THE SPECTRAL SEQUENCE OF A SPLIT EXTENSION AND THE COHOMOLOGY OF AN EXTRASPECIAL GROUP OF ORDER p^{3} AND EXPONENT p

STEPHEN F. SIEGEL

Abstract. Let (E_{r}, d_{r}) be the LHS spectral sequence associated to a split extension 1 → H → G → G/H → 1 of finite groups with coefficients in a field k. We prove a version of a theorem of Charlap and Vasquez which gives an explicit formula for d_{2}. We then apply this to the case where p is an odd prime, k has characteristic p, G is extraspecial of order p^{3} and exponent p, and H is elementary abelian of order p^{2}. We calculate the terms of the spectral sequence in this case and prove E_{3} = E_{∞} (and if p = 3, E_{2} = E_{∞}).

1. Introduction

The Lyndon-Hochschild-Serre spectral sequence associates to any group extension

1 ⟶ H ⟶ G ⟶ Q ⟶ 1  (1)

and field k a sequence (E_{r}, d_{r}) (r ≥ 0) of differential bigraded k-algebras such that E_{r+1} = H(E_{r}, d_{r}). One has E_{2} ≅ H^{∗}(Q, H^{∗}(H, k)), and the sequence converges to the graded object associated to a certain filtration of H^{∗}(G, k) (Hochschild-Serre [7]). The most common case considered in the literature is where (1) is a central extension, for then Q acts trivially on H^{∗}(H, k) and the universal coefficient theorem implies E_{2} ≅ H^{∗}(Q, k) ⊗ H^{∗}(H, k). But if the extension is not central then Q does not necessarily act trivially, and even E_{2} can be quite complicated. Nevertheless, the non-central case is often quite interesting and can sometimes offer computational advantage. In particular if the extension is split and H is abelian then the few examples which have been calculated tend to suggest that the spectral sequence in this case converges quite rapidly. But in general, very little is known about the spectral sequence of a split extension.

The mod-2 cohomology rings of the extraspecial 2-groups were calculated long ago by Quillen [14], but the corresponding problem for p odd remains largely unsolved. Here we analyze the mod-p LHS spectral sequence arising from the extraspecial group of order p^{3} and exponent p presented as a split extension (p an odd prime). The integral cohomology of this group was calculated by Lewis [11], and the mod-p cohomology ring, which is quite complicated, was calculated by Leary [8, 9, 10]. Benson and Carlson [3] have also produced an algebraic version of Leary’s proof. However, both proofs involve spectral sequences which converge at approximately the E_{2p} page. In contrast, we show here that for the split extension, E_{3} = E_{∞} (and if p = 3, E_{2} = E_{∞}).

Date: January 17, 1995.

The key to our calculation is a theorem of Charlap and Vasquez [4, 5] which allows one to calculate d_{2} in a variety of settings. We state a slightly refined version of their theorem which can be easily applied to our case, and give a simplified proof. We next determine the E_{2} term for the split extension described above, and calculate the differentials d_{2} using the Charlap-Vasquez theorem. Finally, we prove that E_{3} = E_{∞} (and if p = 3, E_{2} = E_{∞}) by considering automorphisms of the group extension.

2. A theorem of Charlap and Vasquez

Let p be a prime and k a field of characteristic p. Assume (1) is a split extension of finite groups, and fix a splitting once and for all, so that we may identify Q with a subgroup of G, and G = HQ. Let M be a kG-module (all modules are assumed to be finitely generated left modules) such that M is semisimple as a kH-module. We construct the LHS spectral sequence as follows. Let X → k be a kG-projective resolution, Y → k the kQ-bar resolution. Let

E_{0} = Hom_{kQ}(Y, Hom_{kH}(X, M))

and define differentials d′ : E_{0}^{r,s} → E_{0}^{r+1,s} and d′′ : E_{0}^{r,s} → E_{0}^{r,s+1} by d′(f) = f ∘ ∂ and d′′(f)(y) = (−1)^{r}f(y) ∘ ∂, and let {(E_{r}, d_{r}) | r ≥ 0} be the spectral sequence arising from the double complex (E_{0}, d′′, d′).

