

for some projective module P. This completes the proof. □

Theorem 3.2. Let M be a kG-module of complexity c. Suppose that V_{G}(M) = V_{1} ∪ ⋯ ∪ V_{t} ∪ W, where V_{1}, …, V_{t} are the irreducible components of V_{G}(M) of dimension c and W is the union of all components of smaller dimension. Then there exist kG-modules N_{1}, …, N_{t} ∈ Q_{c} such that V_{i} is the only c-dimensional component of V_{G}(N_{i}) for 1 ≤ i ≤ t and

M ⊕ Ω(M) ≅ N_{1} ⊕ ⋯ ⊕ N_{t}

in Q_{c}.

Proof. For i = 1, …, t choose ζ_{i} ∈ H^{n_{i}}(G, k) such that V_{i} ∪ W ⊆ V_{G}(ζ_{i}) and V_{j} ⊈ V_{G}(ζ_{i}) for j ≠ i. Replacing ζ_{i} by an appropriate power if necessary, we may assume that deg ζ_{i} = deg ζ_{j} for all i and j and that the period of Q_{c} divides this degree. Moreover, we have V_{G}(ζ_{1} ⋯ ζ_{t}) = V_{G}(ζ_{1}) ∪ ⋯ ∪ V_{G}(ζ_{t}) ⊇ V_{G}(M), so there is an integer m > 0 such that (ζ_{1} ⋯ ζ_{t})^{m} annihilates Ext_{kG}^{∗}(M, M). Replacing each ζ_{i} by ζ_{i}^{m}, we may also assume that ζ_{1} ⋯ ζ_{t} annihilates Ext_{kG}^{∗}(M, M). Set L_{i} = L_{ζ_{i}} for i = 1, …, t.

We show by induction on i that L_{ζ_{1}···ζ_{i}} ⊗ M ≅ (L_{1} ⊗ M) ⊕ ⋯ ⊕ (L_{i} ⊗ M) in Q_{c} for 1 ≤ i ≤ t. If i = 1, then the result is trivial, so assume that i > 1. Then

V_{G}(L_{ζ_{1}···ζ_{i−1}} ⊗ L_{ζ_{i}} ⊗ M) = V_{G}(L_{ζ_{1}···ζ_{i−1}}) ∩ V_{G}(L_{ζ_{i}}) ∩ V_{G}(M)
= V_{G}(ζ_{1} ⋯ ζ_{i−1}) ∩ V_{G}(ζ_{i}) ∩ V_{G}(M)

has dimension smaller than c. Let n = deg ζ_{1} = ⋯ = deg ζ_{t}. It follows from Lemma 3.1 that there is a short exact sequence

0 ⟶ L_{ζ_{1}···ζ_{i−1}} ⊗ L_{ζ_{i}} ⊗ M ⟶
Ω^{n}(L_{ζ_{1}···ζ_{i−1}} ⊗ M) ⊕ Ω^{n(i−1)}(L_{ζ_{i}} ⊗ M) ⊕ P ⟶ L_{ζ_{1}···ζ_{i}} ⊗ M ⟶ 0

for some projective module P, so Ω^{n}(L_{ζ_{1}···ζ_{i−1}} ⊗ M) ⊕ Ω^{n(i−1)}(L_{i} ⊗ M) ≅ L_{ζ_{1}···ζ_{i}} ⊗ M in Q_{c}. Since the period of Q_{c} divides n, we conclude by induction that L_{ζ_{1}···ζ_{i}} ⊗ M ≅ (L_{ζ_{1}···ζ_{i−1}} ⊗ M) ⊕ (L_{i} ⊗ M) ≅ (L_{1} ⊗ M) ⊕ ⋯ ⊕ (L_{i} ⊗ M).

