Products in negative cohomology

D. J. Benson Mathematical Institute 24–29 St. Giles Oxford OX1 3LB Great Britain

Jon F. Carlson^{∗} Dept. of Mathematics University of Georgia Athens, GA30602 U.S.A.

Abstract

In this paper, we investigate negative degree elements of group cohomology and their products. We give a definition for arbitrary groups, which generalizes the definitions of Tate for finite groups and Farrell for groups of finite virtual cohomological dimension. We find that for finite groups, if we take coefficients in a field of characteristic p, very often all products between elements of negative degree vanish. In particular, this happens when the depth of the positive cohomology ring is greater than one. The latter is the case, for example, when a Sylow p-subgroup has a non-cyclic center.

1 Introduction

Suppose that G is a finite group and that K is a field of characteristic p > 0. It is a classical theorem that KG, the group algebra, is self injective, and hence that projective KG-modules are injective. This fact has profound influence on the module theory for KG. It implies, for example, that a module which is not itself projective cannot have a finite projective resolution. With respect to homology or cohomology, it implies that the usual dimension shifting can proceed in either a positive or a negative direction and over intervals of any length. Of course, negative dimension shifts require that we use Tate cohomology or a similar theory in studying homology. It is necessary that the homology or cohomology groups have both positive and negative gradings. An advantage to this point of view is that we can use a (Tate) duality. A disadvantage is that some of the constructions such as the cup product seem somewhat less natural or behave peculiarly in this setting. Nevertheless, the cup products are defined and are certainly natural, at least in the technical sense. Also they provide occasionally interesting extensions of the usual ring and module structures on cohomology.

∗Partially supported by a grant from the NSF.

Our purpose in this article is to develop some of the properties of cup products in negative gradings. Our concentration will be mainly on the cohomology with trivial coefficients, Ĥ^{∗}(G, K), although some of the results will apply to or can be generalized to other coefficient modules. Briefly stated, our main results concern the existence of nonzero products in negative cohomology. We relate this to the existence of non-zero products in mixed positive-negative degrees and also to the depth of the cohomology ring. We show that products of elements in negative degrees are zero if the depth is two or more. For groups of p-rank two the theorem has a converse. An example shows that the converse is false for groups of larger rank.

The discussion in Section 6 is focused on dualities on certain spectral sequences. While the topic seems at first to be unrelated to the rest of the paper, it was a major motivation for beginning our consideration of negative cohomology. Also the negative products can be seen to fit neatly into the structure of the dualities. A problem with any study such as this, is that it is very difficult to compute examples. Indeed, even when we assert the existence of non-zero products in negative cohomology in this paper, our justification does not come from explicit calculations. Rather the existence is inferred from the structure of the spectral sequences. In Section 7, we illustrate our techniques in a calculation for a group of order 32. It is hoped that the reader will gain insight into both our techniques and our motivation from this example.

Part of this paper was written while the second author was visiting the University of Warwick. He would like to thank the people at the Mathematics Institute for their hospitality and the S.E.R.C. for financial support.

2 Products and Duality

We begin this section with a brief outline of a treatment of group cohomology from which the desired results on cup product can be easily derived. In fact, our treatment gives a definition of cohomology in positive and negative degrees, for an arbitrary (not necessarily finite) group. For G finite, it is not hard to see that this definition gives the usual Tate cohomology. For G of finite virtual cohomological dimension, it gives the Farrell cohomology [8]. Our point of view is similar to that in the theory of derived categories where modules are replaced by projective resolutions. The reader who is familiar with any of the customary algebraic developments of group cohomology (e.g., Mac Lane [10]) will have little trouble filling in the details. We note that cup products in Tate cohomology of finite groups are also described in Cartan and Eilenberg [6], Section XII.4, and in Farrell cohomology in Brown [3], Section X.2.

Let G be a group, and R a commutative ring of coefficients. The tensor product of RG-modules, M ⊗ N, will always be the tensor of the modules over R. The action of G on M ⊗ N is given by g(m ⊗ n) = gm ⊗ gn. A projective resolution,

⋯ → P_{2} ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} →^{ε} M → 0

of an RG-module M will be denoted by (P, ε). The tensor product of projective resolutions or of chain complexes will be assumed to be taken over the ground ring R, and the usual Künneth formula will apply to homology.

Suppose that M and N are RG-modules and that (P, ε) and (Q, η) are projective resolutions of M and N respectively. An almost-chain map of degree n from (P, ε) to (Q, η) is a set of functions μ = {μ_{i}} such that for each i, μ_{i} : P_{i+n} → Q_{n} is an RG-homomorphism and for all but a finite number of integers i, the diagram

P_{i+n} ⟶^{∂} P_{i+n−1}
μ_{i} ↓  ↓ μ_{i−1}
Q_{i} ⟶^{∂} Q_{i−1}

commutes. We assume that P_{i} = {0} = Q_{i} if i < 0 so that the definition makes sense even if i or n is negative. Two almost-chain maps μ, ν : (P, ε) → (Q, η) are almost-chain homotopic if there exists a set σ = {σ_{i}} of RG-homomorphisms, σ_{i} : P_{i+n−1} → Q_{i}, such that for all but a finite number of indices i, μ_{i} − ν_{i} = σ_{i}∂ + ∂σ_{i+1}. The usual details can be easily checked. The composition of two almost-chain maps is an almost-chain map. Degrees are additive. Almost-chain homotopy is an equivalence relation.

Now we can define Êxt_{RG}^{n}(M, N), for any integer n, positive or negative, to be the R-module of almost-chain homotopy equivalence classes of almost-chain maps of degree n from (P, ε) to (Q, η). The usual independence of the choices of resolutions is easily established. In addition, Êxt_{RG}^{n}( , ) is functorial in both variables. We define Ĥ^{n}(G, M) = Êxt_{RG}^{n}(R, M) = Êxt_{ZG}^{n}(Z, M).

There is another useful formulation of the above definition. If M and N are RGmodules, we define \underline{Hom}_{RG}(M, N) to be the quotient of Hom_{RG}(M, N) by maps that factor through a projective module. There is a natural map

\underline{Hom}_{RG}(M, N) → \underline{Hom}_{RG}(ΩM, ΩN).

For G finite, this is an isomorphism, while for G of finite virtual cohomological dimension it is eventually (after enough iterations) an isomorphism. In any case, we have

(∗)  Êxt_{RG}^{n}(M, N) = lim_{\overrightarrow{m}} \underline{Hom}_{RG}(Ω^{n+m}M, Ω^{m}N).

In this formula, m runs over the integers which are large enough so that both m and n+m are positive.

Next, we turn to products. Yoneda products are easily defined in terms of composition of almost-chain maps, and give us a well defined associative pairing

Êxt_{RG}^{m}(M_{2}, M_{3}) ⊗ Êxt_{RG}^{n}(M_{1}, M_{2}) → Êxt_{RG}^{m+n}(M_{1}, M_{3}).