Let P → k be the minimal kH-projective resolution. Recall that the image of the differential on P is contained in rad_{kH}(P), so since rad_{kH}(M) = 0, we may identify H^{∗}(H, M) with Hom_{kH}(P, M), and then identify

E_{1} = Hom_{kQ}(Y, Hom_{kH}(P, M)).

How does Q act on Hom_{kH}(P, M) under these identifications? Even though Q does not necessarily act on P in a manner consistent with the action of H, it does “act up to homotopy”. To make this precise, for each σ ∈ Q, the Comparison Theorem (cf. Theorem 2.4.2 and the remark following it in Benson [2]) guarantees the existence of a kHchain map A(σ) : P → P^{σ} which commutes with the augmentation.

Here, P^{σ} = P as k-complexes but has H-action h.x = σhσ^{−1}x (h ∈ H, x ∈ P). It is not to difficult to see that the action of Q is given by

σ.F = σ ∘ F ∘ A(σ^{−1}), σ ∈ Q, F ∈ Hom_{kH}(P, M),

where, by a slight abuse of notation, we let σ also denote the automorphism induced by σ on M.

For σ, τ ∈ Q, the uniqueness part of the Comparison Theorem implies that A(στ) is chain homotopic to A(σ)A(τ), i.e. there exists U(σ, τ) ∈ Hom_{kH}(P, P^{στ})_{1} such that

∂U(σ, τ) + U(σ, τ)∂ = A(στ) − A(σ)A(τ).  (2)

The pair (A, U) is called a Q-system for the extension (1). Charlap and Vasquez’s theorem reduces the problem of calculating d_{2} to the calculation of a Q-system. To see how this is done, first define a set of maps D_{2} : E_{1}^{r,s} → E_{1}^{r+2,s−1} by

D_{2}(f)[σ_{1}| ⋯ |σ_{r+2}] = σ_{1}σ_{2} ∘ f[σ_{3}| ⋯ |σ_{r+2}] ∘ U(σ_{2}^{−1}, σ_{1}^{−1}).

Theorem 1. (Charlap-Vasquez) Suppose ζ ∈ E_{2}^{r,s} (r ≥ 0, s ≥ 1) is represented by the Q-cocycle f ∈ E_{1}^{r,s}. Then d_{2}(ζ) is represented by (−1)^{r}D_{2}(f).

Proof. We first show that D_{2}(f) is a Q-cocycle (i.e., is in the kernel of d_{1}). Indeed, since f is a Q-cocycle,

D_{2}(f) ∘ ∂([σ_{1}| ⋯ |σ_{r+3}]) = σ_{1}σ_{2}σ_{3} ∘ f[σ_{4}| ⋯ |σ_{r+3}] ∘ ϕ(σ_{3}^{−1}, σ_{2}^{−1}, σ_{1}^{−1}), (3)

where for ρ, σ, τ ∈ Q,

ϕ(ρ, σ, τ) = U(ρ, σ)A(τ) − U(ρ, στ) + U(ρσ, τ) − A(ρ)U(σ, τ).

Using (2), one checks that ϕ(ρ, σ, τ) : P → P^{ρστ}[−1] is a chain map (if C is a complex then C[n] denotes the complex with C[n]_{i} = C_{i−n} and with differentials given by multiplyling the differentials in C by (−1)^{n}.) The Comparison Theorem then implies that it is chain homotopic to zero; hence its image is contained in rad_{kH}(P^{ρστ}) = rad_{kH}(P), and (3) is zero, as required.

An equally straightforward argument shows that the element of E_{2}^{r+2,s−1} defined by D_{2}(f) is independent of the choice of Q-system (A, U). We can therefore prove the theorem using any Q-system we want.

Since P is the minimal resolution, there exist kH-chain maps ι : P → X and π : X → P, commuting with the augmentations, such that πι = 1. Let W = Ker(π), so X = ι(P)⊕W. W is a kH-projective resolution of 0, hence is kH-contractible. Take any contracting homotopy of W and extend it to a map s ∈ Hom_{kH}(X, X)_{1} by setting s(x) = 0 for x ∈ ι(P). Then ∂s + s∂ = 1 − ιπ and sι = πs = 0. Now define A(σ) = πσι and U(σ, τ) = πσsτι (σ, τ ∈ Q). Since

∂U(σ, τ)+U(σ, τ)∂ = πσ(∂s+s∂)τι = πσ(1−ιπ)τι = A(στ)−A(σ)A(τ),

(A, U) is a Q-system.

