A THEOREM OF JON F. CARLSON ON FILTRATIONS OF MODULES

FATMA ALTUNBULAK ^{†} AND ERGÜN YALÇIN ^{‡}

Abstract. We give an alternative proof to a theorem of Jon F. Carlson [5] which states that if G is a finite group and k is a field of characteristic p, then any kG-module is a direct summand of a module which has a filtration whose sections are induced from elementary abelian p-subgroups of G. We also prove two new theorems which are closely related to Carlson’s theorem.

1. Introduction

Let G be a finite group and let k be a field of characteristic p > 0. Let H be a collection of subgroups of G. We say that a kG-module M is filtered by modules induced from H if there is a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n−1} ⊆ M_{n} = M

such that for each i = 1, 2, …, n, there is a subgroup H_{i} ∈ H and a kH_{i}-module W_{i} such that M_{i}/M_{i−1} ≅ W_{i} ↑_{H_{i}}^{G} . We consider the following theorem of Jon F. Carlson.

Theorem 1.1 (Carlson [5]). Any kG-module M is a direct summand of a module that is filtered by modules induced from elementary abelian p-subgroups.

This theorem provides another way to see the role of elementary abelian p-groups in modular representation theory, and has many applications, including the Chouinard’s theorem for finitely generated modules (see Theorem 8.2.12 in [7]). Carlson proves Theorem 1.1 by first reducing it to p-groups and then by showing that it follows from the following statement by induction.

Theorem 1.2 (Carlson [5]). Suppose that G is a p-group which is not elementary abelian. Then there is a sequence H_{1}, …, H_{n} of maximal subgroups of G such that k⊕Ω^{1−n}(k)⊕(proj) has a filtration

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

where L_{i}/L_{i−1} ≅ Ω^{1−i}(k) ↑_{H_{i}}^{G} for i = 1, …, n.

The maximal subgroups H_{1}, …, H_{n} in the above theorem are not necessarily distinct. In fact, for p > 2 we take n = 2k and H_{2i} = H_{2i−1} for all i = 1, …, k. Carlson proves the existence of such a filtration using Serre’s theorem on vanishing products in group cohomology (see Theorem 3.1 for the statement of Serre’s theorem) and a hypercohomology calculation with coefficients in a chain complex obtained from Serre’s theorem. Theorem 1.2 plays an important role in [6] where the authors use this filtration to find an upper bound for the dimensions of critical endo-trivial modules.

Date: Last updated on July 29, 2005.

Key words and phrases. L_{ζ}-modules, cohomology of groups, varieties of modules.

† Supported by the Scientific and Technical Research Council of Turkey Ph.D. research scholarship (TÜBİTAK-BAYG).

‡ Partially supported by the Turkish Academy of Sciences in the framework of the Young Scientist Award Program (TÜBA-GEBİP/2005-16).

2 FATMA ALTUNBULAK ^{†} AND ERGÜN YALÇIN ^{‡}

In this paper we give an alternative proof to Theorem 1.2 using L_{ζ}-modules. The L_{ζ}- modules are defined as follows: Associated to a cohomology class ζ ∈ H^{n}(G, k), there is a module L_{ζ} defined as the kernel of the representing homomorphism ζ̂: Ω^{n}(k) → k. Modules of this form are called L_{ζ}-modules. They are commonly used to relate cohomology theory with modular representation theory; for example, they appear in many results about varieties of modules (see [1], [2], [3], [7], [8]).

In our proof for Theorem 1.2, we still use Serre’s theorem, but we avoid the hypercohomology calculation. Given ζ_{1}, …, ζ_{n} ∈ H^{1}(G, 𝔽_{p}) satisfying the conclusion of Serre’s theorem, i.e., β(ζ_{1}) ⋯ β(ζ_{n}) = 0 where β is the Bockstein map, we observe that L_{β(ζ_{1})···β(ζ_{n})} = Ω(k) ⊕ Ω^{2n}(k) has a filtration whose sections are isomorphic to Heller shifts of L_{β(ζ_{i})}’s. This is an easy consequence of a known exact sequence for the L_{ζ} when ζ is a product of two cohomology classes (see Proposition 2.3). Next, we show that for every ζ ∈ H^{1}(G, 𝔽_{p}) with kernel H ≤ G, there is a 2-step filtration for L_{β(ζ)} ⊕ (proj) such that the sections are induced from H (see Proposition 2.4 and 2.5). In fact, for p = 2, the argument is much simpler since, in this case, Serre’s theorem is true without Bocksteins, and we have L_{ζ} ≅ k ↑_{H}^{G} for every ζ ∈ H^{1}(G, 𝔽_{2}) with kernel H. We present our alternative proof in Section 3.

In the rest of the paper, we prove two theorems which are variations of Carlson’s theorems. The first one is a generalization of Proposition 2.4 and 2.5, and it is strong enough to imply Theorem 1.2 when it is applied to a suitable extension.

Theorem 1.3. Let ζ be the cohomology class in H^{n}(G, k) which is represented by the extension

E : 0 → k → M_{n−1} → ⋯ → M_{0} → k → 0.

Then, L_{ζ} ⊕ (proj) has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n} = L_{ζ} ⊕ (proj)

with L_{i}/L_{i−1} ≅ Ω^{n−i+1}(M_{i−1}) for i = 1, …, n.

Here the notation (proj) means that the statement is true after adding a suitable projective summand. We will be using this notation throughout the paper.

In section 5, we introduce the varieties of modules, and prove

Theorem 1.4. Let H be a collection of subgroups of G. Then, for a finitely generated kGmodule M, the following are equivalent:

(i) V_{G}(M) = ⋃_{H∈H} res_{G,H}^{∗}(V_{H}(M ↓_{H}^{G})).

(ii) There exists a finitely generated kG-module V such that M ⊕ V is filtered by modules induced from H.

We conclude the paper with the following application:

Corollary 1.5. Let G be an elementary abelian 2-group. If ζ ∈ H^{n}(G, k) is represented by the extension

E : 0 → k → M_{n−1} → ⋯ → M_{0} → k → 0

where M_{i}’s are direct sums of modules induced from proper subgroups, then ζ is a non-zero scalar multiple of a product of one dimensional classes in H^{1}(G, 𝔽_{2}). In particular, E is equivalent to an extension coming from a topological group action on a sphere.

There is a similar result for p > 2 under stronger conditions. This result is also proved in Section 5.

Throughout this paper, G always denotes a finite group, k is a field of characteristic p > 0. We assume that all kG-modules are finitely generated, and all tensor products are over k unless otherwise is stated clearly.