Alternatively, one checks that composition of homomorphisms passes to the limit in equation (∗).

Cup products are slightly trickier, because the tensor product of two almost-chain maps is not necessarily an almost-chain map. So we must use the other formulation, and check that tensor product of maps passes to the limit (using the identification

Ω(M) ⊗ N ≅ Ω(M ⊗ N) ⊕ (projective)

given by Schanuel’s lemma) and defines a pairing

Êxt_{RG}^{m}(M, N) ⊗ Êxt_{RG}^{n}(M′, N′) → Êxt_{RG}^{m+n}(M ⊗ M′, N ⊗ N′).

We may check that the product is well defined and associative. It is also graded commutative in the sense that the following diagram commutes:

Êxt_{RG}^{m}(M, N) ⊗ Êxt_{RG}^{n}(M′, N′) → Êxt_{RG}^{m+n}(M ⊗ M′, N ⊗ N′)
↓≅  ≅ ↓(−1)^{mn}
Êxt_{RG}^{n}(M′, N′) ⊗ Êxt_{RG}^{m}(M, N) → Êxt_{RG}^{m+n}(M′ ⊗ M, N′ ⊗ N)

We are most interested in the case in which M ≅ M′ ≅ R, so that Êxt_{RG}^{n}(R, N) = Ĥ^{n}(G, N). If N ≅ N′ ≅ R, then the identification R ⊗ R ≅ R makes Ĥ^{∗}(G, R) into an associative and graded commutative ring.

As usual, cup products may be expressed in terms of Yoneda products as follows. We write the tensor product α ⊗ β of two maps as the composite of α ⊗ 1 and 1 ⊗ β. In this way, we see for example that cup and Yoneda products on Ĥ^{∗}(G, R) agree, and hence that in this situation Yoneda product is graded commutative. This is not true in general for the Yoneda product on Êxt_{RG}^{∗}(M, M).

From now on, for the rest of this paper, we assume that G is finite and R = K is a field of characteristic p > 0. Furthermore, we only deal with finitely generated KG-modules. We use the symbol M^{∗} to denote the vector space dual Hom_{K}(M, K). The action of G on M^{∗} is given by (gf)(m) = f(g^{−1}m). Also, in this situation, minimal resolutions exist. If (P, ε) is a minimal resolution of M then the KG-module Ker (∂_{n−1}) ≅ ∂_{n}(P_{n}) ≅ Coker (∂_{n+1}) is denoted Ω^{n}(M). Finally, and most importantly, projective KG-modules are injective, so that two actual chain maps are chain homotopic if and only if they are almost-chain homotopic.

Now Ĥ^{−1}(G, K) ≅ K is generated by a class α. Incidentally, α may be regarded as the class of the almost split sequence in Ext_{KG}^{1}(K, Ω^{2}(K)). Tate duality is given by the pairing

⟨ , ⟩ : Ĥ^{i}(G, M) ⊗ Ĥ^{−i−1}(G, M^{∗}) → Ĥ^{−1}(G, M ⊗ M^{∗}) ⟶^{Tr_{∗}} H^{−1}(G, K) → K

where the first map is the cup product, the second is the trace map or evaluation map

Tr : Hom_{K}(M, M) ≅ M ⊗ M^{∗} → K

(sends m ⊗ λ to λ(m)) and the third sends α to 1. If ζ ∈ Ĥ^{ℓ}(G, M^{∗}), γ ∈ Ĥ^{m}(G, K) and θ ∈ Ĥ^{n}(G, M) with ℓ + m + n = −1, then we have (ζγ)θ = ζ(γθ) and hence also

⟨ζγ, θ⟩ = ⟨ζ, γθ⟩.

The following is a most important tool in this study.

Lemma 2.1 The following are equivalent for a KG-module M.

(i) There exists a pair of integers i, j such that i < 0 < j, i + j ≥ 0 and

Ĥ^{i}(G, K) · Ĥ^{j}(G, M) ≠ 0.

(ii) There exists a pair of integers m, n with m < 0, n < 0 and

Ĥ^{m}(G, M^{∗}) · Ĥ^{n}(G, K) ≠ 0.

Moreover if (i) is true then (ii) is true with (m, n) = (−i − j − 1, i) while if (ii) is true then (i) is true with (i, j) = (n, −m − n − 1).

Proof Suppose that (i) holds. Then there exist γ ∈ H^{i}(G, K), θ ∈ H^{j}(G, M) with γθ ≠ 0 and i + j ≥ 0. Then by the duality there exists ζ ∈ H^{−i−j−1} with ⟨ζ, γθ⟩ ≠ 0. In particular, ζ(γθ) and hence ζγ are not zero. So (ii) holds with −i − j − 1 = m < 0 and i = n < 0. The converse is easily obtained by reversing the above argument. □

Lemma 2.2 If the Sylow p-subgroup of G is not cyclic or generalized quaternion then

Ĥ^{m}(G, K) · Ĥ^{−m}(G, K) = 0.

Remark The condition on G is the same as the condition that Ĥ(G, K) is not periodic (Cartan–Eilenberg [6], Section XII.11).

Proof of Lemma If ζ ∈ Ĥ^{m}(G, K), γ ∈ Ĥ^{−m}(G, K), then these elements can be viewed as equivalence classes represented by maps

ζ̂ ∈ Hom_{KG}(Ω^{m}(K), K) and γ̂ ∈ Hom_{KG}(K, Ω^{m}(K)).

Their cup product is represented by the composition ζ̂ ∘ γ̂ : K → K. If m ≠ 0, then the assumptions on the Sylow p-subgroup of G guarantee that Ω^{m}(K) ≅ K̸ and, therefore, ζ̂ ∘ γ̂ = 0. □

3 Depth and Products in Negative Cohomology

In this section we show that non-zero products in negative are somewhat rare and for many groups they do not exist. However the reader should keep in mind that the situation is entirely different if the Sylow p-subgroup of G is cyclic or, when p = 2, quaternion. In these cases the cohomology Ĥ^{∗}(G, K) is periodic and the negative part is simply an extension of the positive part. For example, if G ≅ Z/pZ with p > 2 then

Ĥ^{∗}(G, K) ≅ Λ(u) ⊗ K[x, x^{−1}]

where deg(u) = 1 and deg(x) = 2. Here Ĥ^{∗}(G, K) is a finitely generated ring. We shall see that for a group G of p-rank two or more Ĥ^{∗}(G, K) will not be finitely generated as either a ring or a module over H^{∗}(G, K) = ∑_{n≥0} H^{n}(G, K).

Theorem 3.1 Suppose that the p-rank of G is two or more. If H^{∗}(G, K) is Cohen– Macaulay then

Ĥ^{m}(G, K) · Ĥ^{n}(G, K) = 0

for all m < 0 and n < 0.

