Periodic modules with large period

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.

1 Introduction

One of the goals of modular representation theory is to understand projective resolutions of finitely generated modules for a modular group algebra. Eisenbud [7] has proved that a module whose minimal resolution is bounded in dimension is periodic, in the sense that the modules and maps in the minimal resolution repeat with a certain period. The period is bounded by the largest degree of a non-nilpotent generator of cohomology. If G is a p-group and K is a field of characteristic p, the period of an indecomposable periodic KGmodule is always of the form p^{a} if p = 2, and p^{a} or 2p^{a} if p is odd (see Proposition 2.2). In this paper, we prove that for p = 2 the integer a may be chosen arbitrarily large, by choosing the group and the module appropriately.

Theorem 1.1 Given a positive integer a, and a field K of characteristic 2, there exists a finite 2-group G and a periodic KG-module M whose period is exactly 2^{a}.

The groups G in question may be taken to be extraspecial 2-groups of width 2a. The calculation depends on Quillen’s calculation [8] of the cohomology of the extraspecial 2groups. The only obstruction to proving a similar theorem for p odd by the same techniques is a lack of understanding of the cohomology of odd extraspecial groups. With the current state of knowledge [5] we can prove that there exist modules of period exactly 2p^{a} for a = 0, 1 or 2 for suitable extraspecial groups.

The proof is based on a method of Andrews [2]. He proves that given a group extension

1 → N → G → Ḡ → 1

in which N is a normal cyclic subgroup of G of order p, and given a KḠ-module M which is regarded as a KG-module by inflation, if M is periodic then the exact period of M is determined by the variety VḠ(M). See Section 2 for a brief summary of the theory of varieties for modules. Let ε = 1 if p = 2 and ε = 2 if p is odd, so that since N acts trivially on M, ε must divide the period of M if it is periodic. Define subvarieties V_{r} of VḠ as follows. Let Ñ denote the sum of the elements of N, as an element of the group algebra KN. Writing Ω_{G}^{εr}(K) for the εrth kernel in the minimal resolution of K as a KG-module, we set V_{r} = VḠ(ÑΩ_{G}^{εr}(K)).

∗Partially supported by a grant from the NSF.

Theorem 1.2 (Andrews) Let K be a field of characteristic p. Suppose that

1 → N → G → Ḡ → 1

is a central extension with N cyclic of order p. Let M be an indecomposable KḠ-module regarded as a KG-module by inflation. Set V_{r} = VḠ(ÑΩ_{G}^{εr}(K)) as above. Then M is periodic of period dividing εr if and only if VḠ(M) ∩ V_{r} = {0}.

For a proof of this theorem, see Section 3. Our contribution is to show how to calculate the varieties V_{r}. By a theorem of Alperin and Evens [1], to calculate V_{r} it suffices to consider the case where Ḡ = E is an elementary abelian p-group. In this case, one may restrict ones attention to r a power of p, say r = p^{a}. In Section 5 we prove the following theorem.

Theorem 1.3 Let K be a field of characteristic p. Suppose that

1 → N → G → Ḡ → 1

is a central extension with N cyclic of order p and Ḡ a p-group. Denote by I the kernel of the partial inflation map H^{∗}(Ḡ, K) = E_{2}^{∗,0} → E_{εp^{a}+1}^{∗,0} in the Lyndon–Hochschild–Serre spectral sequence

E_{2}^{∗∗} = H^{∗}(Ḡ, H^{∗}(N, K)) ⇒ H^{∗}(G, K).

Then

V_{p^{a}} = VḠ(I).

In case p = 2, the kernels of the partial inflation maps in the above theorem follow easily from Quillen’s calculation [8] of the cohomology of the extraspecial 2-groups. In particular, V_{2^{a}} has codimension exactly a as a subvariety of V_{E}(K), whenever 2^{a} is at most the index of a maximal elementary abelian subgroup of G. In particular, if G is an extraspecial group of order 2^{1+2n} of the appropriate type, then this allows us to produce examples of inflated modules M, which are periodic as KG-modules and which have the property that V_{E}(M) ∩ V_{2^{a}} ≠ 0 for all a < n. Applying Andrews’ theorem, the modules will have period exactly 2^{n}. Theorem 1.1 then follows.

2 Some preliminaries

Throughout the paper G denotes a finite group and K an algebraically closed field of characteristic p > 0. We set ε = 1 if p = 2 and ε = 2 if p is odd. All KG-modules are assumed to be finitely generated left modules. Unless otherwise noted the tensor product of two KG-modules M and M′ is the tensor product over K, M ⊗ M′ = M ⊗_{K} M′, with diagonal G-action.

