COHOMOLOGY AND INDUCTION FROM

ELEMENTARY ABELIAN SUBGROUPS

Jon F. Carlson

Department of Mathematics University of Georgia Athens, GA 30602 jfc@sloth.math.uga.edu http://www.math.uga.edu/˜ jfc

1. Introduction. A century ago Maschke [15] proved that for a finite group G and a field k, the group algebra kG is semi-simple if and only if the characteristic p of k does not divide the order of G. If kG is semi-simple then every kG-module is projective and is a direct sum irreducible modules. For this reason the early work on group representations concentrated on the study of irreducible module and characters. The same was true of the early work by Brauer and others on modular representations and blocks. There were some efforts aimed at the structure of projective modules, but a serious attempt to develop the homological algebra necessary for a module theory did not come until much later.

From the very beginning it was clear that the restriction of a module to the Sylow p-subgroup of G indicates a great deal about the structure of the module. Quillen’s Theorem [18], written in the early 1970’s, showed that the elementary abelian psubgroups also played a major role in the module theory. The first application of Quillen’s result in this direction was in the thesis of Chouinard (see [10]) where he proved that a kG-module is projective if and only its restriction to every elementary abelian p-subgroup is projective. Other application by Alperin and Evens [1,2], Avrunin [3] and others [4,8] quickly followed.

The purpose of this paper is to present a general proof and unified treatment of role of the elementary abelian p-subgroups. In recent years the approach taken here has gained in relevance because of the interest in infinitely generated modules and the increasing use of the notation and methods of category theory in representation theory (e.g. see [6,7,23]). In categorical terms the statement we prove is that the stable category of finitely generated kG-modules is generated as a triangulated category by modules induced from elementary abelian p-subgroups of G (see Corollary 2.4). This is what our main theorem (Theorem 2.1) really says, although we have couched the theorem in terms of filtrations on modules. The statement about filtrations is very specific and easy to use to derive the consequences. Included among the consequences are theorems of Quillen, Alperin-Evens, and others

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*Partly supported by grants from NSF and the EPSRC

mentioned above. All of this is developed in Section 2 of the paper. The proof of the main theorem is in Section 3.

One advantage to our categorical approach is that it becomes very clear when we need to assume that modules are finitely generated. For example, Theorem 2.5 and Corollary 2.7 on the nilpotency of cohomology products do not require this assumption.

An advantage to the proof is that it makes only minimal use of group cohomology. Of course it uses a lot of homological algebra, but specialized constructions such as spectral sequences are avoided. The proof is based on Serre’s Theorem on the vanishing of products of Bocksteins. Serre’s Theorem is certainly more technical than we would like, however it seems unavoidable. Somewhere in the theory we need a statement which says that the module theory for group algebras of elementary abelian p-groups is different from that of other p-groups. This is the role of Serre’s Theorem.

Over the years I have had many illuminating discussions with Dave Benson on this subject. It is certain that some part of this paper, whether ideas or formulation, is due to him. I can’t thank him enough.

We have tried to make the exposition reasonably self contained. However, the reader may require some general references. For information on group cohomology, varieties and modules we suggest consulting [5] or [11]. For homological algebra, see any of the standard texts such as [13], [14] or [23]. For categories and triangulations see [12] or [23]. Almost all of this surveyed in [9].

2. The main theorem and its consequences. .

Throughout the section and the rest of the paper, let G be a finite group and let k be a field of characteristic p > 0. We also let k denote the trivial kG-module of dimension one. The following is the main theorem of the paper.

Theorem 2.1. There exists an integer τ = τ(G, p), depending only on G and p, and a finitely generated kG-module V such that the direct sum k ⊕V has a filtration

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{τ} = k ⊕ V

with the property that for each i = 1, …, τ, there is an elementary abelian psubgroup E_{i} ⊆ G and a kE_{i}-module W_{i} such that

L_{i}/L_{i−1} ≅ W_{i}^{↑G}.

For notation we let W^{↑G} denote the induced module W^{↑G} = kG ⊗_{kH} W for any kH-module W, with H a subgroup of G. For V as in the theorem we say that k ⊕ V is filtered by modules induced from elementary abelian p-subgroups.

We postpone the proof of the theorem for the next section. For the rest of this section we record some of the implications of the main result. We begin by noting that a similar theorem holds for any kG-module M. In the statements, the corollaries and their proofs, the integer τ and the sequences E_{1}, …, E_{τ} of elementary abelian p-subgroups and W_{1}, …, W_{τ} of modules are meant to be exactly as given in the Theorem.