Proof Suppose that Ĥ^{∗}(G, K) has non-zero products in negative cohomology. Then by Lemma 2.1, there is a pair of integers m, n with m < 0 < n, m + n > 0 such that γθ ≠ 0 for some γ ∈ Ĥ^{m}(G, K), θ ∈ Ĥ^{n}(G, K). By hypothesis H^{∗}(G, K) has a regular sequence ζ_{1}, ζ_{2}, …, ζ_{r} where r is the p-rank of G. Hence ζ_{1} is not a divisor of zero and ζ_{1}(γθ) ≠ 0. It follows that ζ_{i}^{a}γ ≠ 0 for all a > 0. Therefore there must exist an element γ′ ∈ H^{b}(G, K) with −n_{1} < b < 0, n_{1} = deg(ζ_{1}) and ζ_{1}γ′ ≠ 0. But now, 0 < deg(ζ_{1}γ′) < n_{1} and hence ζ_{1}γ′ ∉ ζ_{1}H^{∗}(G, K). That is, ζ_{1}γ′ is not a multiple of ζ_{1} by an element in nonnegative degree. However for c sufficiently large ζ_{2}^{c}(ζ_{1}γ′) = ζ_{1}(ζ_{2}^{c}γ′) ∈ ζ_{1}H^{∗}(G, K). The consequence of this is that ζ_{2} cannot be a regular element on H^{∗}(G, K)/ζ_{1}H^{∗}(G, K). Hence ζ_{1}, ζ_{2} is not a regular sequence for H^{∗}(G, K) and we have a contradiction. □

Observe that the proof of Theorem 3.1 actually shows that H^{∗}(G, K) cannot have a regular sequence of length two if there are to be non-zero products in negative cohomology. So by definition H^{∗}(G, K) must have depth one in this case. We now invoke a theorem of Duflot.

Theorem 3.2 (Duflot [7]) The depth of H^{∗}(G, K) is at least the p-rank of the center of a Sylow p-subgroup of G.

So we have proved the following:

Theorem 3.3 Suppose there exist integers m < 0, n < 0 such that

Ĥ^{m}(G, K) · Ĥ^{n}(G, K) ≠ 0.

Then H^{∗}(G, K) has depth one, and the center of any Sylow p-subgroup of G has rank one.

Remark The theorem indicates that for many classes of group, with p-rank larger than one, H^{∗}(G, K) has no non-zero products in negative cohomology. These include all groups with abelian Sylow p-subgroups and, for p = 2, all groups with dihedral or extraspecial Sylow 2-subgroups. But, more than that, the theorem and Lemma 2.1 imply that no group whose cohomology is detected on abelian subgroups can have non-zero products in negative cohomology. Such groups would include any group whose Sylow p-subgroups are iterated wreath products of abelian groups. Examples are given by the symmetric groups, and the general linear groups GL_{n}(F_{q}) for q not a power of 2 or p.

Definition We say that an element ζ ∈ Ĥ^{ℓ}(G, K) has ideal growth rate n if the variety of the ideal Ann(ζ) has dimension n. Here Ann(ζ) is the annihilator of ζ in H^{∗}(G, K). This can also be phrased in terms of the growth rate of the graded module H^{∗}(G, K) · ζ = A. Then ζ has ideal growth rate n provided n is the least integer such that lim_{t→∞}(dim A_{t}/t^{n}) = 0. It should be clear that if ζ, γ ∈ Ĥ^{∗}(G, K) then the ideal growth rate of ζγ is at most equal to the ideal growth rate of γ.

Proposition 3.4 Suppose that γ ∈ Ĥ^{n}(G, K) for n < 0. Then the ideal growth rate of γ is either zero or one. Moreover if the rate is zero then ζγ = 0 for all ζ ∈ Ĥ^{m}(G, K) with either m ≥ −n or m < 0.

Proof We may suppose that the p-rank of G is at least two, or equivalently that the Krull dimension of H^{∗}(G, K) is at least two. First suppose that there exists an element ζ ∈ H^{m}(G, K), m ≥ −n such that ζγ ≠ 0. Note that deg(ζγ) > 0 by Lemma 2.2. Suppose that α ∈ H^{ℓ}(G, K), ℓ > 0 is a regular element for H^{∗}(G, K). Then α^{t}ζγ ≠ 0 for all t > 0. Therefore the ideal growth rate of γ is at least one since also α^{t}γ ≠ 0 for all t > 0. To continue we need the following.

Lemma 3.5 Suppose that α ∈ H^{ℓ}(G, K), ℓ > 0, is a regular element for H^{∗}(G, K). Then for any m < 0 the map α̂_{m} : Ĥ^{m−ℓ}(G, K) → Ĥ^{m}(G, K), given by multiplication by α, is surjective.

Proof Let V = {μ ∈ Ĥ^{−m−1}(G, K) | μν = 0 for all ν ∈ α̂_{m}(Ĥ^{m−ℓ}(G, K))}. So an element μ is in V if and only if ⟨μ, ν⟩ = 0 for all ν ∈ α̂_{m}(Ĥ^{m−ℓ}(G, K)), by Tate duality. We wish to show that V = {0} and hence that α̂_{m} is surjective. So suppose that μ ∈ H^{−m−1}(G, K) and μ ≠ 0. Then αμ ≠ 0 and there exists φ ∈ Ĥ^{m−ℓ}(G, K) such that ⟨μα, φ⟩ ≠ 0. Then (μα)φ = μ(αφ) ≠ 0. In particular αφ = α̂_{m}(φ) and ⟨μ, αφ⟩ ≠ 0. Therefore μ ∉ V. We conclude that V = {0}. □

To continue with the proof of the proposition, let γ_{t} ∈ Ĥ^{n−tℓ}(G, K) be an element such that α^{t}γ_{t} = γ. Suppose that γ has ideal growth rate two or more. Then there exists a second element β ∈ H^{∗}(G, K) such that the K-homomorphism

K[α, β] → Ĥ^{∗}(G, K),

given by multiplication by γ, is injective. We may suppose that β is a homogeneous element and that α, β ∈ H^{∗}(G, K) are algebraically independent. We also assume that α and β have the same degree since we can replace them by α^{a} and β^{b} for some a and b if necessary. We now see that the elements α^{i}β^{t−i}γ_{t} are K-linearly independent for 0 ≤ i ≤ t. This is because the elements α^{t}α^{i}β^{t−i}γ_{t} = α^{i}β^{t−i}γ are linearly independent. But α^{i}β^{t−i}γ_{t} ∈ Ĥ^{n}(G, K). So Ĥ^{n}(G, K) has dimension greater than t for all t. We conclude that γ cannot have ideal growth rate greater than one.

Suppose now that there exists an element ζ ∈ Ĥ^{m}(G, K), m < 0 such that ζγ ≠ 0. Then there exist θ ∈ Ĥ^{−m−n−1}(G, K) such that ⟨ζγ, θ⟩ ≠ 0 and θγ ≠ 0. But deg(θγ) = −m − 1 > 0 and we have already seen that the ideal growth rate of γ must be one. □

4 Groups of p-Rank two

The main point of this section is to show that for group with p-rank two the converse of Theorem 3.1 is true. In the next section we will demonstrate that this result does not extend to groups of p-rank three. It appears that in the depth one, rank two case the products in negative cohomology are forced to exist because there is no room for the existence of certain matric Massey products defined in terms of systems of parameters.