Every KG-module M has a projective cover P →^{φ} M. That is, P is a projective KG-module, φ is an epimorphism, and the kernel of φ, which we denote Ω_{G}(M), has no projective summands. Inductively we define Ω_{G}^{n}(M) = Ω_{G}(Ω_{G}^{n−1}(M)) for n > 1. We should note that, by Schanuel’s lemma, if

0 → M_{n} → Q_{n−1} → ⋯ → Q_{0} → M → 0

is exact and if each Q_{j} is projective then M_{n} ≅ Ω_{G}^{n}(M) ⊕ Q for some projective module Q. Because KG is self-injective, the modules Ω_{G}^{n}(M) can be defined in a dual fashion for n < 0. When there is no possibility of confusion we write Ω^{n} instead of Ω_{G}^{n}. Recall that if P is projective then for any KG-module M, M ⊗ P is also projective. It follows that M ⊗ Ω^{n}(M′) ≅ Ω^{n}(M ⊗ M′) ⊕ Q for some projective Q.

The cohomology ring H^{∗}(G, K) ≅ Ext_{KG}^{∗}(K, K) is a finitely generated, graded Kalgebra which is commutative modulo its Jacobson radical. Consequently its maximal ideal spectrum is a finite dimensional, homogeneous affine variety which we denote V_{G}(K). If M is a KG-module, let I_{G}(M) denote the annihilator in H^{∗}(G, K) of the cohomology ring Ext_{KG}^{∗}(M, M) ≅ H^{∗}(G, Hom_{K}(M, M)). Let V_{G}(M) be the subvariety of V_{G}(K) associated to I_{G}(M). That is, it is the set of all maximal ideals which contain I_{G}(M). It is a closed homogeneous subvariety. In the following we state some of the properties of varieties which are needed later.

Proposition 2.1 Let M, M_{1}, M_{2} and M_{3} be KG-modules.

(i) If 0 → M_{1} → M_{2} → M_{3} → 0 is an exact sequence then V_{G}(M_{i}) ⊆ V_{G}(M_{j})∪V_{G}(M_{k}) for {i, j, k} = {1, 2, 3}.

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

(iii) V_{G}(Ω^{n}(M)) = V_{G}(M).

(iv) V_{G}(M_{1} ⊕ M_{2}) = V_{G}(M_{1}) ∪ V_{G}(M_{2}).

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

(vi) If 0 ≠ ζ ∈ H^{m}(G, K) is represented by a cocycle ζ̂ : Ω^{m}(K) → K, then ζ̂ is surjective, and the variety of its kernel L_{ζ} = Ker(ζ̂) is the hypersurface V_{G}⟨ζ⟩ determined by regarding ζ as an element of the coordinate ring of V_{G}.

(vii) Let V be a closed homogeneous subvariety of V_{G}. Then there is a KG-module M with V_{G}(M) = V. If V is irreducible then M may be chosen to be indecomposable.

Proof See [6] or [4]. □

Before proceedings further, we should mention one special case. Suppose that G = E = ⟨g_{1}, …, g_{n}⟩ is an elementary abelian group of order p^{n}. If p = 2, then

H^{∗}(E, K) = K[x_{1}, …, x_{n}]

where x_{1}, …, x_{n} are in degree 1. If p > 2 then

H^{∗}(E, K) = K[x_{1}, …, x_{n}] ⊗ Λ(y_{1}, …, y_{n})

where each x_{i} has degree 2, and Λ is the exterior algebra over K generated by y_{1}, …, y_{n} which are in degree one. In the p > 2 case, the Jacobson radical J of H^{∗}(G, K) is the ideal generated by y_{1}, …, y_{n}. In either case H^{∗}(E, K)/J is a polynomial ring K[x_{1}, …, x_{n}] with deg(x_{i}) = ε, and V_{G}(K) is affine n-space A^{n}(K).

The polynomial rate of growth of the minimal projective resolution of a KG-module M is called the complexity of M. It is equal to the dimension of the variety V_{G}(M) (Alperin and Evens [1]).

For example, M has complexity one if and only if the modules in its minimal resolution are bounded in dimension. In this case, V_{G}(M) is a finite union of lines through the origin in V_{G}, and so we may find an element ζ ∈ H^{m}(G, K) so that the hypersurface V_{G}⟨ζ⟩ intersects V_{G}(M) in {0}. Then by parts (ii), (v) and (vi) of the Proposition, V_{G}(L_{ζ} ⊗ M) = V_{G}⟨ζ⟩ ∩ V_{G}(M) = {0} so that L_{ζ} ⊗ M is projective. Since projective KG-modules are injective, the short exact sequence

0 → L_{ζ} ⊗ M → Ω^{m}(K) ⊗ M → M → 0

shows that Ω^{m}(K) ⊗ M ≅ M⊕ (projective). Now Ω^{m}(K) ⊗ M ≅ Ω^{m}(M)⊕ (projective), and so by the Krull–Schmidt theorem M is a direct sum of a projective module and a module M′ with Ω^{m}(M′) ≅ M′.