2. Preliminaries

Given a kG-module M, the Heller shift of M is defined as the kernel of the surjection P (M) → M where P (M) denotes the projective cover of M. We denote the Heller shift of M by Ω(M). The n-th Heller shift of M is defined inductively by Ω^{n}(M) = Ω(Ω^{n−1}(M)) for positive n. Similarly, the negative shift is defined inductively by Ω^{−n}(M) = Ω^{−1}(Ω^{−n+1}(M)) where minus one Heller shift Ω^{−1}(M) of a kG-module M is defined as the cokernel of the injection M → I(M), where I(M) is the injective hull of M. Uniqueness of projective cover and injective hull gives the uniqueness of the modules Ω^{n}(M) up to isomorphism. The details about Heller shifts can be found in many books on modular representation theory. We will use the standard properties of Heller shifts without listing them here. We refer the reader to Proposition 4.4 of [4] for a complete list of these properties.

Given a projective resolution

⋯ → P_{n+1} ⟶^{∂_{n+1}} P_{n} ⟶^{∂_{n}} P_{n−1} ⟶^{∂_{n−1}} ⋯ → P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ε} k → 0

of k as a kG-module, we say that it is minimal if P_{0} is the projective cover of k, P_{1} is the projective cover of ker ε, and P_{n} is the projective cover of ker ∂_{n−1} for n ≥ 2. So, by the above description of Heller shifts, we have Ω(k) = ker ε, and Ω^{n}(k) = ker ∂_{n−1} for all n ≥ 2. Note that the cohomology group H^{n}(G, k) is the n-th cohomology of the cochain complex Hom_{kG}(P_{∗}, k). Let f ∈ Hom_{kG}(P_{n}, k) be a cocycle representing ζ ∈ H^{n}(G, k), then the cocycle condition gives that f restricted to the image of ∂_{n+1} is zero. Thus f gives a map ζ̂: Ω^{n}(k) → k called the representing homomorphism for ζ ∈ H^{n}(G, k). Two homomorphisms ζ̂ and ζ̂′ represent the same cohomology class if they differ by a homomorphism which factors through a projective module. The only homomorphism Ω^{n}(k) → k that factors through a projective module is the zero homomorphism, so the representing homomorphism is unique (see page 140 in [1] or page 16-17 of [4] for details). The L_{ζ}-modules are defined as follows:

Definition 2.1. Let ζ be a cohomology class in H^{n}(G, k)−{0} for n ≥ 1 and let ζ̂: Ω^{n}(k) → k be the homomorphism representing ζ. We define L_{ζ} as the kernel of the homomorphism ζ̂. When ζ = 0, we set L_{ζ} = Ω(k) ⊕ Ω^{n}(k).

Since the representing homomorphism ζ̂ is uniquely defined, L_{ζ} is well defined up to isomorphism. As a consequence of the definition we have the following diagram:

L_{ζ}  L_{ζ}
↓  ↓
0 ⟶ Ω^{n}(k) ⟶ P_{n−1} ⟶ P_{n−2} ⟶ . ⟶ k ⟶ 0
ζ̂↓  ↓  ∥  ∥  ∥
0 ⟶ k ⟶ P_{n−1}/L_{ζ} ⟶ P_{n−2} ⟶ ... ⟶ k ⟶ 0

In particular, we have

Lemma 2.2. For every ζ ∈ H^{n}(G, k) − {0}, there is an exact sequence

0 → k → Ω^{−1}(L_{ζ}) ⊕ (proj) → Ω^{n−1}(k) → 0

with an extension class corresponding to ζ under the isomorphisms

H^{n}(G, k) ≅ Ext_{kG}^{1}(Ω^{n−1}(k), k).

When L_{ζ} ≠ 0, the above sequence is exact without a (proj) summand.

Proof. The above diagram gives the short exact sequence

0 → k → P_{n−1}/L_{ζ} → Ω^{n−1}(k) → 0.

which gives the desired sequence after applying the isomorphism P_{n−1}/L_{ζ} ≅ Ω^{−1}(L_{ζ})⊕(proj).

When L_{ζ} ≠ 0, the module P_{n−1} is the injective hull of L_{ζ}, hence P_{n−1}/L_{ζ} ≅ Ω^{−1}(L_{ζ}). So, when ζ is not a periodicity generator (for example when G is not a periodic group), then the above sequence is exact without a (proj) summand in the middle. □

In our alternative proof for Theorem 1.2, the main ingredient is the following exact sequence:

Proposition 2.3. If ζ_{1} ∈ H^{r}(G, k) and ζ_{2} ∈ H^{s}(G, k), then there is an exact sequence

0 → Ω^{r}(L_{ζ_{2}}) → L_{ζ_{1}·ζ_{2}} ⊕ (proj) → L_{ζ_{1}} → 0.

Proof. See Lemma 5.9.3 on page 191 of [2]. □

For the rest of the section, we assume G is a (finite) p-group. Recall that

H^{1}(G, 𝔽_{p}) ≅ Hom(G, ℤ/p).

Given a one dimensional class ζ ∈ H^{1}(G, 𝔽_{p}), the kernel of the corresponding homomorphism is usually referred as the kernel of ζ.

Lemma 2.4. Let G be a 2-group, and ζ be a cohomology class in H^{1}(G, 𝔽_{2}) considered as a class in H^{1}(G, k). Then L_{ζ} ≅ Ω(k) ↑_{H}^{G} where H is the kernel of ζ.

Proof. We have the following commutative diagram

L_{ζ}  L_{ζ}
↓  ↓
0 ⟶ Ω(k) ⟶ P_{0} ⟶ k ⟶ 0
ζ̂↓  ↓  ∥
0 ⟶ k ⟶ P_{0}/L_{ζ} ⟶ k ⟶ 0.

from which we obtain that the extension class of the sequence 0 → k → P_{0}/L_{ζ} → k → 0 corresponds to ζ under the isomorphism H^{1}(G, k) ≅ Ext_{kG}^{1}(k, k). There is an extension of the form

0 → k → k ↑_{H}^{G}→ k → 0

with extension class equal to ζ. From the equivalence of the exact sequences we get k ↑_{H}^{G} ≅ P_{0}/L_{ζ} ≅ Ω^{−1}(L_{ζ}) ⊕ (proj). Taking the first Heller shift of this isomorphism, we obtain L_{ζ} ≅ Ω(k ↑_{H}^{G}). Note that for p-groups, Ω(k ↑_{H}^{G}) ≅ Ω(k) ↑_{H}^{G}. We conclude that L_{ζ} ≅ Ω(k) ↑_{H}^{G} . □

In the case where p is an odd prime we have a similar result.