Theorem 4.1 Suppose that G is a finite group of p-rank two. If H^{∗}(G, K) is not Cohen– Macaulay then there exists a pair of integers m and n with m < 0, n < 0 and

Ĥ^{m}(G, K) · Ĥ^{n}(G, K) ≠ 0.

Before beginning the proof we need to recall some notation and constructions from [2]. Some of this information is given in greater generality and more detail in Section 6.

Because G has p-rank two, there is a set {ζ_{1}, ζ_{2}} of homogeneous parameters for H^{∗}(G, K) such that ζ_{1}, ζ_{2} is a quasi-regular sequence in the sense of [2]. This means that ζ_{1} is a regular element and if deg(ζ_{i}) = n_{i}, then multiplication by ζ_{2} induces an isomorphism

H^{m}(G, K)/ζ_{1}H^{m−n_{1}}(G, K) → H^{m+n_{2}}(G, K)/ζ_{1}H^{m+n_{2}−n_{1}}(G, K)

provided m ≥ n_{1}. Now let

C = C_{ζ_{1}} ⊗ C_{ζ_{2}} : 0 → C_{t} → ⋯ → C_{0} → 0

be the complex determined by ζ_{1}, ζ_{2} as in [2], and let C^{∗} = Hom_{K}(C, K) be the dual complex. Then t = n_{1} + n_{2} − 2, and H^{∗}(C^{∗}) = Λ(ζ̃_{1}, ζ̃_{2}) the exterior algebra generated by ζ̃_{1}, ζ̃_{2} in degrees n_{1} − 1 and n_{2} − 1 respectively. Let F_{∗} be a doubly infinite projective resolution of K

F : ⋯ → F_{2} → F_{1} → F_{0} →^{∂} F_{−1} → ⋯

where ∂(F_{0}) = K. Consider the spectral sequence whose E_{0} term is

Ê_{0}^{r,s} = Hom_{KG}(F_{r} ⊗ C_{s}, K)

and whose E_{2} term is

Ê_{2}^{r,s} = Ĥ^{r}(G, H^{s}(C^{∗})).

So

Ê_{2}^{∗,∗} ≅ Ĥ^{∗}(G, K) ⊗ Λ(ζ̃_{1}, ζ̃_{2}).

We know that if α ∈ H^{∗}(G, K) and n_{i} = deg(ζ_{i}) then

d_{n_{i}}(αζ̃_{i}) = αζ_{i}

and similarly the coefficient of ζ̃_{j} in d_{n_{i}}(αζ̃_{i}ζ̃_{j}) is αζ_{i}ζ̃_{j}. If n_{i} > n_{j} , so that d_{n_{j}} precedes d_{n_{i}} then the first formula should be taken modulo the image of ζ_{j}, and the second is only defined for α in the kernel of multiplication by ζ_{j}.

We are now ready to prove the theorem. First notice that we may assume that n_{1} < n_{2}. That is, we can replace ζ_{2} by a power of itself without altering the other assumptions. By hypothesis H^{∗}(G, K) is not Cohen–Macaulay. Since ζ_{1} is a regular element it must be that

ζ_{2} : H^{ℓ}(G, K)/ζ_{1}H^{ℓ−n_{1}}(G, K) → H^{ℓ+n_{2}}(G, K)/ζ_{1}H^{ℓ+n_{2}−n_{1}}(G, K)

is not injective for some ℓ and it must also be the case that ℓ < n_{1}. Then for some α ∈ H^{ℓ}(G, K)

d_{n_{2}}((α + Ĥ^{∗}(G, K)ζ_{1})ζ̃_{2}) = 0

and α ∉ Ĥ^{∗}(G, K)ζ_{1}. Now we should recall that this spectral sequence converges to zero. As we see above, all differentials vanish on the class represented by αζ̃_{2} ∈ Ê_{2}^{ℓ,n_{2}−1}. Therefore this class must be in the image of some differential. The only possibility is that

αζ̃_{2} ∈ d_{n_{1}}(Ê_{n_{1}}^{ℓ−n_{1},t}).

That is, for some β ∈ Ĥ^{ℓ−n_{1}}(G, K)

d_{n_{1}}(βζ̃_{1}ζ̃_{2}) = βζ_{1}ζ̃_{2} = αζ̃_{2},

and βζ_{1} = α . Because ℓ − n_{1} < 0 < ℓ we are done by Lemma 2.1. □

Example The following example illustrates why the proof of the proposition works. Let G = SD_{16}, the semi-dihedral group of order 16 and let K be a field of characteristic two. Then

H^{∗}(G, K) = K[x, y, z, w]/(x^{3}, xy, xz, z^{2} − y^{2}w),

where the degrees of x, y, z and w are 1, 1, 3 and 4, respectively. Notice that G has 2-rank two and that {w, y^{2}} is a homogeneous system of parameters with w regular. The dimensions of H^{i}(G, K) , i = 0, 1, … are given by the sequence

1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, …

Let C be the complex determined by w and y^{2} so that

Ê_{2}^{∗,∗} ≅ Ĥ^{∗}(G, K) ⊗ Λ(w̃, ỹ^{2}),

where w̃ and ỹ^{2} are in degrees 3 and 1. So the E_{2} page of the spectral sequence has four rows. The dimensions of the terms Ê_{2}^{r,s} are given in the array

5 4 4 4 3 2 2 2 1 1 2 2 2 3 4 4 4 5 6 ⋯
5 4 4 4 3 2 2 2 1 1 2 2 2 3 4 4 4 5 6 ⋯
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 4 4 4 3 2 2 2 1 1 2 2 2 3 4 4 4 5 6
5 4 4 4 3 2 2 2 1 1 2 2 2 3 4 4 4 5 6

The map d_{2} : Ê_{2}^{r,s} → Ê_{2}^{r+2,s−1} is multiplication by y^{2}. Thus we can check that the dimensions of the terms on the E_{3} page of the spectral sequence are

1 2 2 1 1 2 1 0 1 1 0 0 1 1 0 ⋯
1 1 0 0 1 1 0 1 2 1 1 2 2 1 1 ⋯
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 2 2 1 1 2 1 0 1 1 0 0 1 1 0
1 1 0 0 1 1 0 1 2 1 1 2 2 1 1

The d_{3} differential is zero because of the degrees of w and y^{2}. The next differential is

d_{4} : Ê_{4}^{r,t} → Ê_{4}^{r+4,t−3}

and is multiplication by w. It is necessary that rows number 1 and 3 on the E_{5} page be zero, because there are no other differentials which hit these rows. This means, for example, that the class on the E_{4} page, represented by x^{2} ỹ^{2}, be in the image of d_{4}. We conclude that there are elements u ∈ Ĥ^{−3}(G, K), v ∈ Ĥ^{−2}(G, K) such that wu = x, wv = x^{2}. Then notice that the E_{5} page of the sequence is given by the array

⋯ 0 1 1 0 1 1 0 ⋯
⋯ 0 0 ⋯
⋯ 0 0 ⋯
⋯ 0 0 ⋯
⋯ 0 1 1 0 1 1 0 ⋯

So d_{5} is an isomorphism from the upper left to the lower right in the array.