Corollary 2.2. For any kG-module M there is a kG-module M′ such that M ⊕M′ has a filtration

{0} = N_{0} ⊆ N_{1} ⊆ ⋯ ⊆ N_{τ} = M ⊕ M′

where N_{i}/N_{i−1} ≅ U_{i}^{↑G} for some kE_{i}-module U_{i}. Moreover if M is finitely generated then M′ may also be taken as finitely generated.

proof. Let M′ = M ⊗ V and let N_{i} = M ⊗ L_{i} for i = 1, …, τ. Then by Frobenius reciprocity

N_{i}/N_{i−1} ≅ M ⊗ L_{i}/L_{i−1} ≅ M ⊗ W_{i}^{↑G} = (M_{E_{i}} ⊗ W_{i})^{↑G}

Hence the proof is complete if we let U_{i} = M_{E_{i}} ⊗ W_{i} for all i.

For the next result we require some additional notation. Suppose that H is a subgroup of G and that M is a kG-module. Then the restriction of M to H can be expressed as

M_{H} = ρ_{H}(M) ⊕ f_{H}(M)

where f_{H}(M) is a projective kH-module and ρ_{H}(M) is a kH-module with no projective submodules. Let η_{H}(M) = Dim(ρ_{H}(M)).

Corollary 2.3. [8] There exists a natural number B = B(p, G), depending only on p and G, such that if M is a finitely generated nonprojective indecomposable kG-module, then

Dim(M) ≤ B max_{i}{η_{E_{i}}(M)}.

Proof. . The corollary is a consequence of the fact that projective kG-modules are also injective. We adopt the notation of the last corollary. For each i, (f_{E_{i}}(M) ⊗ W_{i})^{↑G} is a projective kG-module because f_{E_{i}}(M) ⊗ W_{i} is a projective kE_{i}-module. Hence there is a direct summand of N_{i} that is isomorphic to (f_{E_{i}}(M) ⊗ W_{i})^{↑G}. Because this module is also injective it is a projective direct summand of M ⊕ M′ and hence is a projective direct summand of M′. It follows that

Dim M ≤ ∑_{i=1}^{N} Dim (ρ_{E_{i}}(M) ⊗ W_{i})^{↑G}

= ∑_{i=1}^{N} |G : E_{i}| Dim W_{i} · Dim(ρ_{E_{i}}(M))

≤ |G| (∑_{i=1}^{N} Dim W_{i}) max_{i}{η_{E_{i}}(M)}

Hence the proof is complete if we let B = |G| ∑ DimW_{i}.

The stable category, Stmod-kG, has objects consisting of all kG-modules. If M and N are kG-modules then the set of morphisms from M to N in the stable category is the group

Hom_{Stmod−kG}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N).

Here PHom_{kG}(M, N) is the set of all homomorphisms from M to N which factor through a projective module. The stable category is triangulated with the triangles corresponding roughly to exact sequences in the module category (see [9]). A full subcategory of Stmod-kG is said to be thick if it is triangulated and if it is closed under taking direct summands. Let stmod-kG be the stable category of finitely generated kG-modules. The next corollary is immediate from the Theorem.

Corollary 2.4. The stable category Stmod-kG is generated as a triangulated category by modules induced from elementary abelian p-subgroups of G. Likewise stmod-kG is generated by finitely generated modules induced from elementary abelian p-subgroups of G.

The main theorem can also be used to prove several other results on the homological properties of modules. One such is the theorem of Chouinard [10] which asserts that a kG-module is projective if its restriction to every elementary abelian p-subgroup is projective. The proof is a special case of the proof of Corollary 2.3. Similarly we can deduce the theorem of Alperin and Evens [1] that the complexity (or polynomial rate of growth of the minimal projective resolution of a module) is equal to the maximum of the complexities of the restrictions of the module to elementary abelian p-subgroups. Note that the complexity of a module is equal to the dimension of the variety of the module and the module is projective if and only if the dimension of the variety is zero. Consequently both of the theorems can also be derived from the results on varieties which follow.

Theorem 2.5. Suppose that X_{0}, X_{1}, ⋯, X_{τ} are kG-modules and that for each i we have an element ζ_{i} ∈ Ext_{kG}^{n_{i}}(X_{i}, X_{i−1}) for some n_{i} such that

res_{G},_{E_{i}} (ζ_{i}) = 0.