Now set i = t. Because ζ_{1} ⋯ ζ_{t} annihilates Ext_{kG}^{∗}(M, M), Proposition 5.9.5 of [1] shows that L_{ζ_{1}···ζ_{t}} ⊗ M ≅ Ω^{nt}(M) ⊕ Ω(M) in \overline{mod}-kG. Thus in Q_{c} it follows that M ⊕ Ω(M) ≅ L_{ζ_{1}···ζ_{t}} ⊗ M ≅ (L_{1} ⊗ M) ⊕ ⋯ ⊕ (L_{t} ⊗ M). Now let N_{i} = L_{i} ⊗ M, and notice that V_{G}(N_{i}) = V_{G}(L_{i}) ∩ V_{G}(M) has the desired form. □

4. The Endomorphism Ring of the Trivial Module

In general, the generic point of the component V_{E} = res_{G,E}^{∗}(V_{E}(k)) of V_{G}(k) is given by the minimal prime ideal 𝔓 = res_{G,E}^{−1}(Rad H^{∗}(E, k)), and one can think of the construction of Q(G, E) as localizing the category \overline{mod}-kG away from 𝔓. The problem, however, is that it is necessary to use a homogeneous localization process. To obtain the endomorphism ring of the trivial module, for example, only the homogeneous elements of H^{∗}(G, k) that are not in 𝔓 are inverted, and we then take the elements of degree zero in this localization. Because of this additional complication, the important algebraic properties of cohomology rings do not always transfer directly to the endomorphism rings in Q(G, E). This section is devoted to studying these endomorphism rings, particularly that of the trivial module.

For simplicity we will write Q for Q(G, E) throughout this section.

Lemma 4.1. End_{Q}(k) is a Noetherian ring.

Proof. Let R = H^{∗}(G, k). Then there is a multiplicative subset S consisting of homogeneous elements such that End_{Q}(k) is isomorphic to (RS^{−1})^{0}, the elements of degree zero in RS^{−1}. Now R is Noetherian and hence so is RS^{−1}. If I_{1} ⊆ I_{2} ⊆ ⋯ is an ascending chain of ideals in (RS^{−1})^{0}, then I_{1}RS^{−1} ⊆ I_{2}RS^{−1} ⊆ ⋯ is an ascending chain of ideals in RS^{−1}, so there is an integer n with I_{n}RS^{−1} = I_{m}RS^{−1} for all m ≥ n. Hence I_{n} = (I_{n}RS^{−1})^{0} = (I_{m}RS^{−1})^{0} = I_{m} for m ≥ n, and we conclude that (RS^{−1})^{0} is Noetherian. □

Lemma 4.2. End_{Q}(k) is an Artinian local ring.

Proof. Suppose that α, ζ ∈ H^{n}(G, k) are elements such that α/ζ ∈ End_{Q}(k) is not invertible. Then V_{E} ⊆ V_{G}(α) and V_{E} ⊈ V_{G}(ζ). Write V_{G}(k) = V_{E} ∪ W, where W is the union of all components of V_{G}(k) other than V_{E}. Choose a homogeneous element β ∈ H^{∗}(G, k) such that W ⊆ V_{G}(β) but V_{E} ⊈ V_{G}(β). Then β̂ is a Q-isomorphism, and V_{G}(αβ) = V_{G}(α) ∪ V_{G}(β) = V_{G}(k), so αβ is nilpotent. In particular, αβ ∈ Rad H^{∗}(G, k). Since Rad H^{∗}(G, k) is nilpotent, there is an integer n with (Rad H^{∗}(G, k))^{n} = 0. Then in End_{Q}(k) we have (α/ζ)^{n} = (αβ/ζβ)^{n} = 0.

We have now shown that every element of End_{Q}(k) is either nilpotent or invertible, so End_{Q}(k) is local. Moreover, if m denotes the maximal ideal, then it is easy to see that m^{n} = 0. Because End_{Q}(k) is Noetherian by Lemma 4.1, we conclude that it is Artinian. □

Proposition 4.3. There is a maximal subfield F of End_{Q}(k) such that k ⊆ F and End_{Q}(k) is integral over F.

Proof. Let m be the maximal ideal of End_{Q}(k), and let S denote the collection of all k-subalgebras S of End_{Q}(k) such that S ∩m = 0. Note that S is nonempty because k ∈ S. Then Zorn’s Lemma implies that S contains a maximal element F.