A module M is said to be periodic if Ω^{m}(M) ≅ M for some m > 0. The smallest value of m for which this holds is called the period of M. The above argument shows that M has complexity one if and only if it is a direct sum of a periodic module and a projective module; this was first proved by Eisenbud [7] using techniques from commutative algebra. The above argument also gives a method for finding upper bounds on the period of M.

Proposition 2.2 If M is a periodic KG-module then the period of M divides ε|G : E|, where E is a maximal elementary abelian p-subgroup of G of smallest rank.

Proof It follows from Quillen [9] that the images of the Evens norm map from the maximal elementary abelian p-subgroups of G generate a subring over which H^{∗}(G, K) is finitely generated as a module, so that the intersection of the subvarieties of V_{G} defined by these norms is zero. Hence some linear combination of these norms gives an element ζ with degree m dividing ε|G : E| and with V_{G}⟨ζ⟩ ∩ V_{G}(M) = {0}, so that by the argument preceding the lemma we have Ω^{m}(M) ≅ M. □

3 Andrews’ Theorem

In the last section, we described a method for obtaining upper bounds on the period of a periodic KG-module. In general, it is much harder to get lower bounds. However, a method of Andrews [2], which we now describe, enables us to determine the exact period of a periodic inflated module. We begin with a technical lemma.

Lemma 3.1 Suppose N is a normal p-subgroup of G, and M is a KG-module. Denote by Ñ the sum ∑_{g∈N} g of the elements of N, as an element of the group algebra KN ⊆ KG. Then M has a non-zero projective summand as a KG-module if and only if ÑM has a non-zero projective summand as a K(G/N)-module.

Proof If M has a non-zero projective summand P as a KG-module then ÑP is a non-zero projective summand of ÑM as a K(G/N)-module. Conversely, suppose that ÑM has a non-zero projective summand Q as a K(G/N)-module. Denote by π the projection of ÑM onto Q, and write π as a transfer π = Tr_{1,G/N}(α) = Tr_{N,G}(α), α ∈ End_{K}(ÑM). Since N is normal in G, the transfer map Tr_{1,N} maps elements of End_{K}(M) to elements preserving the submodule ÑM, and gives a surjective map from End_{K}(M) onto End_{K}(ÑM). So there is an element β ∈ End_{K}(M) with the property that Tr_{1,N}(β) acts as α on ÑM. Thus Tr_{1,G}(β) acts as π on ÑM, and so Tr_{1,G}(β) is a non-nilpotent KG-module endomorphism of M. So we may choose ϕ ∈ End_{KG}(M) in such a way that ϕ ∘ Tr_{1,G}(β) is a projection onto a KG-module summand of M. Since ϕ ∘ Tr_{1,G}(β) = Tr_{1,G}(ϕ ∘ β), this summand is projective. □

Now suppose we have a group extension

1 → N → G → Ḡ → 1

with N cyclic of order p. If M is a KḠ-module, regarded as a KG-module by inflation, then M ↓_{N} is a module with trivial action, and so M ↓_{N} is periodic with period ε (recall that ε = 1 if p = 2 and ε = 2 if p is odd). So if M is periodic as a KG-module, then its period is certainly divisible by ε.

Let Ñ denote the sum ∑_{g∈N} g of the elements of N, as an element of the group algebra KN. Then for any r > 0, ÑΩ_{G}^{εr}(K) is a KG-module with N acting trivially, so we may regard it as a KḠ-module. We set

V_{r} = VḠ(ÑΩ_{G}^{εr}(K)) ⊆ VḠ.

Theorem 3.2 (Andrews) If M is a KḠ-module regarded as a KG-module by inflation, then the following are equivalent:

(a) dim_{K} Ω_{G}^{εr}(M) = dim_{K} M

(b) ÑΩ_{G}^{εr}(M) = 0

(c) ÑΩ_{G}^{εr}(K) ⊗ M is a projective KḠ-module

(d) Ω_{G}^{εr}(M) ≅ M

(e) VḠ(ÑΩ_{G}^{εr}(K) ⊗ M) = {0}

(f) V_{r} ∩ VḠ(M) = {0}

Proof By Schanuel’s lemma we have

Ω_{G}^{εr}(M)↓_{N} ≅ Ω_{G}^{εr}(M ↓_{N}) ⊕ Q ≅ M ↓_{N} ⊕Q

where Q is a projective KN-module with ÑΩ_{G}^{εr}(M) = ÑQ = Q^{N}. Since N is a p-group, Q = 0 if and only if Q^{N} = 0, and so (a) is equivalent to (b).

Since Ω_{G}^{εr}(M) has no projective summands as a KG-module, the lemma implies that ÑΩ_{G}^{εr}(M) has no projective summands as a KḠ-module. Applying Ñ to the isomorphism

Ω_{G}^{εr}(K) ⊗ M ≅ Ω_{G}^{εr}(M) ⊕ (projective KG-module)

we obtain

ÑΩ_{G}^{εr}(K) ⊗ M ≅ ÑΩ_{G}^{εr}(M) ⊕ (projective KḠ-module)

and so (b) is equivalent to (c).