Lemma 2.5. Let G be a finite p-group, and let β(ζ) be the Bockstein of a one dimensional class ζ ∈ H^{1}(G, 𝔽_{p}). Consider β(ζ) as a class in H^{2}(G, k). Then, L_{β(ζ)} ⊕ (proj) has a filtration 0 = M_{0} ⊆ M_{1} ⊆ M_{2} = L_{β(ζ)} ⊕ (proj) with the property M_{2}/M_{1} ≅ Ω(k) ↑_{H}^{G} and M_{1} ≅ Ω^{2}(k) ↑_{H}^{G} where H is the kernel of ζ.

Proof. By Proposition 5.7.6 in [7], there is an extension of the form

0 → k → k ↑_{H}^{G}→ k ↑_{H}^{G}→ k → 0

with extension class equal to β(ζ). Thus, we have a diagram

0 ⟶ Ω^{2}(k) ⟶ P_{1} ⟶ P_{0} ⟶ k ⟶ 0
β̂(ζ)↓  ↓^{f_{1}}  ↓^{f_{0}}  ∥
0 ⟶ k ⟶ k ↑_{H}^{G} ⟶ k ↑_{H}^{G} ⟶ k ⟶ 0

where the left most homomorphism is the representing homomorphism for β(ζ). Thus, we have the diagram

kerf̃_{1} ⟶^{≅} kerf_{0}
↓  ↓
0 ⟶ k ⟶ P_{1}/L_{β(ζ)} ⟶ P_{0} ⟶ k ⟶ 0
∥  ↓f̃_{1}  ↓^{f_{0}}  ∥
0 ⟶ k ⟶ k ↑_{H}^{G} ⟶ k ↑_{H}^{G} ⟶ k ⟶ 0

where kerf_{0} = Ω(k ↑_{H}^{G})⊕(proj). The first vertical short exact sequence in the above diagram gives us the sequence

0 → Ω(k ↑_{H}^{G}) ⊕ (proj) → Ω^{−1}(L_{β(ζ)}) ⊕ (proj) → k ↑_{H}^{G}→ 0.

If we tensor this exact sequence with Ω(k) over k, we get

0 → Ω^{2}(k ↑_{H}^{G}) ⊕ (proj) → L_{β(ζ)} ⊕ (proj) → Ω(k ↑_{H}^{G}) ⊕ (proj) → 0.

Note that we can cancel projective modules at both ends of the sequence since projective kG-modules are also injective. Thus, we get an exact sequence

0 → Ω^{2}(k ↑_{H}^{G}) → L_{β(ζ)} ⊕ (proj) → Ω(k ↑_{H}^{G}) → 0

which gives the desired filtration. □

Throughout the paper, we will come across the situations, where we will need to cancel projective modules from both ends of exact sequences as we did in the above proof. More generally, we will need to cancel projective modules from sections of filtrations of modules. We quote the following lemma from [7]. The proof easily follows from the fact that projective kG-modules are injective.

Lemma 2.6. Suppose that M is a kG-module which has a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M

with M_{i}/M_{i−1} ≅ X_{i} ⊕ P_{i} for some projective modules P_{i}. Then, M ≅ M′ ⊕ P for some projective module P such that M′ has a filtration

0 = M′_{0} ⊆ M′_{1} ⊆ ⋯ ⊆ M′_{n} = M′

with M′_{i}/M′_{i−1} ≅ X_{i} for all i = 1, …, n.

3. The Alternative Proof

The aim of this section is to give a proof for Theorem 1.2 using L_{ζ}-modules. First, we recall Serre’s theorem on vanishing products in group cohomology.

Theorem 3.1 (Serre [9]). Suppose that G is a p-group which is not elementary abelian. Then there is a sequence ζ_{1}, …, ζ_{n} ∈ H^{1}(G, 𝔽_{p}) of nonzero elements such that

ζ_{1}ζ_{2} ⋯ ζ_{n} = 0 if p = 2,
β(ζ_{1})β(ζ_{2}) ⋯ β(ζ_{n}) = 0 if p > 2.

Now, we are ready to prove Theorem 1.2.

Proof of Theorem 1.2. First lets assume p = 2. Let ζ_{1}, …, ζ_{n} be classes in H^{1}(G, 𝔽_{2}) satisfying the conclusion of Serre’s theorem. Then, we have

L_{ζ_{1}···ζ_{n}} ≅ Ω(k) ⊕ Ω^{n}(k).

By Proposition 2.3, for each i = 1, …, n − 1, there is an exact sequence of the form

0 → Ω^{n−i}(L_{ζ_{i}}) → L_{ζ_{i}···ζ_{n}} ⊕ P_{i} → L_{ζ_{i+1}···ζ_{n}} → 0

where P_{1}, …, P_{n−1} are projective modules. By adding projective summands to the last two terms of the exact sequences above, we find exact sequences of the form

0 → Ω^{n−i}(L_{ζ_{i}}) → L_{ζ_{i}···ζ_{n}} ⊕ Q_{i} → L_{ζ_{i+1}···ζ_{n}} ⊕ Q_{i+1} → 0

for i = 1, …, n − 1 where

Q_{i} = ⊕_{k=i}^{n−1}P_{k}.

Using these exact sequences, we obtain a filtration for

L_{ζ_{1}···ζ_{n}} ⊕ Q_{1} ≅ Ω(k) ⊕ Ω^{n}(k) ⊕ Q_{1}

as follows: Let M_{n} = L_{ζ_{1}···ζ_{n}} ⊕ Q_{1} and M_{1} = Ω^{n−1}(L_{ζ_{1}}). Then, M_{n}/M_{1} ≅ L_{ζ_{2}···ζ_{n}} ⊕ Q_{2}. Choose M_{2} such that M_{2}/M_{1} ≅ Ω^{n−2}(L_{ζ_{2}}). Then the exact sequence

0 → Ω^{n−2}(L_{ζ_{2}}) → L_{ζ_{2}···ζ_{n}} ⊕ Q_{2} → L_{ζ_{3}···ζ_{n}} ⊕ Q_{3} → 0

gives that M_{n}/M_{2} ≅ L_{ζ_{3}···ζ_{n}} ⊕ Q_{3} which will be the middle term of the next exact sequence. Continuing this way we obtain a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = Ω(k) ⊕ Ω^{n}(k) ⊕ Q_{1}

with M_{n}/M_{i} ≅ L_{ζ_{i+1}···ζ_{n}} ⊕ Q_{i+1} and M_{i}/M_{i−1} ≅ Ω^{n−i}(L_{ζ_{i}}) for i = 1, …, n. Tensoring the entire filtration by Ω^{−n}(k), and cancelling the projective summands from sections as in Lemma 2.6, we obtain a filtration