We begin by showing that F is a field. Let x ∈ F −{0}. Since x ∉ m, it follows that x is invertible in End_{Q}(k). If x^{−1} ∉ F, then the maximality of F implies that F [x^{−1}] ∩ m ≠ 0. Thus there are elements a_{0}, …, a_{n} ∈ F such that y = a_{n}x^{−n} + a_{n−1}x^{−n+1} + ⋯ + a_{0} is a nonzero element of m. But then x^{n}y = a_{n} + a_{n−1}x + ⋯ + a_{0}x^{n} ∈ F ∩ m = 0 and hence y = 0. Thus we get a contradiction, and it follows that F is a field.

Now let u ∈ End_{Q}(k). If F [u] ≠ F, then F [u]∩m ≠ 0. It follows that there are elements b_{0}, …, b_{m−1} ∈ F such that u^{m} + b_{m−1}u^{m−1} + ⋯ + b_{0} ∈ m. Since all elements of m are nilpotent, we see that u is integral over F, and this completes the proof. □

Theorem 4.4. Let F ⊇ k be a maximal subfield of End_{Q}(k) such that End_{Q}(k) is integral over F. Then End_{Q}(k) is finite-dimensional over F.

Proof. Set L(G) = End_{Q}(k)/Rad End_{Q}(k). Then the natural map End_{Q}(k) → L(G) maps F onto a subfield of L(G), and L(G) is algebraic over F. Example 5.1 of [3] shows that if r is the p-rank of E, then L(E) = End_{Q(E,E)}(k)/Rad End_{Q(E,E)}(k) is a pure transcendental extension of k of degree r − 1. The restriction map Res_{E}^{G} : Q(G, E) → Q(E, E) induces a map from L(G) onto a subfield of L(E), so we have k ⊆ F ⊆ L(G) ⊆ L(E). Since L(E) is finitely generated over k, it follows by Theorem 4.1.5 of [10] that L(G) is finitely generated over k and hence also over F. Since L(G) is also algebraic over F, we see that it is finite-dimensional. Because End_{Q}(k) has a composition series, it follows that End_{Q}(k) is also finite-dimensional over F. □

Corollary 4.5. Let F be a subfield of End_{Q}(k) satisfying the conditions of Theorem 4.4. If M and N are objects of Q, then Hom_{Q}(M, N) is finite-dimensional over F.

Proof. It suffices to show that Hom_{Q}(M, N) is a finitely generated End_{Q}(k)-module. We may assume that V_{E} ⊆ V_{G}(M), for otherwise Hom_{Q}(M, N) = 0 and the result is trivial.

Let t be the period of Q. Then there exist an integer s > 0 and elements α ∈ H^{st}(G, k) and β ∈ H^{(s−1)t}(G, k) such that the map

Ω^{t}(M) ⟵^{Ω^{t}β̂_{M}} Ω^{st}(M) ⟶^{α̂_{M}} M

is an isomorphism in Q.

Set H^{∗} = ⨁_{n=0}^{∞} H^{nt}(G, k) and E^{∗} = ⨁_{n=0}^{∞} Êxt_{kG}^{nt}(M, N), and let

S = {ζ ∈ H^{∗} | ζ is homogeneous and V_{E} ⊈ V_{G}(ζ)}.

Then H^{∗} is a Noetherian ring and E^{∗} is finitely generated over H^{∗}, so H^{∗}S^{−1} is Noetherian and E^{∗}S^{−1} is finitely generated over H^{∗}S^{−1}. Let f_{1}, …, f_{n} ∈ E^{∗} be homogeneous elements whose images generate E^{∗}S^{−1} as an H^{∗}S^{−1}-module, and let deg f_{i} = k_{i}t. Set f′_{i} = f_{i}∘(β_{M}/α_{M})^{k_{i}} for 1 ≤ i ≤ n. Then f′_{1}, …, f′_{n} generate E^{∗}S^{−1} as an H^{∗}S^{−1}-module, and deg f′_{i} = 0 for all i, so it follows that (E^{∗}S^{−1})^{0} is finitely generated as a module for (H^{∗}S^{−1})^{0}. Thus Hom_{Q}(M, N) is a finitely generated End_{Q}(k)-module, and this completes the proof. □

Corollary 4.6. Let F be a subfield of End_{Q}(k) satisfying the conditions of Theorem 4.4. If r = dim V_{E}, then F has transcendence degree r − 1 over k.