Tensoring the short exact sequence

0 → ÑΩ_{G}^{εr}(K) → Ω_{G}^{εr}(K)^{N} → K → 0

with M, we see that if (c) holds then

Ω_{G}^{εr}(K)^{N} ⊗ M ≅ M ⊕ (projective KḠ-module).

Since (c) also implies that Q = 0 so that N acts trivially on Ω_{G}^{εr}(M), we have

Ω_{G}^{εr}(K)^{N} ⊗ M ≅ (Ω_{G}^{εr}(K) ⊗ M)^{N} ≅ Ω_{G}^{εr}(M) ⊕ (projective KḠ-module)

and hence (c) implies (d). It is easy to see that (d) implies (a).

The equivalence of (c) and (e) and the equivalence of (e) and (f) are given by parts (ii) and (v) of Proposition 2.1 respectively. □

Corollary 3.3 Suppose 1 → N → G →^{π} Ḡ → 1 is a central extension as above, and M is a KḠ-module regarded as a KG-module by inflation. If M is periodic, then the period of M as a KG-module is equal to the maximal period of M ↓_{π^{−1}E}, as E runs over the elementary abelian p-subgroups of Ḡ.

Proof By a theorem of Alperin and Evens [1] and Avrunin [3], we have

VḠ(M) = ⋃_{E≤Ḡ} res_{Ḡ,E}^{∗}V_{E}(M ↓_{E}),
V_{r} = ⋃_{E≤Ḡ} res_{Ḡ,E}^{∗}V_{E,r}

where V_{E,r} = V_{E}(ÑΩ_{π^{−1}E}^{εr}(K)) ⊆ V_{E}. Thus condition (f) of the above theorem is satisfied for M and G if and only if it is satisfied for the restriction of M to π^{−1}E for each elementary abelian subgroup E of Ḡ. □

4 Cohomology of central extensions

In this section, we collect some well known facts about the spectral sequence of a central extension, in preparation for the calculations in the next section.

Suppose that K is a field of characteristic p, and

1 → N → G → Ḡ → 1

is a central extension with N cyclic of order p. Then we have a Lyndon–Hochschild–Serre spectral sequence with

E_{2}^{rs} = H^{r}(Ḡ, H^{s}(N, K))

and converging to H^{∗}(G, K).

If p = 2, then H^{∗}(N, K) is a polynomial ring in a generator γ in degree one, while if p is odd, H^{∗}(G, K) has an element of degree one, squaring to zero, and a polynomial generator γ in degree two. Set ε = 1 if p = 2 and ε = 2 if p is odd, so that γ ∈ H^{ε}(N, K).

Also denote by γ the corresponding element of E_{2}^{0,ε}.

Proposition 4.1 For any a ≥ 0, let ζ = γ^{p^{a}} ∈ E_{2}^{εp^{a}}. Then for 2 ≤ m ≤ εp^{a}, d_{m}(ζ) = 0, so that γ lives to E_{εp^{a}+1}^{0,εp^{a}}. For 2 ≤ m ≤ εp^{a} + 1 and any j ≥ 0 and any values of r and s, multiplication by ζ^{j} is an isomorphism from E_{m}^{r,s} to E_{m}^{r,s+εp^{a}j}. Furthermore, for the same range of values of m, j and r, and any s with εp^{a}j ≤ s ≤ εp^{a}j + m − 1, the differentials d_{m} : E_{m}^{r,s} → E_{m}^{r+m,s−m+1} are zero. In other words, no non-zero differential crosses the horizontal line between rows εp^{a}j − 1 and εp^{a}j.

Proof We prove this by induction on a. First of all, since the Bockstein map and the Steenrod operations commute with the transgressions in the spectral sequence (Serre [10], Section II.9 c), induction on a shows that ζ = γ^{p^{a}} = P^{p^{a−1}}P^{p^{a−2}} ⋯ P^{1}γ (respectively Sq^{2^{a−1}}Sq^{2^{a−2}} ⋯ Sq^{1}γ in case p = 2) transgresses. In other words, d_{m}(ζ) = 0 for 2 ≤ m ≤ εp^{a}, so that ζ lives to E_{εp^{a}+1}^{0,εp^{a}}, and d_{εp^{a}+1}(ζ) = P^{p^{a−1}}P^{p^{a−2}} ⋯ P^{1}d_{3}(γ) (respectively Sq^{2^{a−1}}Sq^{2^{a−2}} ⋯ Sq^{1}d_{2}(γ)).