We now calculate d_{2}(ζ). Clearly, ζ is represented on E_{0}^{r,s} by the map

x^{r,s} : [σ_{1}| ⋯ |σ_{r}] ↦ f[σ_{1}| ⋯ |σ_{r}] ∘ π

(but note: this defines x^{r,s} on a kQ-basis of Y_{r}; it is not necessarily the case that x^{r,s}(y) = f(y) ∘ π for all y ∈ Y_{r}). Hence

\begin{aligned}d′(x^{r,s})[σ_{1}| ⋯ |σ_{r+1}] & = & σ_{1} ∘ f[σ_{2}| ⋯ |σ_{r+1}] ∘ πσ_{1}^{−1} \\ & & +(−f[σ_{1}σ_{2}| ⋯ |σ_{r+1}] + ⋯ ± f[σ_{1}| ⋯ |σ_{r}]) ∘ π. \end{aligned}

Notice that in general, if F : X → M is a coboundary, then F ∘ ι = 0, hence

F = F ∘ (1 − ιπ) = F ∘ (∂s + s∂) = (F s) ∘ ∂.

It follows that if we define x^{r+1,s−1} ∈ E_{0}^{r+1,s−1} by

\begin{aligned}x^{r+1,s−1}[σ_{1}| ⋯ |σ_{r+1}] & = & (−1)^{r}d′(x^{r,s})[σ_{1}| ⋯ |σ_{r+1}] ∘ s \\ & = & (−1)^{r}σ_{1} ∘ f[σ_{2}| ⋯ |σ_{r+1}] ∘ πσ_{1}^{−1}s \end{aligned}

(using πs = 0), then d′(x^{r,s}) + d′′(x^{r+1,s−1}) = 0. By definition, d_{2}(ζ) is represented on E_{0} by z = d′(x^{r+1,s−1}), and

z[σ_{1}| ⋯ |σ_{r+2}] = (−1)^{r}σ_{1}σ_{2} ∘ f[σ_{3}| ⋯ |σ_{r+2}] ∘ πσ_{2}^{−1}sσ_{1}^{−1} + (⋯) ∘ s.

Using the fact that sι = 0, we see that the image of z in E_{1} is the map

[σ_{1}| ⋯ |σ_{r+2}] ↦ (−1)^{r}σ_{1}σ_{2} ∘ f[σ_{3}| ⋯ |σ_{r+2}] ∘ πσ_{2}^{−1}sσ_{1}^{−1}ι,

completing the proof.

3. The cyclic case

It is unfortunate that Theorem 1 is formulated using the bar resolution Y, since any kQ-projective resolution can be used to construct the LHS spectral sequence. However, at least in a special case, we can find a similar expression for d_{2} using the minimal resolution in place of Y.

As before, let p be a prime and k a field of characteristic p, but now let

1 ⟶ H ⟶ G = H ⋊ B ⟶ B ⟶ 1

be a split extension of finite groups with B = ⟨b | b^{p} = 1⟩. Let Z → k be the minimal kB-resolution; say Z_{n} = kBe_{n} (n ≥ 0), ∂(e_{n}) = N(b)e_{n−1} (if n > 0 is even), where N(b) = ∑_{i=1}^{p} b^{i}, and ∂(e_{n}) = (b − 1)e_{n−1} (if n is odd). It follows that for any kB-module V,

H^{r}(B, V) ≅ \begin{cases}soc_{kB}(V) & if r = 0 \\ soc_{kB}^{p−1}(V)/rad_{kB}(V) & if r is odd \\ soc_{kB}(V)/rad_{kB}^{p−1}(V) & if r > 0 is even. \end{cases}  (4)

For the general definitions of “soc” and “rad” see Alperin [1]. In this case, soc_{kB}(V) is just V^{B}, the set of invariant elements of V under the action of B, soc_{kB}^{p−1}(V) is the kernel of N(b) on V, rad_{kB}(V) is the image of b − 1, and rad_{kB}^{p−1}(V) is the image of N(b).