5 Vanishing Products in Large Rank

Our aim in this section is to show with a single example that all products in negative cohomology can be zero even though the cohomology ring H^{∗}(G, K) has elements with ideal growth rate one. Hence the rank two hypothesis in Theorem 4.1 cannot be removed. In the case of groups of larger rank the elements of ideal growth rate one can be accounted for the spectral sequence by matric Massey products. This will be seen more clearly in Section 7 where we consider this same group in greater detail. As in the previous example we do not actually compute the cup products. Rather the vanishing of the products is implied by general principals.

Example Suppose that p = 2 and that G is the group Γ_{7}a_{2} in the notation of Hall and Senior [9]. The group has order 32 and is presented by generators and relations as G = ⟨a, b, c | ∗⟩ where ∗ consists of the relations a^{4}b = ba^{4}, a^{4}c = ca^{4}, bc = cb, a^{8} = b^{2} = c^{2} = 1, aba^{−1} = bc and aca^{−1} = a^{4}c. Let K be a field of characteristic two. The cohomology ring is generated by elements z, y, x, w, v, u, t, s having degrees 1, 1, 2, 2, 3, 3, 4, 4, respectively. This information is all given in Rusin [11]. The ring relations for H^{∗}(G, K) are

0 = zy = y^{2} = yx = yw = yv = yu = yt
= xw + zu = z^{2}w + w^{2} = wv + zt = zxw + wu

= z^{2}t + wt = x^{3} + zxv + z^{2}s + v^{2} = vu + xt
= x^{2}u + zxt + zws + vt = x^{3}w + z^{2}xt + z^{2}ws + t^{2}
= zxt + ut = x^{2}w + u^{2}.

Thus as a module over the polynomial subring A = K[z, x, s], H^{∗}(G, K) is a direct sum of 5 copies of A generated by 1, w, v, u and t, modulo the relation zu + xw = 0, and one copy of K[s] generated by y.

Proposition 5.1 (a) H^{∗}(G, K) has an element (y) of ideal growth rate one. (b) If m < 0 and n < 0 then

Ĥ^{m}(G, K) · Ĥ^{n}(G, K) = 0.

Proof The first statement is obvious from the ring relations. To prove the second we need to notice that the elements λs^{i}y for λ ∈ K, i ≥ 0 are the only elements of ideal growth rate one. The element s ∈ H^{4}(G, K) is regular for H^{∗}(G, K). Hence if the second statement in the Proposition is false then there exists an element σ^{∗} ∈ Ĥ^{−3}(G, K) such that sσ^{∗} = y. Choose y^{∗} ∈ Ĥ^{−2}(G, K) so that y^{∗}y = α where α ∈ Ĥ^{−1}(G, K) is the dualizing class. Then α = y^{∗}σ^{∗}s and σ = y^{∗}s is an element in H^{2}(G, K) which is dual to σ^{∗}. But then σ has ideal growth rate one. But from Rusin’s calculation [11] we see that there is no such element. □

Actually the proof indicates a more general result. Namely we have the following.

Lemma 5.2 Let G have p-rank at least two. Suppose that H^{n}(G, K) contains a regular element ζ for H^{∗}(G, K). Suppose that Ĥ^{ℓ}(G, K), ℓ < 0, contains an element γ of ideal growth rate one. Then there exist two integers m_{1}, m_{2} with −n < m_{1} < m_{2} < 0 such that H^{m_{i}}(G, K) · H^{n}(G, K) ≠ 0. (In fact, we can take m_{2} = −m_{1} − n − 1).

Proof The existence of an element of ideal growth rate one requires that H^{∗}(G, K) have depth one. Hence by Theorem 3.2, the center of the Sylow p-subgroup of G is cyclic. Now if p > 2 then clearly n = deg(ζ) is even since elements in odd degree are nilpotent. But suppose that p = 2. Then the variety W ⊆ V_{G}(K) corresponding to the ideal generated by the non-regular homogeneous elements has dimension exactly one. Hence it must be equal to the image of V_{Z}(K), the variety of the center Z = Z(P) of a Sylow p-subgroup P of G, under the map induced by restriction. In particular the restriction of any regular element to Z must not be zero. So in the Lyndon–Hochschild–Serre spectral sequence for the pair (P, Z), the restriction of ζ to P must be represented by an element on the fiber

E_{2}^{0,∗} = H^{0}(P/Z, H^{∗}(Z, K)) ≅ K.

But the extension of P/Z by Z is a nontrivial extension and hence d_{2}(E_{2}^{0,s}) ≠ 0, for s an odd integer. Therefore at the E_{3} page, E_{3}^{0,s} = 0 if s is an odd integer. It now follows that any regular element in H^{∗}(G, K) must have even degree.

Now for all t > 0, ζ^{t}γ ≠ 0, and for some t > 0, deg(ζ^{t}γ) > 0. Hence we may replace γ by ζ^{s}γ if necessary, to get that deg(γ) = ℓ < 0 and deg(ζγ) = ℓ + n > 0. By duality there exists an element b ∈ H^{−ℓ−n−1}(G, K) so that b · ζγ ≠ 0. Then bζ ≠ 0 and deg(bζ) = −ℓ − 1 > 0. Because n is even, −ℓ − n − 1 ≠ ℓ. Therefore if we let m_{1} be the smaller of the two numbers ℓ or −ℓ − n − 1 and let m_{2} be the larger, then we derive the conclusion of the proposition. □

6 Duality for the spectral sequences

In this section, we discuss the dualities on the spectral sequences which were used in Section 4. We obtain these dualities by combining the Poincaré duality introduced in [2] with Tate duality. In the interest of completeness we give some details of the constructions used in [2]. If 0 ≠ ζ ∈ H^{n}(G, K) ≅ Ext_{KG}^{n}(K, K) with n ≥ 2, we can represent ζ by a cocycle ζ̂: Ω^{n}(K) → K. We define L_{ζ} to be the kernel of the surjective map ζ̂, and form the pushout diagram

0  0
↓  ↓
L_{ζ}  =  L_{ζ}
↓  ↓
0 → Ω^{n}(K) → P_{n−1} → P_{n−2} → ⋯ → P_{0} → K → 0
↓ζ̂  ↓  ↓  ↓  ∥
0 → K → P_{n−1}/L_{ζ} → P_{n−2} → ⋯ → P_{0} → K → 0.
↓  ↓
0  0

The bottom row is an n-fold extension representing the element ζ ∈ Ext_{KG}^{n}(K, K). We denote by C_{ζ} the chain complex

C_{ζ} : 0 → P_{n−1}/L_{ζ} → P_{n−2} → ⋯ → P_{0} → 0

formed by truncating the bottom row of this diagram. We denote by C′_{ζ} the dual complex, shifted in degree by n−1, so that (P_{n−1}/L_{ζ})^{∗} occurs in degree zero and P_{i}^{∗} occurs in degree n − 1 − i.