Now suppose that 2 ≤ m ≤ εp^{a}. Then the fact that d_{m}(ζ) = 0 shows that for any j ≥ 0, multiplication by ζ^{j} commutes with the differentials d_{m}, so that by induction on m, multiplication by ζ^{j} is an isomorphism from E_{m}^{r,s} to E_{m}^{r,s+εp^{a}j}. Finally, if s is in the range εp^{a}j ≤ s ≤ εp^{a}j + m − 1, then any element of E_{m}^{r,s} is the product of ζ^{j} with an element in E_{m}^{r,s−εp^{a}j}, and d_{m} vanishes on such an element. □

5 The varieties of ÑΩ^{εp^{a}}(K)

Let K be a field of characteristic p, and let

1 → N → G → Ḡ → 1

be a central extension with N cyclic of order p and Ḡ a p-group. Denote by Ñ the sum of the elements of N as an element of KN, and by U the module ÑΩ_{G}^{εp^{a}}(K), where ε = 1 if p = 2 and ε = 2 if p is odd. In this section we determine the varieties VḠ(U) in terms of what is happening on the bottom row of the Lyndon-Hochschild-Serre spectral sequence of the group extension

H^{∗}(Ḡ, H^{∗}(N, K)) ⇒ H^{∗}(G, K).

The idea is to compare the spectral sequences of various double complexes obtained as tensor products. Denote by P a resolution of K as a KG-module,

P : ⋯ → P_{1} → P_{0} → 0

and by Q a resolution of K as a K(G/N)-module,

Q : ⋯ → Q_{1} → Q_{0} → 0.

For any complex C of KG-modules, we denote by E(C) the spectral sequence of the double complex

E_{0}^{r,s}(C) = Hom_{KG}(Q_{r} ⊗ C_{s}, K)
≅ Hom_{KḠ}(Q_{r}, Hom_{KN}(C_{s}, K)).

The E_{1} and E_{2} terms are given by

E_{1}^{r,s}(C) = Hom_{KḠ}(Q_{r}, H^{s}(Hom_{KN}(C_{∗}, K)))
E_{2}^{r,s}(C) = H^{r}(Ḡ, H^{s}(Hom_{KN}(C_{∗}, K))).

Thus for example E(P) is the usual Lyndon-Hochschild-Serre spectral sequence of the group extension, with E_{2}^{r,s}(P) = H^{r}(Ḡ, H^{s}(N, K)) and converging to the cohomology H^{∗}(G, K). Of course, since N is cyclic we have H^{s}(N, K) ≅ K for all s ≥ 0.

Now the spectral sequence E(P) comes with a multiplicative structure induced by the diagonal maps of complexes Δ_{Q} : Q → Q ⊗ Q and Δ_{P} : P → P ⊗ P in the usual way.

In general, the spectral sequence E(C) is not a module over E(P). However, it does come with an action of E_{2}^{∗,0} = H^{∗}(Ḡ, K) via the map of complexes Δ_{Q} ⊗ 1 : Q ⊗ C → Q ⊗ Q ⊗ C.

Denote by θ : U → P_{εp^{a}−1} the composite of the inclusion maps

U = ÑΩ^{εp^{a}}(K) ↪ Ω^{εp^{a}}(K) ↪ P_{εp^{a}−1}.

Define X to be the complex

X : ⋯ → P_{1} → P_{0} ⟶^{ϵ^{∗}ϵ} P_{0}^{∗} → P_{1}^{∗} → ⋯ → P_{εp^{a}−1}^{∗} →^{θ^{∗}} U^{∗} → 0

where ϵ : P_{0} → K is the augmentation map. Here, P_{i} is in degree i and P_{i}^{∗} is in degree −i − 1. Thus X is exact except in degree −εp^{a}, where its homology is (Ω^{εp^{a}}(K)/U)^{∗}. Denote by X′ the complex obtained by truncating X at degree zero.

X′ : 0 → P_{0}^{∗} → P_{1}^{∗} → ⋯ → P_{εp^{a}−1}^{∗} → U^{∗} → 0.

Proposition 5.1 For r > εp^{a} we have isomorphisms

H^{r+εp^{a}+1}(Ḡ, U) ≅ H^{r}(Hom_{KG}(Q_{∗} ⊗ X′_{∗}, K)),

respecting the action of H^{∗}(Ḡ, K).

Proof We compute the cohomology of Hom_{KG}(Q⊗X′, K) by using the spectral sequence in which the homology is taken first with respect to the horizontal differential (∂ ⊗ 1)^{∗} coming from Q (the “second spectral sequence” of the double complex). If −εp^{a} ≤ s < 0, then X′_{s} = P_{−s−1}^{∗} is projective, so the homology with respect to (∂ ⊗1)^{∗} is Hom_{KG}(X_{s}, K) in degree zero, and zero in higher degrees. For s = −εp^{a} − 1, X′_{s} = U^{∗} is a module on which N acts trivially, so that

Hom_{KG}(Q_{r} ⊗ U^{∗}, K) = Hom_{KḠ}(Q_{r} ⊗ U^{∗}, K).