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

with L_{i}/L_{i−1} ≅ Ω^{−i}(L_{ζ_{i}}). Note that by Proposition 2.4, we have L_{ζ_{i}} ≅ Ω^{1}(k ↑_{H_{i}}^{G}). Thus,

L_{i}/L_{i−1} ≅ Ω^{−i}(Ω(k ↑_{H_{i}}^{G})) ≅ Ω^{1−i}(k ↑_{H_{i}}^{G}) ≅ Ω^{1−i}(k) ↑_{H_{i}}^{G}

where the last isomorphism is true because G is a p-group. So, the proof for p = 2 is complete. Now, assume p > 2. As above, we can obtain a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = Ω(k) ⊕ Ω^{2n}(k) ⊕ (proj)

such that M_{i}/M_{i−1} ≅ Ω^{2n−2i}(L_{β(ζ_{i})}) for i = 1, …, n. By Proposition 2.5, L_{β(ζ_{i})} ⊕ (proj) has a filtration with sections isomorphic to Ω^{2}(k) ↑_{H}^{G} and Ω^{1}(k) ↑_{H}^{G}. After adding projective modules to each M_{i}, we can assume M_{i}/M_{i−1} has a filtration with sections isomorphic to Ω^{2n−2i+2}(k) ↑_{H}^{G} and Ω^{2n−2i+1}(k) ↑_{H}^{G}. For each i = 1, …, n, let N_{i} be the kG-module satisfying M_{i−1} ⊆ N_{i} ⊆ M_{i} with N_{i}/M_{i−1} ≅ Ω^{2n−2i+2}(k) ↑_{H_{i}}^{G} and M_{i}/N_{i} ≅ Ω^{2n−2i+1}(k) ↑_{H_{i}}^{G}. By taking L_{2i} = M_{i} and L_{2i−1} = N_{i}, tensoring everything with Ω^{−2n}(k), and cancelling the projective summands on the sections, we obtain a filtration

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

where

L_{j}/L_{j−1} ≅ Ω^{1−j}(k) ↑_{H_{i}}^{G}

when j = 2i or j = 2i − 1. This completes the proof. □

Now, we explain briefly how Theorem 1.1 follows from Theorem 1.2. The details of this argument can be found on page 166 of [7]. First note that it is enough to prove Theorem 1.1 for M = k. The general case follows by tensoring everything with M. Also note that if P is a Sylow p-subgroup of G, then k is a summand of k ↑_{P}^{G}. So, it is enough to prove Theorem 1.1 for p-groups. To see this, suppose that there is a kP -module V such that k ⊕ V has a filtration whose sections are induced from elementary abelian p-subgroups. Inducing the entire filtration to G, we get a filtration for k ↑_{P}^{G} ⊕V ↑_{P}^{G}, and hence conclude that k is a direct summand of a finitely generated module which has a filtration with the desired properties.

When G is a p-group, and M is the trivial module k, Theorem 1.1 follows from Theorem 1.2 by an induction. Note that by Theorem 1.2, there is a filtration

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

where L_{i}/L_{i−1} ≅ Ω^{1−i}(k) ↑_{H_{i}}^{G} for i = 1, …, n. If any of the subgroups H_{i} is not an elementary abelian p-group, then we can apply Theorem 1.2 to H_{i} and refine the above sequence further until we reach the stage that all the subgroups involved are elementary abelian p-subgroups.

We note that the following version of Theorem 1.1 is also true.

Theorem 3.2 (Carlson [5]). There exists a finitely generated kG-module V such that k ⊕ V has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n} = k ⊕ V

where for every i = 1, …, n, the sections L_{i}/L_{i−1} are isomorphic to Ω^{n_{i}}(k) ↑_{E_{i}}^{G} for some integer n_{i} and some maximal elementary abelian p-subgroup E_{i} of G.

We will use this version later in Section 5.

4. A Filtration Theorem for L_{ζ}-Modules

The main purpose of this section is to prove Theorem 1.3 stated in the introduction. We also prove an important corollary which will be useful later in Section 5.

Definition 4.1. Let E be an n-fold extension of kG-modules with extension class α in Ext_{kG}^{n}(A, B). Suppose that Ẽ : 0 → Ω^{−n+1}(B) → M → A → 0 is an extension whose class is associated to α under the isomorphism

Ext_{kG}^{n}(A, B) ≅ Ext_{kG}^{1}(A, Ω^{−n+1}(B)).

Then, we say Ẽ is a contraction of E.

Lemma 4.2. Let E be an n-fold extension of kG-modules

E : 0 → B → M_{n−1} → M_{n−2} → ⋯ → M_{0} → A → 0

and let

Ẽ : 0 → Ω^{−n+1}(B) → M → A → 0

be a contraction of E. Then, M ⊕ (proj) has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n} = M ⊕ (proj)

with L_{i}/L_{i−1} ≅ Ω^{1−i}(M_{i−1}) for i = 1, …, n.

Proof. Consider the following commutative diagram

0 ⟶ B ⟶ M_{n−1} ⟶ M_{n−2} ⟶ M_{n−3} ⟶ ...
∥  ↓  ↓  ∥
0 ⟶ B ⟶ I(M_{n−1}) ⟶ K_{n−2} ⟶ M_{n−3} ⟶ ...
↓  ↓
Ω^{−1}(M_{n−1})  Ω^{−1}(M_{n−1})

where K_{n−2} is the push out and I(M_{n−1}) is the injective hull of M_{n−1}. From the second horizontal exact sequence, we obtain an extension

E′ : 0 → Ω^{−1}(B) ⊕ (proj) → K_{n−2} → M_{n−3} → ⋯ → M_{0} → A → 0

whose extension class corresponds to α under the isomorphism

Ext_{kG}^{n−1}(A, Ω^{−1}(B)) ≅ Ext_{kG}^{n}(A, B).

Let

Ẽ : 0 → Ω^{−n+1}(B) → M → A → 0

be a contraction of E. Note that Ẽ is also a contraction of E′. So, by induction, M ⊕ (proj) has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n−1} = M ⊕ (proj)

with L_{i}/L_{i−1} ≅ Ω^{1−i}(M_{i−1}) for i = 1, …, n − 2 and L_{n−1}/L_{n−2} ≅ Ω^{2−n}(K_{n−2}).