Proof. Recall first that H^{∗}(E, k) = P ⊗_{k} Λ, where P is a polynomial ring in r variables x_{1}, …, x_{r}, and Λ is an exterior algebra. If p = 2, then Λ = k and deg(x_{i}) = 1 for all i; if p > 2, then Λ is generated by the elements of degree one, and deg(x_{i}) = 2 for all i. The augmentation homomorphism ε : Λ → k induces a homomorphism H^{∗}(E, k) → P whose kernel is the radical of H^{∗}(E, k), and a homogeneous element of H^{∗}(E, k) is a Qisomorphism if and only if it does not lie in the radical. Thus if T ⊆ P is the set of nonzero homogeneous polynomials, then in the notation of the proof of Theorem 4.4 we have L(E) = (P · T^{−1})^{0}. Hence L(E) ≅ k(x_{1}/x_{r}, …, x_{r−1}/x_{r}) is a pure transcendental extension of k of degree r − 1.

Now let

S = {ζ ∈ H^{∗}(G, k) | ζ is homogeneous and res_{G,E}(ζ) is not nilpotent}.

Because V_{E} ⊈ V_{G}(ζ) if and only if res_{G,E}(ζ) is not nilpotent, it follows that

Hom_{Q}(k, k_{E}^{↑G}) = (Êxt_{kG}^{∗}(k, k_{E}^{↑G}) · S^{−1})^{0}.

Let S_{E} = res_{G,E}(S), and recall that we have embeddings k ⊆ F ⊆ L(G) ⊆ L(E) as in the proof of Theorem 4.4. By the Eckmann–Shapiro Lemma there is an isomorphism Ext_{kG}^{∗}(k, k_{E}^{↑G}) ≅ Ext_{kE}^{∗}(k, k) as H^{∗}(G, k)-modules, so it follows that there is an isomorphism

Hom_{Q}(k, k_{E}^{↑G}) = (Êxt_{kG}^{∗}(k, k_{E}^{↑G}) · S^{−1})^{0} ≅ (H^{∗}(E, k) · S_{E}^{−1})^{0}

of vector spaces over F. Corollary 4.5 now implies that (H^{∗}(E, k) · S_{E}^{−1})^{0} is finitedimensional over F.

Let s ∈ S_{E} be an element of positive degree, and note that if p > 2, then deg s is even because s is not nilpotent. Let n = deg s if p = 2, and let 2n = deg s if p > 2. Then for 1 ≤ i ≤ r it follows that x_{i}^{n}/s ∈ (H^{∗}(E, k) · S_{E}^{−1})^{0} and hence x_{i}^{n}/s is integral over F. Thus the field

K = F (x_{1}^{n}/s, …, x_{r}^{n}/s) ⊆ L(E)

is a finite extension of F. Moreover, (x_{i}/x_{r})^{n} = (x_{i}^{n}/s)(s/x_{r}^{n}) ∈ K for 1 ≤ i ≤ r − 1, so L(E) = k(x_{1}/x_{r}, …, x_{r−1}/x_{r}) is a finite extension of K. It follows that L(E) is a finite extension of F and that L(E) and F have the same transcendence degree over k. □

Remark 4.7. (i) It should be the case that there is a maximal subfield F of End_{Q}(k) with

F ≅ End_{Q}(k)/Rad End_{Q}(k)

under the quotient map. But this does not appear to be so easy to prove.