C′_{ζ} : 0 → P_{0}^{∗} → P_{1}^{∗} → ⋯ → (P_{n−1}/L_{ζ})^{∗} → 0.

Then (cf. [2], Section 5) there is a homology equivalence C_{ζ} → C′_{ζ}

0 → P_{n−1}/L_{ζ} → P_{n−2} → ⋯ →  P_{0}  → 0
↓  ↓  ↓
0 →  P_{0}^{∗}  → P_{1}^{∗} → ⋯ → (P_{n−1}/L_{ζ})^{∗} → 0.

If ζ_{1}, …, ζ_{m} are homogeneous elements then we may form the tensor products C = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{m}} and C′ = C′_{ζ_{1}} ⊗ ⋯ ⊗ C′_{ζ_{m}} and obtain a homology equivalence θ : C → C′. Note that C′ = C^{∗}[t], where t = ∑_{i=1}^{m}(n_{i} − 1).

Now the modules C′_{i} are projective for i ≠ 0, because they are direct sums of tensor products involving at least one projective module. Letting M = C′_{0}, it is shown in [1] Section 3 (see also [2] Section 4) that M is projective if and only if H^{∗}(G, K) is finitely generated as a module over the subring generated by ζ_{1}, …, ζ_{m} (i.e., if and only if ζ_{1}, …, ζ_{m} “cover” the cohomology variety V_{G}). In case M is projective, the map C → C′ described above is a chain homotopy equivalence.

Now let

F : ⋯ → F_{1} → F_{0} ⟶^{δ_{0}} F_{−1} ⟶^{δ_{−1}} F_{−2} → ⋯

be a doubly infinite projective resolution of K. That is, δ_{0}(F_{0}) = Ker (δ_{−1}) ≅ K. Then Tate duality is given by a chain homotopy equivalence between F and its dual (shifted in degree by −1)

⋯ → F_{−2}^{∗} → F_{−1}^{∗} → F_{0}^{∗} → F_{1}^{∗} → ⋯
↓  ↓  ↓  ↓
⋯ → F_{1} → F_{0} → F_{−1} → F_{−2} → ⋯

Indeed, one usually takes F_{−n−1} = F_{n}^{∗}, so that this map is an isomorphism of complexes. Putting these dualities together, we obtain a map of double complexes

F^{∗} ⊗ C → F ⊗ C^{∗}

of bidegree (−1, −t), or in other words a map

(F ⊗ C′)^{∗} → F ⊗ C′

of bidegree (−1, t).

Now consider the spectral sequences which begin with the E_{0} term

Ê_{0}^{r,s} = Hom_{KG}(F_{r} ⊗ C′_{s}, K).

If we first take homology with respect to the boundary homomorphism on (F, δ), then we get

(†)  Ẽ_{1}^{r,s} = \{ \begin{matrix}Êxt_{KG}^{r}(M, K) & if s = 0 \\ 0 & if s ≠ 0. \end{matrix}

Of course, there are no further differentials in this direction. On the other hand, suppose we first take homology with respect to the boundary on C′. Then we get

Ê_{1}^{r,s} = Hom_{KG}(F_{r} ⊗ H_{s}(C′), K)

The next differential is that induced by the boundary on F and we get

Ê_{2}^{r,s} = Êxt_{KG}^{r}(H_{s}(C′), K).

The spectral sequence must converge to the cohomology of the total complex, which is Êxt_{KG}^{∗}(M, K).

The above map (F ⊗ C′)^{∗} → F ⊗ C′ of bidegree (−1, t) is given by a chain homotopy equivalence in the first variable and a homology equivalence in the second. So passing to G-fixed points, we obtain a map

Ê_{0}^{r,s} → (Ê_{0}^{−1−r,t−s})^{∗}.

This induces a map

Ê_{1}^{r,s} → (Ê_{1}^{−1−r,t−s})^{∗}

which is a chain homotopy equivalence on each row (by the above identification of Ê_{1}^{r,s}). So it induces an isomorphism on the E_{2} page

Ê_{2}^{r,s} →^{≅} (Ê_{2}^{−1−r,t−s})^{∗},

and hence on all further pages.

The adjoints of these maps are non-singular pairings

⟨ , ⟩_{q} : Ê_{q}^{∗,∗} ⊗ Ê_{q}^{∗,∗} → K[−1, t] (q ≥ 2)

where the right-hand side consists of the trivial module K in bidegree (−1, t).

On the other hand, if we follow this pairing through the other (Ẽ) spectral sequence (†) of this double complex, we see that on the total complex it is given as follows. We have M^{∗} ≅ Ω^{t}(M), so that

Êxt_{KG}^{−s−1}(M^{∗}, K) ≅ Êxt_{KG}^{−s−1}(Ω^{t}(M), K) ≅ Êxt_{KG}^{t−s−1}(M, K)

and the pairing on the cohomology of the total complex agrees with the Tate duality pairing

Êxt_{KG}^{s}(M, K) ⊗ Êxt_{KG}^{−s−1}(M^{∗}, K) → K

via this isomorphism.

Next, we discuss products on the above spectral sequence. For this purpose, we make the assumptions that n_{i} = deg(ζ_{i}) is an even integer if p > 2, and that ζ_{i} annihilates the cohomology ring Ext_{KG}^{∗}(L_{ζ_{i}}, L_{ζ_{i}}) for every i = 1, …, n. This latter assumption is assured if p > 2, but may fail for careless choices if p = 2. It is shown in [4] that under these assumptions on the ζ_{i}’s, there is a diagonal map Δ_{i} : C′_{ζ_{i}} → C′_{ζ_{i}} ⊗ C′_{ζ_{i}} which is a KG-chain map. It makes the ordinary cohomology of C′_{ζ_{i}} into an exterior algebra H^{∗}(C_{ζ_{i}}^{∗}) = H^{∗}(Hom_{K}(C_{ζ_{i}}, K)) = Λ(ζ̃_{i}), on a single generator ζ̃_{i} in degree n_{i} − 1. By the Künneth theorem we have

H^{∗}(C^{′∗}) = Λ(ζ̃_{1}, …, ζ̃_{n}).

Combining this product structure with the cup product in Tate cohomology, we have a ring structure on the E_{2} page,

Ê_{2}^{∗,∗} = Ĥ^{∗}(G, H^{∗}(C^{∗})) = Ĥ^{∗}(G, K) ⊗ Λ(ζ̃_{1}, …, ζ̃_{n}).