To finish the proof, we need to refine the above filtration at the L_{n−1}/L_{n−2} section. For this, consider the exact sequence 0 → M_{n−2} → K_{n−2} → Ω^{−1}(M_{n−1}) → 0. After tensoring this exact sequence with Ω^{2−n}(k), and cancelling projective summands from both ends of the sequence, we get a filtration for Ω^{2−n}(K_{n−2}) ⊕ (proj) with sections isomorphic to Ω^{2−n}(M_{n−2}) and Ω^{1−n}(M_{n−1}). By adding projective summands to L_{n} if necessary, we can assume L_{n−1}/L_{n−2} also have a similar filtration. So, there exists a kG-module L̃_{n−1} such that

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n−2} ⊆ L̃_{n−1} ⊆ L_{n−1} = M ⊕ (proj)

is a filtration having the desired properties. □

Now, we are ready to prove Theorem 1.3.

Proof of Theorem 1.3. Let ζ be the cohomology class in H^{n}(G, k) which is represented by the extension

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

Tensoring the exact sequence in Lemma 2.2 with Ω^{1−n}(k), and cancelling the projective summands from both ends, we obtain a short exact sequence

Ẽ : 0 → Ω^{1−n}(k) → Ω^{−n}(L_{ζ}) ⊕ (proj) → k → 0

with extension class corresponding to ζ under the isomorphism

Ext_{kG}^{n}(k, k) ≅ Ext_{kG}^{1}(k, Ω^{1−n}(k)).

So, Ẽ is a contraction of E. Applying Lemma 4.2, we obtain a filtration

0 = T_{0} ⊆ T_{1} ⊆ ⋯ ⊆ T_{n} = Ω^{−n}(L_{ζ}) ⊕ (proj)

with T_{i}/T_{i−1} = Ω^{1−i}(M_{i−1}) for i = 1, …, n. Tensoring the entire system with Ω^{n}(k), and eliminating the projective summands if necessary, gives the desired filtration. □

Note that Theorem 1.2 follows from Theorem 1.3 as a corollary. To see this, first observe that for a p-group which is not elementary abelian, there is a sequence of maximal subgroups H_{1}, …, H_{n} and an exact sequence

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

such that C_{2i−2} ≅ C_{2i−1} ≅ k ↑_{H_{i}}^{G} for i = 1, …, n and the class of E in Ext_{kG}^{2n}(k, k) is zero. This is just a consequence of Serre’s theorem (see Corollary 3.4 in Carlson [5]). We now apply Theorem 1.3 to this sequence and conclude Theorem 1.2.

In the rest of the section, we study some consequences of Theorem 1.3. We first introduce some more terminology: Given a kG-module M, let J_{G}(M) denote the kernel of the homomorphism

M ⊗_{k} − : Ext_{kG}^{∗}(k, k) → Ext_{kG}^{∗}(M, M)

defined by tensoring an extension with M (over k). Note that J_{G}(M) can also be considered as the annihilating ideal of Ext_{kG}^{∗}(M, M) as a Ext_{kG}^{∗}(k, k)-module. We now recall the following well known theorem.

Theorem 4.3. Let ζ ∈ H^{n}(G, k) and let M be a kG-module. Then, ζ ∈ J_{G}(M) if and only if

L_{ζ} ⊗ M ≅ Ω^{n}(M) ⊕ Ω(M) ⊕ (proj).

Proof. See Proposition 9.7.5 in [7]. □

Combining this theorem with Theorem 1.3, we obtain

Corollary 4.4. Let M be a finitely generated kG-module such that J_{G}(M) includes a product of Bocksteins of one dimensional classes. Then, for each integer s, there exists a finitely generated kG-module V such that Ω^{s}(M) ⊕ V has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{m} = Ω^{s}(M) ⊕ V

where for every i = 1, …, m, we have L_{i}/L_{i−1} ≅ Ω^{s+i−1}(M ↓_{H_{i}}^{G}) ↑_{H_{i}}^{G} for some maximal subgroup H_{i} of G.

Proof. Suppose that ζ = β(u_{1}) ⋯ β(u_{n}) ∈ J_{G}(M) where u_{i}’s are one dimensional classes in H^{1}(G, 𝔽_{p}). Note that ζ is represented by an extension which is the Yoneda splice of extensions of the form

0 → k → k ↑_{H_{i}}^{G}→ k ↑_{H_{i}}^{G}→ k → 0

where H_{i} is the kernel of u_{i}. So, by Theorem 1.3, there is a filtration for L_{ζ} ⊕ (proj)

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{2n} = L_{ζ} ⊕ (proj)

where the jth section L_{j}/L_{j−1} is isomorphic to Ω^{n+1−j}(k) ↑_{H_{i}}^{G} when j = 2i or j = 2i − 1. Tensoring L_{i}’s with M, we obtain a filtration

0 = (L_{0} ⊗ M) ⊆ (L_{1} ⊗ M) ⊆ ⋯ ⊆ (L_{2n} ⊗ M) = (L_{ζ} ⊗ M) ⊕ (proj)

such that

(L_{j} ⊗ M)/(L_{j−1} ⊗ M) ≅ (Ω^{n+1−j}(k) ↑_{H_{i}}^{G}) ⊗ M ≅ Ω^{n+1−j}(M ↓_{H_{i}}^{G}) ↑_{H_{i}}^{G} ⊕(proj)

where j = 2i or j = 2i − 1. By Theorem 4.3, we have L_{ζ} ⊗ M ≅ Ω^{n}(M) ⊕ Ω(M) ⊕ (proj). So, tensoring the entire system with Ω^{s−n}(k), and eliminating the projective summands, we obtain the desired filtration. □

Note that the filtration length m in Corollary 4.4 depends on the number of one dimensional classes whose product of Bocksteins is in J_{G}(M). This suggests the following definition:

Definition 4.5. The cohomology length of a kG-module M, denoted by chl_{G}(M), is defined as the smallest positive integer n such that there exist non-zero elements u_{1}, u_{2}, …, u_{n} ∈ H^{1}(G, 𝔽_{p}) such that

u_{1}u_{2} ⋯ u_{n} ∈ J_{G}(M) if p = 2,
β(u_{1})β(u_{2}) ⋯ β(u_{n}) ∈ J_{G}(M) if p > 2.

If no such integer exists, then we set chl_{G}(M) = ∞.

It is easy to see that if chl_{G}(M) = n, then there is a filtration as in Corollary 4.4 of length 2n. Note that for p = 2, we have a filtration of length n. So, the cohomology length of a module is an interesting invariant to consider if one is interested in finding filtrations like in Corollary 4.4 of shortest length.