(ii) In the quotient categories M_{c}/M_{d} with c − d > 1, it seems likely that the sets of morphisms from one module to another should be finite-dimensional over some field F. If r is the p-rank of G, then F should be a subfield of Hom_{M_{r}/M_{d}}(k, k) having transcendence degree d over k. However, it remains to be seen what the endomorphism ring of the trivial module is like in these cases.

5. Equivalences of Categories

Much of the recent work in modular representation theory has centered on equivalences of stable or derived categories. It seems natural, therefore, to look for similar relationships between quotient categories, especially since some of the results on decompositions hint directly at equivalences coming from induction or restriction. As motivation we recall a result from [2]. Assume that E is a maximal elementary abelian p-subgroup of G, and let D_{G}(E) be the set of all elements of N_{G}(E) whose conjugation action on E ≅ 𝔽_{p}^{r} is by scalar multiplication. That is, if x ∈ D, then there is some integer a with 1 ≤ a ≤ p − 1 such that xyx^{−1} = y^{a} for all y ∈ E. Set C = C_{G}(E), D = D_{G}(E), N = N_{G}(E), and m = |N : D|. Then C ⊆ D ◁ N, and D/C has order dividing p − 1. Theorem 4.2 of [2] says that for some n > 0 and any ℓ > 0 there is a projective module P and an exact sequence

0 ⟶ U ⟶ (Ω^{nℓ}(k))^{m} ⊕ P ⟶^{η} k_{D}^{↑G} ⟶ 0

such that V_{G}(U)∩V_{E} ≠ V_{E}. In other words, U ∈ N (G, E) and η is a Q(G, E)-isomorphism. Because Q(G, E) is periodic, we get the following result.

Proposition 5.1. k_{D}^{↑G} ≅ k^{m} in Q(G, E).

Frobenius reciprocity says that if M is any kG-module, then M ⊗k_{D}^{↑G} ≅ (M_{↓D})^{↑G}. This fact seems to indicate that except for the multiplicity m, there should be an equivalence of categories between Q(D, E) and Q(G, E) given by induction. Unfortunately, the situation does not appear to be quite so straightforward. Indeed, the appropriate subgroup to consider seems to be N rather than D.

But the passage from D to N comes at a price. If H is any subgroup of G containing E, then the restriction functor between the stable categories induces a functor Res_{H}^{G} : Q(G, E) → Q(H, E). With the induction functor, however, the situation is not so simple. For example, if E_{1} is a subgroup of H that is conjugate to E in G but not in H, then res_{H,E}^{∗}(V_{E}(k)) ≠ res_{H,E_{1}}^{∗}(V_{E_{1}}(k)) but res_{G,E}^{∗}(V_{E}(k)) = res_{G,E_{1}}^{∗}(V_{E_{1}}(k)). Thus a kHmodule M might satisfy res_{H,E}^{∗}(V_{E}(k)) ⊈ V_{H}(M) and res_{H,E_{1}}^{∗}(V_{E_{1}}(k)) ⊆ V_{H}(M). Then M ≅ 0 in Q(H, E), but res_{G,E}^{∗}(V_{E}(k)) ⊆ V_{G}(M^{↑G}), so M^{↑G} ≅ 0̸ in Q(G, E). In fact, induction will induce a functor Ind_{H}^{G} : Q(H, E) → Q(G, E) if and only if every G-conjugate of E contained in H is actually an H-conjugate of E. In particular, Ind_{N}^{G} : Q(N, E) → Q(G, E) is defined provided that E^{x} ⊈ N whenever x ∈ G − N. We begin by studying this case.

Lemma 5.2. Suppose G has the property that E^{x} ⊈ N whenever x ∈ G − N. Then N contains a Sylow p-subgroup of G.

Proof. Let Q be a Sylow p-subgroup of N, and let P be a Sylow p-subgroup of G containing Q. Let x ∈ N_{P} (Q). Since E ⊆ Q, it follows that E^{x} ⊆ Q^{x} = Q ⊆ N and hence x ∈ N. Then ⟨Q, x⟩ is a p-subgroup of N containing Q, so x ∈ Q. Hence Q = P. □

The next lemma is again similar to Theorem 4.2 of [2], although the proof is much easier in this case.