Then ζ_{τ} ⋯ ζ_{1} = 0 in Ext_{kG}^{n}(X_{τ} , X_{0}), where n = n_{1} + ⋯ + n_{τ}.

Proof. Let M = X_{0} ⊕ ⋯ ⊕ X_{τ}. By the Theorem, or more precisely by Corollary 2.2, there exists a kG-module M′ such that M ⊕ M′ = Y is filtered by modules induced from elementary abelian p-subgroups. Specifically we have submodules

{0} = N_{0} ⊆ N_{1} ⊆ ⋯ ⊆ N_{τ} = M ⊕ M′ = Y

such that N_{i}/N_{i−1} ≅ U_{i}^{↑G} for some kE_{i}-module U_{i}. We note that for each i, Ext_{kG}^{∗}(X_{i}, X_{i−1}) is a direct summand of Ext_{kG}^{∗}(Y, Y), and so we identify each ζ_{i} with its image in Ext_{kG}^{n_{i}}(Y, Y). We need the following.

Lemma 2.6. . For each i = 1, …, τ, we have that Ext_{kG}^{∗}(Y, N_{i})·ζ_{i} ⊆ j_{i∗}(Ext_{kG}^{∗}(Y, N_{i−1})) where j_{i} is the inclusion map N_{i−1} ↪ N_{i}.

Proof of Lemma. For each i and any r we have a commutative diagram

Ext_{kG}^{r}(Y, N_{i−1}) ⟶^{j_{i∗}} Ext_{kG}^{r}(Y, N_{i}) ⟶^{π_{i∗}} Ext_{kG}^{r}(Y, U_{i}^{↑G})
ζ_{i}↓  ^{ζ_{i}}↓  ^{ζ_{i}}↓
Ext_{kG}^{r+n_{i}}(Y, N_{i−1}) ⟶^{j_{i∗}} Ext_{kG}^{r+n_{i}}(Y, N_{i}) ⟶^{π_{i∗}} Ext_{kG}^{r+n_{i}}(Y, U_{i}^{↑G})

where the vertical maps are right multiplication by ζ_{i}. By the Eckmann-Shapiro Lemma we have a commutative diagram

Ext_{kG}^{r}(Y, U_{i}^{↑G}) ≅ Ext_{kE_{i}}^{r}(Y, U_{i})
ζ_{i}↓  ↓^{res_{G,E_{i}} (ζ_{i})}
Ext_{kG}^{r+n_{i}}(Y, U_{i}^{↑G}) ≅ Ext_{kE_{i}}^{r+n_{i}}(Y, U_{i})

Because res_{G},_{E_{i}} (ζ_{i}) = 0, we have that ζ_{i}π_{i} = 0. This proves the Lemma.

To finish the proof of the theorem we apply the Lemma inductively to get that

Ext_{kG}^{∗}(Y, Y) · ζ_{n} ⋯ ζ_{i+1} ⊆ ℓ_{i∗}(Ext_{kG}^{∗}(Y, N_{i}))

where ℓ_{i} : N_{i} → Y is the inclusion. But N_{0} = {0} and hence ζ_{τ} ⋯ ζ_{1} = 0 as desired.

Corollary 2.7. Suppose that M is a kG-module. Let I ⊆ Ext_{kG}^{∗}(M, M) be the ideal consisting of all elements ζ such that res_{G,E}(ζ) = 0 for all elementary abelian p-subgroups E ⊆ G. Then I^{τ} = {0}.

The corollary is an immediate consequence of the previous theorem. Next we notice that many of the results on varieties of modules are also consequences of the theorem. We know that the cohomology ring H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) is a finitely generated graded-commutative algebra. So its maximal ideal spectrum V_{G}(k) is a finite dimensional homogeneous affine variety. If H ⊆ G is a subgroup 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) on varieties. If M is a finitely generated kG-module, let J_{G}(M) be the annihilator in H^{∗}(G, k) of Ext_{kG}^{∗}(M, M). Then the variety of M is V_{G}(M) = V_{G}(J_{G}(M)) ⊆ V_{G}(k), is the closed set of all maximal ideals containing J_{G}(M).

Corollary 2.8. [1,3,18,19] Let A denote the set of all maximal elementary abelian p-subgroups of G. Then

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

Moreover, for any finitely generated kG-module M,

V_{G}(M) = ⋃_{E∈A} res_{G,E}^{∗}(V_{E}(M)).