Note that there is a notion of cohomology length for p-groups (which are not elementary abelian) as a consequence of Serre’s theorem. The cohomology length of a p-group G, denoted by chl(G), is defined as the minimal m such that the product in Serre’s theorem vanish. We can declare chl(G) = ∞ for groups where the Serre’s theorem does not hold. Then, it is clear that chl(G) = chl_{G}(k). Also note that for any kG-module M, we have chl_{G}(M) ≤ chl(G). In general, the cohomology length of a kG-module M can be much smaller then the cohomology length of G. For example, if G is an elementary abelian p-group and H a maximal subgroup of G, then chl(G) = ∞ whereas chl_{G}(k ↑_{H}^{G}) = 1.

5. Varieties of Modules

In this section we introduce the varieties of modules, and prove Theorem 1.4 and Corollary 1.5 stated in the introduction. As in the previous sections, k denotes a field of characteristic p > 0. We do not assume that k is algebraically closed and denote the algebraic closure of k by K. Let V_{G}(k) denote the maximal ideal spectrum of H^{•}(G, k) where H^{•}(G, k) = H^{∗}(G, k) for p = 2, and H^{•}(G, k) = H^{ev}(G, k), the ring of even dimensional classes, for p > 2. Since H^{•}(G, k) is a finitely generated commutative k-algebra, V_{G}(k) is a finite dimensional homogeneous affine variety, where a point in V_{G}(k) can be viewed as a k-linear ring homomorphism H^{•}(G, k) → K. Two such ring homomorphisms correspond to the same point in V_{G}(k) if and only if they are in the same orbit under the Galois action of Aut_{k}(K).

¿From the above description, it is easy to see that any ring homomorphism f : H^{•}(G, k) → H^{•}(H, k) induces a continuous map f^{∗} : V_{H}(k) → V_{G}(k) of corresponding varieties. In particular, for every H ≤ G, the restriction homomorphism res_{G,H} : H^{•}(G, k) → H^{•}(H, k) induces a map res_{G,H}^{∗} : V_{H}(k) → V_{G}(k) on varieties.

For a finitely generated kG-module M, the support variety of M, denote by V_{G}(M), is defined as the variety of the ideal J_{G}(M) where J_{G}(M) is the annihilator of Ext_{kG}^{∗}(M, M) in H^{•}(G, k). Observe that we have res_{G,H}^{∗}(V_{H}(M)) ⊆ V_{G}(M) as a consequence of the obvious inclusion for ideals res_{G,H}(J_{G}(M)) ⊆ J_{H}(M). So, res_{G,H}^{∗} induces a map res_{G,H}^{∗} : V_{H}(M) → V_{G}(M) on the varieties of the module M.

The following is a list of properties of varieties which we will need in this section.

Lemma 5.1. Suppose that k is a field of characteristic p > 0, G is a finite group, and H is a subgroup of G. Let M, M_{1}, M_{2}, M_{3} be kG-modules, and N a kH-module.

(i) V_{G}(M) = 0 if and only if M is projective.

(ii) V_{G}(M) = V_{G}(Ω^{n}(M)) for all integers n.

(iii) If 0 → M_{1} → M_{2} → M_{3} → 0 is exact, then V_{G}(M_{2}) ⊆ V_{G}(M_{1}) ⋃ V_{G}(M_{3}). In particular V_{G}(M_{1} ⊕ M_{2}) = V_{G}(M_{1}) ⋃ V_{G}(M_{2}).

(iv) V_{G}(N ↑_{H}^{G}) = res_{G,H}^{∗}(V_{H}(N)).

(v) (res_{G,H}^{∗})^{−1}(V_{G}(M)) = V_{H}(M ↓_{H}^{G}).

(vi) V_{G}(M_{1} ⊗ M_{2}) = V_{G}(M_{1}) ∩ V_{G}(M_{2}).

(vii) Let ζ ∈ Ext_{kG}^{n}(k, k). Then V_{G}(L_{ζ}) = V_{G}(ζ) where V_{G}(ζ) is the variety of the ideal generated by ζ.

Proofs of these statements can be found in [2], [4], [7], and [8]. Note that the above properties hold for an arbitrary field if and only if they hold for an algebraically closed field. To see this, observe that the ring homomorphism ϕ : H^{•}(G, k) → H^{•}(G, K) ≅ H^{•}(G, k)⊗_{k} K defined by ϕ(ζ) = 1 ⊗ ζ induces a map ϕ^{∗} : V_{G}(K) → V_{G}(k) on varieties which is finite to one. For a kG-module M, we have J_{G}(K ⊗ M) ≅ K ⊗ J_{G}(M), which gives V_{G}(K ⊗ M)) = (ϕ^{∗})^{−1}(V_{G}(M)). Thus proving these results for V_{G}(M ⊗ K) will give corresponding results for V_{G}(M). For details of this argument, we refer the reader to Theorem 10.4.2 and Remark 10.4.3 in [7].

Lemma 5.2. Let E be an elementary abelian p-group, and M be a collection of the maximal subgroups of E. Suppose M is a finitely generated kG-module such that

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

Then, J_{E}(M) includes a product of Bocksteins of one dimensional classes in H^{1}(E, 𝔽_{p}) whose kernels are in M.

Proof. Applying (ϕ^{∗})^{−1} to the given equality, we obtain

V_{E}(K ⊗ M) ⊆ ⋃_{D∈M} res_{E,D}^{∗}(V_{D}(K ⊗ M ↓_{D}^{E})) ⊆ ⋃_{D∈M} res_{E,D}^{∗}(V_{D}(K)).

For each D ∈ M, choose a one dimensional class x_{D} ∈ H^{1}(E, 𝔽_{p}) such that the kernel of x_{D} is D. Then, it is clear that β(x_{D}) is in the kernel of the restriction map res_{E,D}, so we have

res_{E,D}^{∗}(V_{D}(K)) = V_{E}(ker res_{E,D}) ⊆ V_{E}(β(x_{D}))

for each D. This gives

V_{E}(K ⊗ M) ⊆ V_{E}( ∏_{D∈M} β(x_{D})).

By Hilbert’s Nullstellensatz,

u = ( ∏_{D∈M} β(x_{D}))^{r} ∈ J_{E}(K ⊗ M)

for some r > 0. Since J_{E}(K ⊗ M) = K ⊗ J_{E}(M) and u ∈ H^{•}(E, 𝔽_{p}), u lies in J_{E}(M). □

Now, we are ready to prove Theorem 1.4, which is our main result in this section.