Lemma 5.3. Suppose G has the property that E^{x} ⊈ N whenever x ∈ G − N. Let n > 0 be an integer such that there is a Q(G, E)-isomorphism ζ̂ : Ω^{n}(k) → k. Then there is a Q(G, E)-isomorphism η : Ω^{n}(k) → k_{N}^{↑G}.

Proof. Because N contains a Sylow p-subgroup of G by Lemma 5.2, we know that k is a summand of k_{N}^{↑G}. But

(k_{N}^{↑G})_{↓E} ≅ ∑_{x∈N\G/E} k_{N^{x}∩E}^{↑E}.

Since E does not normalize any of its distinct conjugates by assumption, it follows that N^{x} ∩ E < E unless x ∈ N. Thus k_{N}^{↑G} ≅ k ⊕ M, where res_{G,E}^{∗}(V_{E}(k)) ⊈ V_{G}(M) so that M ≅ 0 in Q(G, E). Let ι : k → k_{N}^{↑G} be a split monomorphism, and set η = ιζ̂. Then η satisfies the required conditions. □

For any kN-module M let ρ_{M} : M → (M^{↑G})_{↓N} denote the kN-homomorphism satisfying ρ_{M}(m) = m ⊗ 1. Then ρ : 1 → Res_{N}^{G}Ind_{N}^{G} is a natural transformation.

Lemma 5.4. Suppose G has the property that E^{x} ⊈ N whenever x ∈ G − N. Then ρ : 1 → Res_{N}^{G}Ind_{N}^{G} is a natural isomorphism of functors on Q(N, E).

Proof. Let S be a set of representatives for the right cosets of N contained in G − N, and let T be a set of representatives for the (N, N)-double cosets contained in G − N. For any kN-module M we have (M^{↑G})_{↓N} = (M ⊗ 1) ⊕ ∑_{x∈S} M ⊗ x, and

∑_{x∈S} M ⊗ x ≅ ∑_{x∈T}(M_{↓N^{x}∩N}^{x})^{↑N}.

Thus in order to show that ρ is an isomorphism, it suffices to prove that if the component res_{N,E}^{∗}(V_{E}(k)) is contained in V_{N}((M_{↓N^{x}∩N}^{x})^{↑N}), then x ∈ N. But the containment implies that E ⊆ N ∩ N^{x}, so we get the desired result. □

Theorem 5.5. Suppose G has the property that E^{x} ⊈ N whenever x ∈ G − N. Then Q(G, E) is triangle-equivalent to Q(N, E).

Proof. We already know by Lemma 5.4 that there is a natural isomorphism ρ : 1 → Res_{N}^{G}Ind_{N}^{G}, so it suffices to show that there is also a natural isomorphism 1 → Ind_{N}^{G}Res_{N}^{G}.

By Proposition 2.3 and Lemma 5.3 there exist an integer n > 0 and Q(G, E)-isomorphisms ζ̂ : Ω^{n}(k) → k and η : Ω^{n}(k) → k_{N}^{↑G}. Let μ_{M} be the natural isomorphism k_{N}^{↑G} ⊗ M → (M_{↓N})^{↑G}, and let ν_{M} be the natural isomorphism k ⊗ M → M. Then there is a natural transformation γ : 1 → Ind_{N}^{G}Res_{N}^{G} of functors on Q(G, E) such that γ_{M} is given by

M ⟵^{ν_{M} (ζ̂⊗1_{M})} Ω^{n}(k) ⊗ M ⟶^{μ_{M} (η⊗1_{M})} (M_{↓N})^{↑G}.

It is easy to see that γ is an isomorphism, as desired. □

Next we consider what can be said without the assumption that E^{x} ⊈ N whenever x ∈ G − N. Although there is no longer a functor Ind_{N}^{G} : Q(N, E) → Q(G, E), the subgroup D contains a unique G-conjugate of E, so there is a well-defined functor I_{N}^{G} = Ind_{D}^{G}Res_{D}^{N} : Q(N, E) → Q(G, E). If we set m = |N : D|, then we get the following result.