Proof. The ideal I ⊆ H^{∗}(G, k) consisting of all ζ with res_{G,E}(ζ) = 0 for all E ∈ A is nilpotent by the last corollary. Hence it is in every maximal ideal of H^{∗}(G, k). It follows that every maximal ideal of H^{∗}(G, k) is the kernel of a homorphism H^{∗}(G, k) → K which factors through the restriction to some E ∈ A, where K is the algebraic closure of k. This proves the first statement.

For the second, notice that ζ ∈ J_{G}(M) if and only if ζ · Id_{M} ∈ Ext_{kG}^{∗}(M, M) is zero. So if I is the set of all ζ ∈ H^{∗}(G, k) such that

res_{G,E}(ζ)Id_{M_{E}} = res_{G,E}(ζId_{M}) = 0,

then I · Id_{M} ⊆ Ext_{kG}^{∗}(M, M) is nilpotent. That is,

⋂_{E∈A} res_{G,E}^{−1}(J_{E}(M_{E})) ⊆ \sqrt{J(M)}.

This implies that V_{G}(M) ⊆ ⋃_{E∈A} res_{G,E}^{∗}(V_{E}(M)). The other containment is easy.

3. Proof of the Main Theorem.

As indicated by the title our objective in this section is to give a proof of Theorem 2.1. The first observation is that it is sufficient to prove the Theorem in the case that the group, G, is a p-group.

Lemma 3.1. If Theorem 2.1 is true for all finite p-groups, then it is true for all finite groups.

Proof. Let G be any finite group and let S be a Sylow p-subgroup of G. By hypothesis there is an integer τ and a kS-module V such that k_{S} ⊕V has a filtration

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{τ} = k_{S} ⊕ V,

where for each i, L_{i}/L_{i−1} ≅ W_{i}^{↑S} for some kE_{i}-module W_{i} and some elementary abelian p-subgroup E_{i} of S. But now we can induce the entire system to G. That is, k_{S}^{↑G} ⊕ V^{↑G} has a filtration

{0} = L_{0}^{↑G} ⊆ L_{1}^{↑G} ⊆ ⋯ ⊆ L_{τ}^{↑G} = k_{S}^{↑G} ⊕ V^{↑G}

such that L_{i}^{↑G}/L_{i−1}^{↑G} = (W_{i}^{↑S})^{↑G} = W_{i}^{↑G}. Then the lemma follows from the fact that k is a direct summand of k_{S}^{↑G}.

For the remainder of the section we assume that G is a finite p-group. As was stated in the introduction, the key to the proof of Theorem 2.1 is Serre’s Theorem [21] which distinguishes the representation theory of an elementary abelian p-group from that of a p-group which is not elementary abelian. Other proofs and some upper bounds for the length of the sequence in the theorem have been given in [16, 17, 22].

Theorem 3.2. [21]. Suppose that G is a p-group which is not elementary abelian. Then there is a sequence γ_{1}, γ_{2}, …, γ_{n} ∈ H^{1}(G, ℤ/pℤ) of nonzero elements such that

β(γ_{1}) ⋯ β(γ_{n}) = 0.

Here β : H^{1}(G, ℤ/pℤ) → H^{2}(G, ℤ/pℤ) is the Bockstein map. The Bockstein is adequately explained in several standard texts, for example [5] or [11]. For our purposes it is only necessary to understand the nature of representative extensions in Ext_{kG}^{2}(k, k) = H^{2}(G, k). First notice that H^{1}(G, ℤ/pℤ) ≅ Hom(G, ℤ/pℤ). So any nonzero γ in H^{1}(G, ℤ/pℤ) corresponds to a homomorphism whose kernel is a maximal subgroup H ⊆ G. Then β(γ) is represented by an exact sequence of the form

0 ⟶ k ⟶ k_{H}^{↑G} ⟶^{ϕ} k_{H}^{↑G} ⟶ k ⟶ 0,

where ϕ is multiplication by x − 1 for some element x ∉ H.

We should also recall that the cup product of elements of cohomology is represented by the Yoneda splice of the representing exact sequence. Also the field of coefficients can be extended without affecting the inductions. Hence Serre’s theorem can be interpreted as saying the following. The sequence E, given below, is the splice of the sequences representing the Bocksteins.

Corollary 3.3. Suppose that G is a p-group which is not elementary abelian. Then there is a sequence of maximal subgroups H_{1}, …, H_{n} of G and an exact sequence

E : 0 → k → C_{2n−1} → ⋯ → C_{1} → C_{0} → k → 0

such that (i) C_{2i−2} ≅ C_{2i−1} ≅ k_{H_{i}}^{↑G} for i = 1, …, n, and (ii) the class of E in Ext_{kG}^{2n}(k, k) is zero.