Proof of Theorem 1.4. (ii) ⇒ (i) Suppose that there exists a kG-module V such that M ⊕ V has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n} = M ⊕ V

where for each i = 1, …, n, the ith section L_{i}/L_{i−1} is isomorphic to W_{i} ↑_{H_{i}}^{G} for some subgroup H_{i} ∈ H and some kH_{i}-module W_{i}. Applying the properties (ii) − (iv) listed in Lemma 5.1, we obtain

V_{G}(M) ⊆ ⋃_{i} V_{G}(W_{i} ↑_{H_{i}}^{G}) = ⋃_{i} res_{G,H_{i}}^{∗}(V_{H_{i}}(W_{i}))

which gives

V_{G}(M) ⊆ ⋃_{H∈H} res_{G,H}^{∗}V_{H}(k).

Note that by property (v), we have

V_{G}(M) ∩ res_{G,H}^{∗}(V_{H}(k)) = res_{G,H}^{∗}(V_{H}(M ↓_{H}^{G}))

for all H ∈ H, so we obtain

V_{G}(M) = ⋃_{H∈H} res_{G,H}^{∗}(V_{H}(M ↓_{H}^{G}))

as desired.

(i) ⇒ (ii) Note that by Theorem 3.2, there exists a finitely generated kG-module V such that k ⊕ V has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n} = k ⊕ V

where for every i = 1, …, n, the sections L_{i}/L_{i−1} are isomorphic to Ω^{n_{i}}(k) ↑_{E_{i}}^{G} for some integer n_{i} and some maximal elementary abelian p-subgroup E_{i} of G. Tensoring this system with M, we obtain a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M ⊕ (M ⊗ V)

where M_{i} = M ⊗ L_{i}, and hence

M_{i}/M_{i−1} ≅ (L_{i}/L_{i−1}) ⊗ M ≅ Ω^{n_{i}}(k) ↑_{E_{i}}^{G} ⊗M ≅ (Ω^{n_{i}}(M ↓_{E_{i}}^{G}) ⊕ P_{i}) ↑_{E_{i}}^{G}

where P_{i} is a projective kE_{i}-module. Since E_{i} is a p-group, P_{i} is a free kE_{i}-module. Thus,

M_{i}/M_{i−1} ≅ Ω^{n_{i}}(M ↓_{E_{i}}^{G}) ↑_{E_{i}}^{G} ⊕F_{i}

for some free kG-module F_{i}.

If for some i ∈ {1, …, n}, the ideal J_{E_{i}}(M ↓_{E_{i}}^{G}) includes a product of Bocksteins of one dimensional classes, then by Corollary 4.4 there is a finitely generated kE_{i}-module W such that Ω^{n_{i}}(M ↓_{E_{i}}^{G}) ⊕ W has a filtration

0 = T_{0} ⊆ T_{1} ⊆ ⋯ ⊆ T_{m} = Ω^{n_{i}}(M ↓_{E_{i}}^{G}) ⊕ W

with

T_{j}/T_{j−1} ≅ Ω^{m_{j}}(M ↓_{D_{j}}^{G}) ↑_{D_{j}}^{E_{i}}

where for j = 1, …, m, we have m_{j} = n_{i} + j − 1 and D_{j} is a maximal subgroup of E_{i}. Inducing the entire filtration to G, we obtain a filtration

0 = S_{0} ⊆ S_{1} ⊆ ⋯ ⊆ S_{m} = Ω^{n_{i}}(M ↓_{E_{i}}^{G}) ↑_{E_{i}}^{G} ⊕W ↑_{E_{i}}^{G}

where S_{j} = T_{j} ↑_{E_{i}}^{G} and

S_{j}/S_{j−1} ≅ Ω^{m_{j}}(M ↓_{D_{j}}^{G}) ↑_{D_{j}}^{G} .

Finally, adding the free summand F_{i} to the last section, we obtain a filtration

0 = U_{0} ⊆ U_{1} ⊆ ⋯ ⊆ U_{m} = (M_{i}/M_{i−1}) ⊕ W ↑_{E_{i}}^{G}

where

U_{j}/U_{j−1} ≅ Ω^{m_{j}}(M ↓_{D_{j}}^{G}) ↑_{D_{j}}^{G} ⊕F_{j}

for some free summand F_{j} (only the last section has a nontrivial free summand).

We can use this filtration to refine the original filtration for M ⊕ (M ⊗ V) as follows: Let Û_{j} denote the pre-image of U_{j} under the quotient map M_{i} ⊕ W′ → (M_{i} ⊕ W′)/M_{i−1} where W′ = W ↑_{E_{i}}^{G}. We have

M_{i−1} = Û_{0} ⊆ Û_{1} ⊆ ⋯ ⊆ Û_{m} = M_{i} ⊕ W′

with

Û_{j}/Û_{j−1} ≅ U_{j}/U_{j−1} ≅ Ω^{m_{j}}(M ↓_{D_{j}}^{G}) ↑_{D_{j}}^{G} ⊕F_{j}.

Splicing this into the filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{i−1} ⊆ M_{i} ⊕ W′ ⊆ M_{i+1} ⊕ W′ ⊆ ⋯ ⊆ M_{n} ⊕ W′ = M ⊕ V′

where V′ = (M ⊗ V) ⊕ W′, we obtain a new filtration for M ⊕ V′ where the ith section is replaced by a sequence of modules with quotients induced from maximal subgroups of E_{i}. Applying this process to every section, we can assume that there is a kG-module V such that M ⊕ V has a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M ⊕ V

with

M_{i}/M_{i−1} ≅ Ω^{n_{i}}(M ↓_{E_{i}}^{G}) ↑_{E_{i}}^{G} ⊕F_{i}

where F_{i} is a free kG-module and E_{i} is an elementary abelian p-group such that the ideal J_{E_{i}}(M ↓_{E_{i}}^{G}) does not have any element which is a product of Bocksteins of one dimensional classes. Note that by Proposition 5.2, this implies that for any collection of maximal subgroups M_{i} of E_{i}

V_{E_{i}}(M ↓_{E_{i}}^{G}) ≠ ⋃_{D∈M_{i}} res_{E_{i},D}^{∗}(V_{D}(M ↓_{D}^{E_{i}})).

We claim that this forces E_{i} to be conjugate to a subgroup H in H. To see this, consider the following calculation. We have

V_{E_{i}}(M ↓_{E_{i}}^{G}) = (res_{G,E_{i}}^{∗})^{−1}(V_{G}(M)) ⊆ ⋃_{H∈H}(res_{G,E_{i}}^{∗})^{−1}(res_{G,H}^{∗}(V_{H}(k)))

which gives

V_{E_{i}}(M ↓_{E_{i}}^{G}) ⊆ ⋃_{H∈H} V_{E_{i}}(k ↑_{H}^{G}↓_{E_{i}}^{G}) = ⋃_{H∈H}( ⋃_{H\G/E_{i}} V_{E_{i}}(k ↑_{^{g}H∩E_{i}}^{E_{i}})).