So the homology with respect to (∂ ⊗ 1)^{∗} is Ext_{KḠ}^{∗}(U^{∗}, K) = H^{∗}(Ḡ, U). Hence the E_{2} term of this spectral sequence takes the following form:

r = 0  r = 1  r = 2

s = 0  0  0  0  ⋯
s = −1  Ĥ^{−1}(G, K)  0  0
Ĥ^{−2}(G, K)  0  0
⋮  ⋮
Ĥ^{−εp^{a}}(G, K)  0  0
s = −εp^{a} − 1  H^{0}(Ḡ, U)  H^{1}(Ḡ, U) H^{2}(Ḡ, U) ⋯
0  0  0

The higher differentials map E_{m}^{rs} → E_{m}^{r−m+1,s+m} and so for r > εp^{a}, we have

H^{r+εp^{a}+1}(Ḡ, U) ≅ E_{2}^{r+εp^{a}+1,−εp^{a}−1}
≅ E_{∞}^{r+εp^{a}+1,−εp^{a}−1}
≅ H^{r}(Hom_{KG}(Q ⊗ X′, K)). □

Our strategy now is to show that the (first) spectral sequence for the double complex Hom_{KG}(Q ⊗ X′, K) looks like a suitably truncated and shifted version of the spectral sequence E(P) of the group extension. The first stage of this process is to replace the complex X′ by X. In order to do this, we formulate a lemma which gives conditions under which differentials can never cross a horizontal line in a spectral sequence.

Lemma 5.2 Suppose that E_{0}^{∗,∗} is a double complex with the property that for some value of t, the vertical differential d_{0} : E_{0}^{r,t−1} → E_{0}^{r,t} is the zero map for all values of r. Then for m ≥ 2, t ≤ s < t + m − 1 and all values of r, the differentials d_{m} : E_{m}^{r,s} → E_{m}^{r+m,s−m+1} are zero. That is, no non-zero differential crosses the horizontal line between levels t − 1 and t in the spectral sequence.

Proof Suppose a ∈ E_{m}^{r,s} is represented by a_{0} ∈ E_{0}^{r,s}. Then there are elements a_{i} ∈ E_{0}^{r+i,s−i}, 1 ≤ i ≤ m − 1, such that

d_{1}(a_{i−1}) = (−1)^{r+i}d_{0}(a_{i}),

and d_{m}(a) is the class in E_{m}^{r+m,s−m+1} represented by d_{1}(a_{m−1}) ∈ E_{0}^{r+m,s−m+1}. In particular, for a to survive to E_{m+1}^{r,s}, it is necessary that d_{1}(a_{m−1}) = (−1)^{r+m}d_{0}(a_{m}) for some a_{m} ∈ E_{0}^{r+m,s−m}.

If the vertical differential d_{0} : E_{0}^{∗,t−1} → E_{0}^{∗,t} is the zero map then d_{0}(a_{s−t+1}) = 0 and so we can replace a_{i} by zero for s − t + 1 ≤ i < m. Thus d_{m}(a) = 0. □

Proposition 5.3 For all m ≥ 2, all values of r, and all s < 0, we have isomorphisms E_{m}^{r,s}(X′) ≅ E_{m}^{r,s}(X), respecting the action of H^{∗}(Ḡ, K).

Proof The map ϵ^{∗}ϵ : P_{0} → P_{0}^{∗} has image K ⊆ Ñ.P_{0}^{∗}. For any value of r, since Ñ acts as the zero map on Q_{r}, the image of 1 ⊗ ϵ^{∗}ϵ : Q_{r} ⊗ P_{0} → Q_{r} ⊗ P_{0}^{∗} is zero. Thus

d_{0} = (1 ⊗ ϵ^{∗}ϵ)^{∗} : Hom_{KG}(Q_{r} ⊗ P_{0}^{∗}, K) → Hom_{KG}(Q_{r} ⊗ P_{0}, K)

is the zero map from E_{0}^{r,−1}(X) to E_{0}^{r,0}(X). Applying Lemma 5.2, we see that no differential in E(X) can cross the horizontal line between s = −1 and s = 0. Since the obvious map of complexes X → X′ induces an isomorphism E_{2}^{r,s}(X′) ≅ E_{2}^{r,s}(X) for all s < 0, we conclude that it also induces an isomorphism E_{m}^{r,s}(X′) ≅ E_{m}^{r,s}(X) for all s < 0. □

Now denote by γ a non-zero element of the one dimensional vector space E_{2}^{0,ε}(P) = H^{0}(Ḡ, H^{ε}(N, K)) ≅ H^{ε}(N, K). Let b be a positive integer such that d_{i}(γ^{p^{b}}) = 0 for all i ≥ 2; that is, γ^{p^{b}} lives to E_{∞}(P). Let η be an element of H^{εp^{b}}(G, K) representing γ^{p^{b}} ∈ E_{∞}(P).

We can represent η by a map η̂ : Ω^{εp^{b}}(K) → K, and we write L_{η} for the kernel of η̂, so that there is a short exact sequence

0 → L_{η} → Ω^{εp^{b}}(K) →^{η̂} K → 0.