At this point we fix a group G which is not elementary abelian with associated long exact sequence E, with sequence of subgroups H_{1}, …, H_{n}, as in the Corollary. Let C_{∗} be the complex obtained by truncating the ends of the exact sequence E. That is let C_{i} = C_{i} for i = 0, …, 2n − 1, while C_{i} = 0 if either i < 0 or i > 2n − 1. The boundary maps C_{i} → C_{i−1} are as in E for 1 ≤ i ≤ 2n − 1. Hence H_{0}(C_{∗}) ≅ k ≅ H_{2n−1}(C_{∗}), and H_{j}(C_{∗}) = 0 if j ≠ 0 or 2n − 1. Let (P_{∗}, ϵ) be a minimal projective resolution of the trivial kG-module k. Then each P_{i} is projective and H_{i}(P_{∗}) = 0 except when i = 0 in which case H_{0}(P_{∗}) = k. It follows from the Künneth formula that H_{i}(C_{∗} ⊗ P_{∗}) ≅ H_{i}(C_{∗}) for all i. Moreover, (C_{∗} ⊗ P_{∗})_{i} = ∑_{j=0}^{i} C_{j} ⊗ P_{i−j} is a projective module for all i. We should emphasize here that the tensor product is taken over the base field k. Hence the modules C_{j} ⊗P_{i−j} are projective kG-modules (see [5] or [11]).

Now let t be any natural number such that t ≥ 2n − 1. Let Γ(C_{∗} ⊗ P_{∗}) be the truncated complex

Γ(C_{∗} ⊗ P_{∗}) : ⋯ → (C_{∗} ⊗ P_{∗})_{t+1} → (C_{∗} ⊗ P_{∗})_{t} → 0.

It can be deduced that H_{i}(Γ(P_{∗}⊗C_{∗})) = 0 unless i = t. That is, Γ(C_{∗}⊗P_{∗}) is zero in degrees less than t. Hence H_{i}(Γ(C_{∗} ⊗P_{∗})) = 0 for i < t. If i > t, then Γ_{i}(C_{∗} ⊗P_{∗}) = (C_{∗} ⊗ P_{∗})_{i}, and hence H_{i}(Γ(C_{∗} ⊗ P_{∗})) ≅ H_{i}(C_{∗} ⊗ P_{∗}) ≅ ⊕H_{j}(C_{∗}) ⊗ H_{i−j}(P_{∗}) = 0.

The major part of the remainder of the section concerns the nature of the homology group H_{t}(Γ(P_{∗} ⊗ C_{∗})). There are two things to prove.

Proposition 3.4. If i ≠ t, then H_{i}(Γ(C_{∗} ⊗ P_{∗})) = 0. Also, H_{t}(Γ(C_{∗} ⊗ P_{∗})) is filtered by modules induced from the subgroups H_{1}, …, H_{n}.

Proof. The proof of the first statement has already been noted. For the rest, let C_{∗}^{(0)} be the complex with only one nonzero term C_{0}^{(0)} = C_{0} concentrated in degree zero. We have an exact sequence of complexes

0 ⟶ C_{∗}^{(0)} ⟶^{μ} C_{∗} ⟶^{ν} D_{∗}^{(0)} ⟶ 0

where μ is the identity map in degree zero and D_{∗}^{(0)} = C_{∗}/C_{∗}^{(0)} coincides with C_{∗} except in degree zero where D_{0}^{(0)} = 0. Similarly, we have an exact sequence

0 → Γ(C_{∗}^{(0)} ⊗ P_{∗}) → Γ(C_{∗} ⊗ P_{∗}) → Γ(D_{∗}^{(0)} ⊗ P_{∗}) ⟶ 0,

which induces an exact sequence on homology

0 → H_{t}(Γ(C_{∗}^{(0)} ⊗ P_{∗})) → H_{t}(Γ(C_{∗} ⊗ P_{∗})) → H_{t}(Γ(D_{∗}^{(0)} ⊗ P_{∗})) → 0.