The boundary map on the E_{2} term is a differential in the sense that

d_{2}(ηθ) = d_{2}(η)θ + (−1)^{deg(η)}η d_{2}(θ).

The product passes down to a product on each page Ê_{q}, and the boundary homomorphism d_{q} is also a differential. Of course, these products are compatable with the multiplication

Êxt_{KG}^{∗}(M, K) ⊗ Êxt_{KG}^{∗}(M, K) → Êxt_{KG}^{∗}(M, K)

given by the diagonal approximation M → M ⊗ M as in [4].

Now let ε : C → K be the augmentation, and let ε^{∗} : K[t] → C′ be the dual map. We also have a map from K regarded as a chain complex concentrated in degree zero to the chain complex of homomorphisms Hom_{K}(C, C) sending 1 to the identity homomorphism. This gives rise to a map η : K[t] → C′ ⊗C, dual to the evaluation map η^{∗} : C ⊗C′ → K[t]. It is easy to check that the following diagram commutes.

K[t] →^{ε^{∗}}  C′
↓η  ↓Δ
C′ ⊗ C ⟶^{1⊗θ} C′ ⊗ C′

It follows that the pairings given above on Ê_{q} are related to the product as follows. We write α_{q} for the element in Ê_{q}^{−1,t} dual to the identity element, and note that α_{q} is non-zero provided Ê_{q} is not identically zero. Then the pairing is given by the formula

η · θ = ⟨η, θ⟩_{q} α_{q}.

Proposition 6.1 Let ζ_{1}, …, ζ_{n} ∈ H^{∗}(G, K) be homogeneous elements such that ζ_{i} annihilates Ext_{KG}^{∗}(L_{ζ_{i}}, L_{ζ_{i}}) for each i (we don’t know whether the latter hypothesis is really necessary; see the above discussion). Let t = ∑(n_{i} − 1), where n_{i} = deg ζ_{i}. Then for the spectral sequence Ê given as above we have the following:

(i) There is a nondegenerate pairing

⟨ , ⟩_{q} : Ê_{q}^{r,s} ⊗ Ê_{q}^{−r−1,t−s} → K

given by η · θ = ⟨η, θ⟩_{q} α_{q}.

(ii) The pairing on the E_{∞} page agrees up to filtration with the Tate duality

Êxt_{KG}^{r}(M, K) ⊗ Êxt_{KG}^{−r−1}(M^{∗}, K) → K.

In particular, If Êxt_{KG}^{∗}(M, K) is non-zero, then α_{q} ≠ 0 for all q ≥ 2.

(iii) Suppose that ζ_{1}, …, ζ_{m} is an irredundent homogeneous system of parameters for H^{∗}(G, K). That is, H^{∗}(G, K) is finitely generated as a module over K[ζ_{1}, …, ζ_{m}] and m is the Krull dimension of H^{∗}(G, K). Then α_{t+1} ≠ 0, α_{t+2} = 0.

Proof We have already proved part (i) and the first part of (ii). The second part of (ii) follows immediately.

Part (iii) is explained by the fact that the image of the class α_{t+1} under d_{t+1} must be the last survivor [2]. Recall that the last survivor is an element, or actually a class of elements, in E_{∞}^{t,0} of the spectral sequence for

E_{0}^{r,s} = Hom_{KG}(P_{r} ⊗ C′_{s}, K)

where (P, ε) is a projective resolution of K (non-negative degrees only). Because it is not zero on the E_{∞} page, it can only be the image of α_{t+1} and only by the differential d_{t+1} in the spectral sequence Ê_{q}^{r,s}. □

There are several variations which can be made on the above development. One would be to force all of the ζ_{i}’s into degree two by degree shifting. Then the E_{2} page would look like the Koszul complexes as in [5]. Another variation is the concatenation of the spectral sequences as in the following.

Let ζ_{1}, …, ζ_{m} ∈ H^{∗}(G, K) be a homogeneous set of parameters for H^{∗}(G, K) as in Proposition 6.1 (iii). Then C′ = C^{(1)} ⊗ C^{(2)} where C^{(1)} = C′_{ζ_{1}} ⊗ ⋯ ⊗ C′_{ζ_{m_{1}}} and C^{(2)} = C′_{ζ_{m_{1}+1}} ⊗ ⋯ ⊗ C′_{ζ_{m}}. Let t_{1} = ∑_{i=1}^{m_{1}}(n_{i} − 1), t_{2} = ∑_{i=m_{1}+1}^{m}(n_{i} − 1), so that t = t_{1} + t_{2}. Now consider the spectral sequence which begins with

Ê_{0}^{r,s} = Hom_{KG}((F ⊗ C^{(1)})_{r} ⊗ C_{s}^{(2)}, K).

To get the E_{1} page, take the cohomology with respect to the differential induced by the boundary on C^{(2)}. We get

Ê_{1}^{r,s} = Hom_{KG}((F ⊗ C^{(1)})_{r} ⊗ H_{s}(C^{(2)}), K) ≅ Hom_{KG}((F ⊗ C^{(1)})_{r}, H^{s}(C^{(2)∗})).

Then the E_{2} page has terms

Ê_{2}^{r,s} = Êxt_{KG}^{r}(M^{(1)}, H^{s}(C^{(2)∗}))

where M^{(1)} = C_{0}^{(1)}. Under the same assumptions as before, this is a ring

Ê_{2}^{∗,∗} = Êxt_{KG}^{∗}(M^{(1)}, K) ⊗ Λ(ζ̃_{m_{1}+1}, …, ζ̃_{m}),

and the spectral sequence converges to zero. Using the same techniques as before we can prove the following.

Proposition 6.2 There is a nondegenerate duality pairing Ê_{2}^{r,s} ⊗ Ê_{2}^{−r−1+t_{1},t_{2}−s} → K obtained by combining the duality pairing on H^{∗}(C^{(2)}) with Tate duality on Êxt_{KG}^{∗}(M, K). Under the assumptions giving rise to a coproduct on C^{(1)} and C^{(2)} as above, the successive pages of the spectral sequence have a ring structure, relative to which the boundary maps are differentials. Moreover each successive page of the spectral sequence inherits a nondegenerate duality pairing:

Ê_{q}^{r,s} ⊗ Ê_{q}^{−r−1+t_{1},t_{2}−s} → K.

7 Γ_{7}a_{2} revisited

In the light of the duality results of the last section, we look again at the example of the group G = Γ_{7}a_{2}, p = 2, discussed in Section 5.

By [11], H^{∗}(G, K) has Poincaré series

p(t) = \frac{1 − t + t^{2}}{(1 − t)^{3}(1 + t^{2})}.

We choose the homogeneous set of parameters

ζ_{1} = s, ζ_{2} = x, ζ_{3} = z^{2}

in degrees 4, 2 and 2, respectively. Note that s is a regular element and so H^{∗}(G, K)/s · H^{∗}(G, K) has Poincaré series

(1 − t^{4})p(t) = (1 + t^{3})/(1 − t)^{2}.