Theorem 5.6. The functors Res_{N}^{G} : Q(G, E) → Q(N, E) and I_{N}^{G} : Q(N, E) → Q(G, E) satisfy I_{N}^{G}Res_{N}^{G} ≅ k^{m} ⊗ − and Res_{N}^{G}I_{N}^{G} ≅ k^{m} ⊗ −.

Proof. We first show that there is a natural isomorphism I_{N}^{G}Res_{N}^{G} = Ind_{D}^{G}Res_{D}^{G} → k^{m} ⊗−. By Proposition 2.3 there exist an ℓ > 0 and a Q(G, E)-isomorphism ζ̂ : Ω^{ℓ}(k) → k. As we observed at the beginning of this section, Theorem 4.2 of [2] shows that there exist an integer n > 0 and a Q(G, E)-isomorphism β : (Ω^{nℓ}(k))^{m} → k_{D}^{↑G}. Because ζ̂: Ω^{ℓ}(k) → k is a Q(G, E)-isomorphism, it follows that there is a Q(G, E)-isomorphism γ : (Ω^{nℓ}(k))^{m} → k^{m}. Let M be a kG-module, and let μ_{M} denote the natural isomorphism k_{D}^{↑G} ⊗M → (M_{↓D})^{↑G}. Then there is an isomorphism (M_{↓D})^{↑G} ≅ k^{m} ⊗ M in Q(G, E) given by

(M_{↓D})^{↑G} ⟵^{μ_{M} (β⊗1_{M})} (Ω^{nℓ}(k))^{m} ⊗ M ⟶^{γ⊗1_{M}} k^{m} ⊗ M,

and it is easy to check that the isomorphism is natural. Thus we conclude that I_{N}^{G}Res_{N}^{G} = Ind_{D}^{G}Res_{D}^{G} ≅ k^{m} ⊗−, as desired. In particular, if G = N, then it follows that I_{N}^{N} ≅ k^{m} ⊗−.

Next we show that there is a natural isomorphism Res_{N}^{G}I_{N}^{G} → k^{m} ⊗ −. Let S be a set of representatives for the right cosets of D in N, and let T be a set of representatives for the right cosets of D contained in G − N. Fix an object M of Q(N, E), and set M_{0} = ∑_{s∈S} M ⊗ s and M_{1} = ∑_{t∈T} M ⊗ t. Then Res_{N}^{G}I_{N}^{G}(M) = M_{0} ⊕ M_{1}, and

M_{1} ≅ ⨁_{x}(M_{↓D^{x}∩N}^{x})^{↑N},

where x runs over a set of representatives for the (D, N)-double cosets contained in G − N. In particular, for each term occurring in this sum we have x ∉ N so that E ⊈ D^{x} ∩ N and hence V_{N}((M_{↓D^{x}∩N}^{x})^{↑N}) ⊉ res_{N,E}^{∗}(V_{E}(k)). Thus M_{1} ≅ 0 in Q(N, E). If π_{M} : Res_{N}^{G}I_{N}^{G}(M) → M_{0} denotes the projection with kernel M_{1}, then it follows that π_{M} is a Q(N, E)-isomorphism. But M_{0} = (M_{↓D})^{↑N}, so π gives a natural isomorphism between the functors Res_{N}^{G}I_{N}^{G} : Q(N, E) → Q(N, E) and Ind_{D}^{N}Res_{D}^{N} : Q(N, E) → Q(N, E). Thus it follows from the first part of the proof that Res_{N}^{G}I_{N}^{G} ≅ Ind_{D}^{N}Res_{D}^{N} = I_{N}^{N} ≅ k^{m} ⊗ −. This completes the proof. □

Acknowledgments

Most of the work presented here was done while the second author was visiting the Department of Mathematics at the University of Georgia, and he would like to express his gratitude for the hospitality shown during this visit.

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