Now notice that C_{∗}^{(0)} ⊗ P_{∗} is, in fact, a projective resolution

… ⟶^{∂_{2}} C_{0} ⊗ P_{1} ⟶^{∂_{1}} C_{0} ⊗ P_{0} ⟶^{ϵ} C_{0} ⟶ 0

of the kG-module C_{0} ≅ k_{H_{1}}^{↑G}. Therefore H_{t}(Γ(C_{∗}^{(0)}⊗P_{∗})) = ker(∂_{t−1}) ≅ Ω^{t}(k_{H_{1}}^{↑G})⊕Q, where Q is some projective kG-module. We remind the reader that for M any kGmodule the module Ω^{t}(M) is the t^{th} Heller translate of M. That is, it is the kernel of ∂_{t−1} where (R_{∗}, ∂_{∗}) is a minimal projective resolution of M.

For the rest of the proof of the proposition, the above process is repeated several times. That is, we next let C_{∗}^{(1)} be the complex with one only one nonzero term C_{1}^{(1)} ≅ C_{1} in degree 1. As before there is an exact sequence

0 → H_{t}(Γ(C_{∗}^{(1)} ⊗ P_{∗})) ⟶^{μ_{∗}} H_{t}(Γ(D_{∗}^{(0)} ⊗ P_{∗})) → H_{t}(Γ(D_{∗}^{(1)} ⊗ P_{∗})) → 0

where μ: C_{∗}^{(1)} → D_{∗}^{(1)} is the identity in degree one and D_{∗}^{(1)} is the quotient. This time we have that C_{∗}^{(1)} ⊗ P_{∗} is a projective resolution of C_{1} but translated back one degree. So H_{t}(Γ(C_{∗}^{(1)} ⊗ P_{∗})) ≅ Ω^{t−1}(k_{H_{1}}^{↑G})⊕ (proj) for some projective module (proj).

It follows that H_{t}(Γ(C_{∗} ⊗ P_{∗})) has a filtration

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{2n} ≅ H_{t}(Γ(C_{∗} ⊗ P_{∗})),

where for j = 0, 1 and i = 2m − 1 − j, we have that

L_{i}/L_{i−1} ≅ Ω^{t−i+1}(k_{H_{m}}^{↑G}).

The proof is complete when it is noticed that Ω^{s}(M^{↑G}) ≅ (Ω^{s}(M))^{↑G} whenever M is a kH-module for H a subgroup of G and G a p-group. This proves the proposition.

Proposition 3.5.

H_{t}(Γ(C_{∗} ⊗ P_{∗})) ≅ Ω^{t}(k) ⊕ Ω^{t−2n+1}(k) ⊕ (proj).

The proof of the proposition is a consequence of the following lemma.

Lemma 3.6. Suppose that C_{∗} and C′_{∗} are finite complexes of kG-modules and that θ : C_{∗} → C′_{∗} is a chain map which induces an isomorphism on homology. As before let Γ(C_{∗} ⊗ P_{∗}) be the truncation of C_{∗} ⊗ P_{∗} at a point t that is large enough so that H_{i}(C) = 0 for i ≥ t. Here P_{∗} is a projective resolution of the trivial module k. Then the homomorphism

θ̂: Γ(C_{∗} ⊗ P_{∗}) → Γ(C′_{∗} ⊗ P_{∗})

has the property that

θ̂_{∗} : H_{i}(Γ(C_{∗} ⊗ P_{∗})) → H_{i}(Γ(C′_{∗} ⊗ P_{∗}))

is a stable isomorphism. In particular (and most importantly) there exist projective modules Q, Q′ such that

H_{t}(Γ(C_{∗} ⊗ P_{∗})) ⊕ Q ≅ H_{t}(Γ(C′_{∗} ⊗ P_{∗})) ⊕ Q′.

Proof. Suppose that D_{∗} is the mapping cone of the chain map θ : C_{∗} → C′_{∗} (see [13] or [14]). That is, for each i, D_{i} ≅ C′_{i} ⊕ C_{i−1} and the boundary map

∂ : D_{i+1} → D_{i}

is given by the matrix ( \begin{matrix}∂′_{i+1} & θ_{i} \\ 0 & ∂_{i} \end{matrix} ) where ∂′_{∗} is the boundary map on C′_{∗} and ∂_{∗} is

the boundary map on C. Then we have an exact sequence of complexes

0 ⟶ C′_{∗} ⟶^{μ} D_{∗} ⟶^{ν} C_{∗}[−1] ⟶ 0

where C_{∗}[−1] is the complex obtained by shifting C_{∗} by one degree. In each degree i we have a split exact sequence