Therefore the dimensions of the cohomology H^{i}(G, K)/s H^{i−4}(G, K) are

1, 2, 3, 5, 7, 9, 11, 13, …

for i = 0, 1, 2, … respectively. Let

C^{(1)} = C′_{ζ_{1}}, C^{(2)} = C′_{ζ_{2}} ⊗ C′_{ζ_{3}}.

We want to look at the spectral sequence of the double complex Hom((F ⊗C^{(1)})⊗C^{(2)}, K) as in Proposition 6.2. In the first diagram we show the dimensions of the K-spaces

Ê_{2}^{r,s} = Êxt_{KG}^{r}(M, H^{s}(C^{(2)∗}))

where M = C_{0}^{(1)}. Notice that the left side is offset in order to reflect the asymmetry in the duality with respect to Êxt_{KG}^{∗}(M, K). Then the rows of the diagram are given as follows.

row  j < 0  j ≥ 0
5  Ĥ^{j}(G, K)z̃^{2} x̃s̃
4 Ĥ^{j}(G, K)x̃s̃ ⊕ Ĥ^{j}(G, K)z̃^{2} s̃
3  Ĥ^{j}(G, K)s̃
2  Ĥ^{j}(G, K)z̃^{2}x̃
1  Ĥ^{j}(G, K)x̃ ⊕ Ĥ^{j}(G, K)z̃^{2}
0  Ĥ^{j}(G, K)

The dimensions are given by the following.

11 9 7 5 3 2 1  Ê_{2}^{∗,∗}
⋯ 22 18 14 10 6 4 2
11 9 7 5 3 2 1
1 2 3 5 7 9 11
2 4 6 10 14 18 22 ⋯
1 2 3 5 7 9 11

The d_{2} differential is a combination of multiplication by x and by z^{2}. For convenience we take these one at a time. After multiplication by x we have

4 4 4 4 2 2 1
⋯ 4 4 4 4 2 3 1
0 0 0 0 0 1 0
0 1 0 0 0 0 0
1 3 2 4 4 4 4 ⋯
1 2 2 4 4 4 4

Then after multiplication by z^{2} we get the E_{3} page of the sequence

\begin{aligned}& 0 & 1 & 2 & 3 & 1 & 2 & 1 \\ ⋯ & 0 & 0 & 0 & 1 & 0 & 2 & 0 \\ & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ & & & & & & & & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ & & & & & & & & 0 & 2 & 0 & 1 & 0 & 0 & 0 & ⋯ \\ & & & & & & & & 1 & 2 & 1 & 3 & 2 & 1 & 0 \end{aligned}  (7.1)

which is a finite complex. The following diagram shows the elements which represent the classes in the E_{3} page on the right side of (7.1). The classes on the left are represented by the dual elements

column 0  1  2  3  4 5

row

2  yx̃ z̃^{2}
1  y z̃^{2}, yx̃  u z̃^{2} + zwx̃
0  1 z, y w zw, v, u zv, t zt

Notice now that for the right side of Diagram (7.1), the total dimensions are

1, 2, 3, 4, 3, 1

respectively in degrees 0, …, 5. The next differential is a matric Massey product

d_{3}(−) = ⟨−| x, z^{2}⟩.

After this differential the cohomology on the left side must be H^{n}(Hom_{KG}(C, K)) where C = C^{(1)} ⊗ C^{(2)}. Because C satisfies a duality it is necessary that

dim H^{n}(Hom_{KG}(C, K)) = dim H^{5−n}(Hom_{KG}(C, K)).

We conclude from this that the d_{3} differential cannot be zero on the class represented by the element y z̃^{2} x̃. Hence the E_{4} page looks like the following.

0 1 1 3 1 2 1  Ê_{4}^{∗,∗} = Ê_{5}^{∗,∗}
⋯ 0 0 0 1 0 2 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 2 0 1 0 0 0 ⋯
1 2 1 3 1 1 0

Notice now that the total dimensions on the left side of the diagram are given by the sequence 1, 2, 3, 3, 2, 1, as required by the duality. A quick check reveals that ⟨y | x, z^{2}⟩ ≡ vz modulo the ideal generated by z^{2} and x. This is because z must annihilate the Massey product modulo the ideal of z^{2} and x.

There is no d_{4} differential because of the degrees of the homogeneous parameters. So the E_{4} page coincides with the E_{5} page. Because the spectral sequence converges to zero, the classes on the first and fourth rows cannot survive to the E_{6} page. So we must have that

d_{5}(y^{∗}s̃x̃) = ⟨y | s, x⟩ ≡ u modulo (s, x)
d_{5}(u^{∗}s̃x̃) = ⟨u | s, x⟩ ≡ y modulo (s, x)
d_{5}(y^{∗}s̃z̃^{2}) = ⟨y | s, z^{2}⟩ ≡ wz modulo (s, z^{2}).

The E_{6} page then looks like

1 1 1 1 1 1  Ê_{6}^{∗,∗}

0 0

0 0 1 1 1 1 1 1 Of course E_{7}^{∗,∗} = 0. That is, d_{6}(−) = ⟨− | s, x, z^{2}⟩ is given by

\begin{matrix}z^{∗}t^{∗}γ → 1 & v^{∗}γ → w \\ t^{∗}γ → z & z^{∗}γ → t \\ w^{∗}γ → v & 1^{∗}γ → zt \end{matrix}

where γ = s̃x̃ z̃^{2} and the images are taken modulo the ideal (s, x, z^{2}).

References

[1] D. J. Benson and J. F. Carlson. Complexity and multiple complexes. Math. Zeit. 195 (1987), 221–238.

[2] D. J. Benson and J. F. Carlson. Projective resolutions and Poincaré duality complexes. To appear.

[3] K. S. Brown. Cohomology of groups. Graduate texts in mathematics 87, Springer-Verlag, Berlin-New York 1982.

[4] J. F. Carlson. Products and projective resolutions. Arcata Conference on Representations of Finite Groups. Proc. Sym. Pure Math. vol. 47 (1987), 399–408.

[5] J. F. Carlson. Projective resolutions and degree shifting for cohomology and group rings. Proceedings of the Tsukuba workshop on Representations of Algebras. To appear.

[6] H. Cartan and S. Eilenberg. Homological Algebra. Princeton University Press, Princeton 1956.

[7] J. Duflot. Depth and equivariant cohomology. Comm. Math. Helvetici 56 (1981), 627– 637.

[8] F. T. Farrell. An extension of Tate cohomology to a class of infinite groups. J. Pure Appl. Algebra 10 (1977), 153–161.

[9] M. Hall, Jr. and J. K. Senior. The Groups of Order 2^{n} (n ≤ 6). Macmillan, New York 1964.

[10] S. Mac Lane. Homology. Springer-Verlag, Berlin-New York, 1963.

[11] D. Rusin. The cohomology of groups of order 32. Math. Comp. 53 (1989), 359–385