From this we obtain

V_{E_{i}}(M ↓_{E_{i}}^{G}) = ⋃_{L} res_{E_{i},L}^{∗}(V_{L}(M ↓_{L}^{E_{i}}))

where L runs over the subgroups of the form ^{g}H ∩ E_{i} over the set of double coset H\G/E_{i} and over H ∈ H. But, no such equality exists for a collection of proper subgroups, so we must have E_{i} = ^{g}H ∩ E_{i} ≤ ^{g}H for every g ∈ H\G/E_{i} and H ∈ H. Thus E_{i} is conjugate to a subgroup of some H ∈ H for all i = 1, …, n.

To complete the proof, we observe that Ω^{n_{i}}(M ↓_{E_{i}}^{G}) ↑_{E_{i}}^{G}≅ Ω^{n_{i}}(M ↓_{E_{i}^{g}}^{G}) ↑_{E_{i}^{g}}^{G} for all g ∈ G, so by replacing E_{i}’s with their conjugates, we can assume E_{i}’s are subgroups of H. Thus M ⊕ V has a filtration with sections induced from subgroups in H. □

¿From the above proof it is clear that the following is also true.

Corollary 5.3. Let M be a kG-module, and H a collection of subgroups in G. Suppose that there is a kG-module V such that M ⊕ V has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{m−1} ⊆ L_{m} = M ⊕ V

such that for each i = 1, 2, …, m, the quotient module L_{i}/L_{i−1} is induced from subgroups in H. Then there exists a kG-module U such that M ⊕ U has a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M ⊕ U

such that for i = 1, …, n,

M_{i}/M_{i−1} ≅ Ω^{n_{i}}(M ↓_{E_{i}}^{G}) ↑_{E_{i}}^{G} ⊕F_{i}

where F_{i} is a free kG-module, and E_{i} is an elementary abelian p-group included in a subgroup H in H.

In particular this tells us that if we have a filtration for a kG-module M where the sections are induced modules, then there is a kG-module U such that M ⊕ U has a filtration whose sections are given in terms of M up to a free module. This could be useful for proving theorems by induction for modules with complexity strictly smaller than the p-rank of the group.

We conclude the paper with the proof of Corollary 1.5 stated in the introduction.

Proof of Corollary 1.5. It is enough to prove the result under the assumption that k is algebraically closed. By Theorem 1.3, we have a filtration for L_{ζ} with sections isomorphic to Heller shifts of M_{i}’s. Thus

V_{G}(L_{ζ}) ⊆ ⋃_{i=0}^{n−1} V_{G}(M_{i}).

Since the modules M_{0}, …, M_{n−1} are direct sums of modules induced from proper subgroups, there is a collection M = {H_{1}, …, H_{m}} of maximal subgroups of G such that

V_{G}(L_{ζ}) ⊆ ⋃_{j} res_{G,H_{j}}^{∗}(V_{H_{j}}(k)).

As in the proof of Lemma 5.2, we can choose one dimensional classes x_{1}, …, x_{m} ∈ H^{1}(G, 𝔽_{2}) such that the kernel of x_{j} is H_{j} for each j = 1, …, m, and replace each res_{G,H_{j}}^{∗}(V_{H_{j}}(k)) with V_{G}(x_{j}). Note that by property (vii), we have V_{G}(ζ) = V_{G}(L_{ζ}), so we obtain

V_{G}(ζ) ⊆ V_{G}(x_{1}x_{2} ⋯ x_{m}).

Since H^{•}(G, k) = H^{∗}(G, k) is a polynomial algebra, we can apply Hilbert’s Nullstellensatz, and conclude that there exists an integer t > 0, such that (x_{1}x_{2} ⋯ x_{n})^{t} ∈ (ζ). This means (x_{1}x_{2} ⋯ x_{n})^{t} = a · ζ for some a ∈ H^{∗}(G, k). Thus, ζ = λx_{1}^{t_{1}} ⋯ x_{n}^{t_{n}} for some scalar 0 ≠ λ ∈ k and some integers t_{i} ≥ 0.

Let X_{i} be a two point G-set with isotropy H_{i}. Now, consider the G-sphere X defined as the join of G-spheres X_{i} where we take t_{i} copies of X_{i} for each i. The cellular homology of X with coefficients in k gives an extension

0 → H_{m}(X, k) → C_{m}(X, k) → ⋯ → C_{1}(X, k) → C_{0}(X, k) → H_{0}(X, k) → 0

where each C_{m}(X, k) is a kG-module and m = ∑_{i} t_{i}. Note that both H_{0}(X, k) and H_{m}(X, k) are isomorphic to k. We say a k by k extension E comes from a geometric group action on a sphere if after choosing appropriate isomorphism for H_{0}(X, k) ≅ k and H_{m}(X, k) ≅ k, the resulting extension is equivalent to E. Since the Euler class of join of spheres is the product of Euler classes, we can choose these isomorphisms to make the extension class equal to ζ. □

For p > 2, the situation is more complicated. In this case H^{•}(G, k) is not a polynomial algebra, so we do not have unique factorization. For example, if x_{1} and x_{2} are two one dimensional classes then, we have (β(x_{1})+x_{1}x_{2})^{p} = β(x_{1})^{p}. On the other hand, if we assume that ζ is a class lying in the subalgebra generated by Bocksteins of one dimensional classes, then there is a similar conclusion for ζ. Namely, ζ is a non-zero scalar multiple of a product of Bocksteins of one dimensional classes in H^{1}(G, 𝔽_{p}). A linear action on a sphere such that the k-invariant is equal to this product can easily be constructed using a direct sum of one dimensional complex representations. Note that since for every ζ ∈ H^{•}(G, k) the p-th power ζ^{p} lies in the polynomial subalgebra generated by Bocksteins of one dimensional classes, we can also conclude that for every ζ satisfying the conditions of Corollary 1.5, the class ζ^{p} is a non-zero scalar multiple of a product of Bocksteins of one dimensional classes in H^{1}(G, 𝔽_{p}).

Acknowledgments: We thank the referee for careful reading of the manuscript and pointing out a mistake in the earlier version of Corollary 1.5.

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Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

   E-mail address: fatma@fen.bilkent.edu.tr, yalcine@fen.bilkent.edu.tr