(3.7)  0 ⟶ C′_{i} ⟶^{μ_{i}} D_{i} ⟶^{ν_{i}} C_{i−1} ⟶ 0

where μ_{i} is the natural inclusion and ν_{i} is the projection. In particular, in the long exact sequence on homology,

⋯ → H_{i}(C′_{∗}) → H_{i}(D_{∗}) → H_{i}(C_{i}[−1]) ⟶^{δ_{i}} H_{i+1}(C′_{i}) → …

the connecting homomorphism δ_{i} is precisely the map θ_{i+1}^{∗} : H_{i}(C_{i}[−1]) ≅ H_{i+1}(C_{i}) → H_{i+1}(C′_{i}). All of this is easily checked, but the book [14] will also serve as a reference.

Because θ induces an isomorphism on homology, we have that the complex D_{∗} is exact. It follows that P_{∗} ⊗ D_{∗} is also exact by the Künneth formula. Because P_{∗} ⊗ D_{∗} is a complex of projective modules which is bounded from below (i.e. (P_{∗} ⊗ D_{∗})_{−1} = 0) we have that H_{∗}(Γ(P_{∗} ⊗ D_{∗})) is a projective module. By (3.7) we have an exact sequence

0 → H_{t}(Γ(P_{∗} ⊗ C′_{∗})) → H_{t}(Γ(P_{∗} ⊗ D_{∗})) → H_{t}(Γ(P_{∗} ⊗ C_{∗}[−1])) → 0.

Therefore H_{t}(Γ(P_{∗} ⊗ C′_{∗})) ≅ Ω(H_{t}(Γ(P_{∗} ⊗ C_{∗}[−1]))) ≅ H_{t}(Γ(P_{∗} ⊗ C_{∗})) as desired.

Proof of Proposition 3.5. Consider the sequence E in Corollary 3.3. The fact that the class of E is zero in Ext_{kG}^{2n}(k, k) means that there is a sequence of exact sequences F_{0}, ⋯, F_{r} such that

1. F_{0} = E and for each i there is a chain map γ_{i} such that

F_{0} ⟶^{γ_{1}} F_{1} ⟵^{γ_{2}} F_{2} ⟶^{γ_{3}} F_{3} ⟵ ⋯ ⟵ F_{r},

2. each sequence F_{i} begins and ends with the trivial module k and each chain map γ_{i} is the identity on the ends of the sequence, and

3. the sequence F_{r} has the form

0 → k → k → 0 ⋯ 0 → k → k → 0.

This is a consequence of the standard equivalence relation on exact sequences [14].

Now the complex C′ that we would get by truncating the ends of the sequence F_{r} has the form

0 → k → 0 → ⋯ → 0 → k → 0

where the first (rightmost) k is in degree 0 and the second is in degree 2n−1. Then P_{∗} ⊗ C′ is actually the direct sum of two complexes, P_{∗} ⊗ k = P_{∗} in degree 0 and P_{∗}[2n − 1] shifted by degree 2n − 1. Hence

H_{t}(Γ(P_{∗} ⊗ C′)) ≅ Ω^{t}(k) ⊕ Ω^{t−2n+1}(k) ⊕ (proj).

So by the lemma (3.6) this is isomorphic to H_{t}(Γ(P_{∗} ⊗ C)) as asserted.

Proof of Theorem 2.1. By Lemma 3.1, we may assume that G is a p-group. From that we proceed by induction on the order of G. Notice that if G is elementary abelian, then k = k_{G}^{↑G} is induced from an elementary abelian p-subgroup. Consequently we assume that G is a p-group which is not elementary abelian. Then by Propositions 3.4 and 3.5 there exists a kG-module M(≅ H_{t}(Γ(C_{∗} ⊗ P_{∗}))) such that M ≅ Ω^{t}(k) ⊕ Ω^{t−2n+1}(k) ⊕ (proj) and such that M has a filtration

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{2n} = M

such that L_{i}/L_{i−1} ≅ W_{i}^{↑G} where W_{i} is a kH_{i}-module for a maximal subgroup H_{i} ⊆ G. Now notice that Ω^{t}(k) ⊗ Ω^{−t}(k) ≅ k ⊕ (proj) where Ω^{−t}(k) is the k-dual of Ω^{t}(k) and is also the cokernel of ∂_{−t+1} where

o ⟶ k ⟶^{ϵ} Q_{0} ⟶^{∂_{0}} Q_{−1} ⟶^{∂_{−1}} Q_{−2} ⟶ ⋯

is a minimal injective resolution of k (the k-dual of a minimal projective resolution of k). More specifically the calculus is that