The composite of the inclusion maps

L_{η} ↪ Ω^{εp^{b}}(K) ↪ P_{εp^{b}−1}

and the map P_{εp^{b}−1} → P_{εp^{b}−2} give us a map P_{εp^{b}−1}/L_{η} → P_{εp^{b}−2} with one dimensional kernel isomorphic to K. Dualising this map gives us the last map in a complex

Z : ⋯ → P_{1} → P_{0} ⟶^{ϵ^{∗}ϵ} P_{0}^{∗} → P_{1}^{∗} → ⋯ → (P_{εp^{b}−1}/L_{η})^{∗} → 0.

Here, P_{i} is in degree i and P_{i}^{∗} is in degree −i − 1. Thus Z is exact except in degree −εp^{b}, where its homology is K.

Proposition 5.4 There is a map of double complexes E_{0}^{r,s−εp^{b}}(Z) → E_{0}^{r,s}(P) which induces isomorphisms E_{m}^{r,s−εp^{b}}(Z) → E_{m}^{r,s}(P) for all m ≥ 1, and respecting the action of H^{∗}(Ḡ, K) for m ≥ 2.

Proof Since Z is exact except in degree −εp^{b}, where its homology is K, there is a map of complexes P_{s} → Z_{s−εp^{b}} which induces an isomorphism on homology. This induces the desired map from E_{0}^{r,s−εp^{b}}(Z) → E_{0}^{r,s}(P).

Since the restriction of η to H^{∗}(N, K) is a power of the generator γ of degree ε, the restriction of Z to a complex of KN-modules is a projective resolution of the trivial module K in degree −εp^{b}. Thus the map of complexes P_{s} → Z_{s−εp^{b}} induces an isomorphism

H^{s−εp^{b}}(Hom_{KN}(Z_{∗}, K)) → H^{s}(Hom_{KN}(P_{∗}, K)),

and hence an isomorphism E_{1}^{r,s−εp^{b}}(Z) → E_{1}^{r,s}(P). Since this isomorphism commutes with the differentials, we see by induction on m that this map induces an isomorphism E_{m}^{r,s−εp^{b}}(Z) → E_{m}^{r,s}(P) for all m ≥ 1. It is clear that these isomorphisms respect the action of H^{∗}(Ḡ, K) = E_{2}^{∗,0}(P) for m ≥ 2. □

Proposition 5.5 Suppose that a ≤ b. Then there is a map of double complexes E_{0}^{r,s}(X) → E_{0}^{r,s}(Z) which induces an isomorphism E_{m}^{r,s}(X) → E_{m}^{r,s}(Z) for 1 ≤ m ≤ εp^{a} + 1 and s ≥ −εp^{a}. These isomorphisms respect the action of H^{∗}(Ḡ, K) for 2 ≤ m ≤ εp^{a} + 1.

Proof We first note that

Ñ.P_{εp^{a}} = (P_{εp^{a}})^{N} = Hom_{KN}(P_{εp^{a}}^{∗}, K),

so that the image of the map

Hom_{KN}(P_{εp^{a}}^{∗}, K) → Hom_{KN}(P_{εp^{a}−1}^{∗}, K)

is the same as the image of the map Ñ.P_{εp^{a}} → Ñ.P_{εp^{a}−1}, namely

ÑΩ_{G}^{εp^{a}}(K) = U = Hom_{KN}(U^{∗}, K).

Since the Q_{r} are projective KG̃-modules, this says that the image of E_{0}^{r,−εp^{a}}(Z) in E_{0}^{r,−εp^{a}+1}(Z) under the vertical differential d_{0} is exactly E_{0}^{r,−εp^{a}+1}(X). In particular, there is a map of double complexes E_{0}^{r,s}(X) → E_{0}^{r,s}(Z) which is an isomorphism for s > −εp^{a}. The induced map E_{1}^{r,s}(X) → E_{1}^{r,s}(Z) is an isomorphism for s ≥ −εp^{a}, and E_{1}^{r,s}(X) = 0 for s < −εp^{a}.

By Proposition 5.4, we have E_{m}^{r,s}(Z) ≅ E_{m}^{r,s+εp^{b}}(P), so that by Proposition 4.1, no differential d_{m} on E_{m}(Z) crosses the horizontal line between E_{m}^{r,−εp^{a}−1}(Z) and E_{m}^{r,−εp^{a}}(Z), for 2 ≤ m ≤ εp^{a}. So by induction on m, we see that the induced map E_{m}^{r,s}(X) → E_{m}^{r,s}(Z) is an isomorphism for 1 ≤ m ≤ εp^{a} + 1. It is clear that these isomorphisms respect the action of H^{∗}(Ḡ, K) = E_{2}^{∗,0}(P) for 2 ≤ m ≤ εp^{a} + 1. □

Theorem 5.6 Denote by I the ideal in H^{∗}(Ḡ, K) generated by the images of the differentials d_{m} for 2 ≤ m ≤ εp^{a} in the spectral sequence E(P). In other words, I is the kernel of the partial inflation map E_{2}^{∗,0}(P) → E_{εp^{a}+1}^{∗,0}(P). Then

VḠ(ÑΩ_{G}^{εp^{a}}(K)) = VḠ(I).

Proof Recall that U = ÑΩ_{G}^{εp^{a}}(K). By Proposition 5.1, the annihilator in H^{∗}(Ḡ, K) of H^{∗}(Ḡ, U) is equal in large enough degrees to the annihilator of E_{∞}(X′) = E_{εp^{a}+1}(X′). By Propositions 5.3, 5.5 and 5.4, we have isomorphisms of H^{∗}(Ḡ, K)-modules

E_{εp^{a}+1}(X′) ≅ ∑_{−εp^{a}≤s<0} E_{εp^{a}+1}^{r,s}(X)
≅ ∑_{−εp^{a}≤s<0} E_{εp^{a}+1}^{r,s}(Z)
≅ ∑_{ε(p^{b}−p^{a})≤s<εp^{b}} E_{εp^{a}+1}^{r,s}(P).

Let ζ be a representative of γ^{p^{a}} in E_{m}^{0,εp^{a}}(P) for 2 ≤ m ≤ εp^{a}+1. Then by Proposition 4.1, multiplication by ζ^{p^{b−a}−1} induces an isomorphism of H^{∗}(Ḡ, K)-modules

∑_{0≤s<εp^{a}} E_{εp^{a}+1}^{r,s}(P) ≅ ∑_{ε(p^{b}−p^{a})≤s<εp^{b}} E_{εp^{a}+1}^{r,s}(P),

so that H^{∗}(Ḡ, U) has the same annihilator as the left hand side of this equation in large enough degrees. Now this left hand side is finitely generated as a module over E_{εp^{a}+1}^{∗,0}(P), and so its annihilator has the same radical as E_{εp^{a}+1}^{∗,0}(P). It follows that the variety defined by the annihilator of H^{∗}(Ḡ, U) in VḠ is the same as the variety defined by the annihilator I of E_{εp^{a}+1}^{∗,0}(P). □

6 The main theorem

Theorem 6.1 Let K be a field of characteristic 2. Suppose that

1 → N → G → E → 1

is a central extension with N cyclic of order 2 and E an elementary abelian 2-group, so that V_{E} = A^{n}(K) is affine n-space, with Frobenius morphism F : V_{E} → V_{E} given by the squaring map on H^{∗}(E, K). Then V_{2^{a}} = V_{E}(ÑΩ_{G}^{ε2^{a}}(K)) is the subvariety of V_{E} consisting of those vectors v ∈ V_{E} for which the linear span of v, F (v), …, F^{a−1}(v) is isotropic (with respect to the quadratic form corresponding to the extension). This is a subvariety of codimension exactly min(a, h) in V_{E}, where h is the codimension of a maximal isotropic F -stable subspace of V_{E}.

Proof This follows from Theorem 5.6, together with the explicit description of the behaviour of the Lyndon–Hochschild–Serre spectral sequence for the central extension given in Quillen [8] (see also Benson and Carlson [5]). □

Corollary 6.2 Let K be a field of characteristic 2. Suppose that G is an extraspecial 2-group of order 2^{1+2n} and of Arf invariant zero (i.e., the index of a maximal elementary abelian subgroup is 2^{n}). Then there are periodic KG-modules of period 2^{a} for a ≤ n, and of no higher period.

Proof By Proposition 2.2, the only possible periods are 2^{a} for a ≤ n. Let N = Z(G) and E = G/N as usual. By Theorem 6.1, for a ≤ n we may find a closed homogeneous subvariety V of V_{E} with V ∩ V_{a−1} ≠ {0} and V ∩ V_{a} = {0}. By part (vii) of Proposition 2.1, we may find a KE-module M with V_{E}(M) = V. Then by Andrews’ Theorem 3.2, Ω_{G}^{2^{a−1}}(M) ≅ M̸ but Ω_{G}^{2^{a}}(M) ≅ M, so that as a KG-module M has period exactly 2^{a}. □

For p odd, some information about the behaviour of the Lyndon–Hochschild–Serre spectral sequence is given in Tezuka and Yagita [11]. However, the problem is that there is no reason why the elements hit by the Kudo transgressions in this spectral sequence should not have been hit much earlier, without more information being available. In the case of the central product of an extraspecial group of exponent p with a cyclic group of order p^{2}, the E_{2p+1}^{∗,∗} page of this spectral sequence is described in Benson and Carlson [5]. This is enough to show that V_{1} has codimension one and V_{p} has codimension two in V_{E}. Thus for a group of this form of order p^{6}, there are periodic inflated modules of period 2, 2p and 2p^{2}.

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