Ω^{r}(k) ⊗ M ≅ Ω^{r}(M) ⊕ (proj)

for any r and any M. So we have that Ω^{−t}(k) ⊗ M ≅ k ⊕ Ω^{−2n+1}(k) ⊕ (proj) has a filtration

(3.8)  {0} = N_{0} ⊆ N_{1} ⊆ ⋯ ⊆ N_{2n} = k ⊕ Ω^{−2n+1}(k) ⊕ (proj)

where

N_{i}/N_{i−1} ≅ Ω^{−t}(k) ⊗ L_{i}/(Ω^{−t}(k) ⊗ L_{i−1}) ≅ (Ω^{−t}(k) ⊗ W_{i})^{↑G}.

It might be worth pointing out here that from the construction, W_{i} ≅ Ω^{r_{i}}(k_{H_{i}}) for some integer r_{i}. So Ω^{−t}(k) ⊗ W_{i} ≅ Ω^{r_{i}−t}(k_{H_{i}}) ⊕ P where P is a projective kH_{i}-module. Then P^{↑G} is a projective kG-module. Because P^{↑G} is also injective it splits completely from the filtration and becomes a summand of the “⊕(proj)” in right hand side of the expression (3.8). We can sum all of this up in the following.

Lemma 3.9. Suppose that G is a p-group which is not elementary abelian. Then there are integers m and n > 0 and sequences H_{1}, …, H_{n} of maximal subgroup of G and t_{1}, …, t_{n} of integers such that k ⊕ Ω^{m}(k) ⊕ (proj) has a filtration

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n} ≅ k ⊕ Ω^{m}(k) ⊕ (proj)

where L_{i}/L_{i−1} ≅ (Ω^{t_{i}}(k_{H_{i}}))^{↑G}.

If all of the maximal subgroups H_{1}, …, H_{n} are elementary abelian, then the proof of Theorem 2.1 is complete for G. Otherwise Lemma 3.9 is just the induction step. That is, suppose that H_{i} is not elementary abelian. Then by induction there exists a kH_{i}-module V such that k_{H_{i}} ⊕ V has a filtration.

{0} = U_{0} ⊆ U_{1} ⊆ ⋯ ⊆ U_{r} ≅ k_{H_{i}} ⊕ V

where for each j there is an elementary abelian subgroup E_{j} ⊆ H_{i} and a KE_{j}- module S_{j} such that U_{j}/U_{j−1} ≅ S_{j}^{↑H_{i}}. So k_{H_{i}}^{↑G} ⊕ V^{↑G} is filtered by the submodules U_{i}^{↑G} where U_{j}^{↑G}/U_{j−1}^{↑G} ≅ S_{j}^{↑G}. Likewise Ω^{t_{i}}(K_{H_{i}})^{↑G} ⊕ Ω^{t_{i}}(V)^{↑G} is filtered by the modules Ω^{t_{i}}(S_{j})^{↑G}. Consequently, T = k ⊕ Ω^{m}(k) ⊕ Ω^{t_{i}}(V)^{↑G} ⊕ (proj) has a filtration

{0} = L_{o} ⊆ ⋯ ⊆ L_{i−1} ⊆ Û_{1} ⊆ ⋯ ⊆ Û_{r} = L_{i} ⊕ Ω^{t_{i}}(V) ⊆ ⋯ ⊆ L_{n} ⊕ Ω^{t_{i}}(V) = T

where for each j, Û_{j} is an extension

0 → L_{i−1} → Û_{i} → Ω^{t_{i}}(U_{j}) → 0

and Û_{j}/Û_{j−1} ≅ Ω^{t_{i}}(U_{j})/Ω^{t_{i}}(U_{j−1}) ≅ (Ω^{t_{i}}(S_{j}))^{↑G}. Continuing in this way we can refine the filtration to eventually satisfy the conclusion of the theorem.

Remark 3.10. From the proof of the theorem we can deduce several things about the terms in the filtration in theorem 2.1. For example we can assume that the module V has the form Ω^{n}(k) ⊕ U for some n and a module U which a direct sum of a projective module and modules of the form (Ω^{m}(k_{H}))^{↑G} for some integers m and some proper subgroups H ⊆ G which is not elementary abelian. Moreover each of the factors L_{i}/L_{i−1} can be assumed to be isomorphic to (Ω^{m}(k_{E}))^{↑G} for some m and some elementary abelian group E.

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