Let M be a kG-module on which H acts trivially, X → k a kGresolution, P → k the minimal kH-resolution, and {(E_{r}, d_{r})} the spectral sequence of the double complex E_{0} = Hom_{kB}(Z, Hom_{kH}(X, M)).

Corollary 2. Let α : P → P^{b^{−1}} be a kH-chain map commuting with the augmentation, and υ ∈ Hom_{kH}(P, P)_{1} a map satisfying ∂υ + υ∂ = 1 − α^{p}. Suppose ζ ∈ E_{2}^{r,s} (r ≥ 0, s ≥ 1) is represented by f ∈ Hom_{kH}(P_{s}, M). Then d_{2}(ζ) is represented by (−1)^{r}f ∘ υ.

Proof. Define, for 0 ≤ i, j ≤ p − 1,

A(b^{−i}) = α^{i}, U(b^{−i}, b^{−j}) = \begin{cases}0 & if i + j < p \\ υα^{i+j−p} & otherwise. \end{cases}

It follows that (A, U) is a B-system.

We next construct explicit comparisons between the minimal resolution Z and the bar resolution Y. Define a kB-chain map θ : Y → Z by setting, for n ≥ 0 and 0 ≤ i_{1}, …, i_{2n+1} ≤ p − 1,

θ[b^{i_{1}}|b^{i_{2}}| ⋯ |b^{i_{2n}}] = \begin{cases}e_{2n} & if i_{2j−1} + i_{2j} ≥ p for all 1 ≤ j ≤ n \\ 0 & otherwise \end{cases}

θ[b^{i_{1}}|b^{i_{2}}| ⋯ |b^{i_{2n+1}}] = \begin{cases}∑_{i=0}^{i_{1}−1} b^{i}e_{2n+1} & if i_{2j} + i_{2j+1} ≥ p for all 1 ≤ j ≤ n \\ 0 & otherwise \end{cases} (5)

(the empty sum is understood to be 0). Define a kB-chain map ϕ : Z → Y by ϕ(e_{0}) = [], ϕ(e_{1}) = [b], and for n ≥ 1,

\begin{aligned}ϕ(e_{2n}) & = & ∑_{0≤i_{1},... ,i_{n}≤p−1}[b^{i_{1}}|b| ⋯ |b^{i_{n}}|b] \\ ϕ(e_{2n+1}) & = & ∑_{0≤i_{1},... ,i_{n}≤p−1}[b|b^{i_{1}}| ⋯ |b|b^{i_{n}}|b]. \end{aligned}  (6)

One can check that both (5) and (6) actually define chain maps.

Assume n ≥ 0 and r = 2n (the case where r is odd is handled similarly). By (5), ζ is represented via the bar resolution by

[b^{i_{1}}| ⋯ |b^{i_{2n}}] ↦ \begin{cases}f & if i_{2j−1} + i_{2j} ≥ p for all 1 ≤ j ≤ n \\ 0 & otherwise, \end{cases}

so by Theorem 1, d_{2}(ζ) is represented by

γ : [b^{i_{1}}| ⋯ |b^{i_{2n+2}}] ↦ \begin{cases}b^{i_{1}+i_{2}} ∘ f ∘ U(b^{−i_{2}}, b^{−i_{1}}) & if i_{2j−1} + i_{2j} ≥ p for all 2 ≤ j ≤ n \\ 0 & otherwise. \end{cases}

And

γ(ϕ(e_{2n+2})) = ∑_{i=0}^{p−1} γ[b^{i}|b|b^{p−1}|b| ⋯ |b^{p−1}|b] = ∑_{i=0}^{p−1} b^{i+1}∘f∘U(b^{−1}, b^{−i}) = f∘υ,

completing the proof.

Of course if the map α (or υ) is defined only on degrees 0 through n (for some n > 0), then it can always be extended to a map on all of P. Hence to calculate d_{2} on E_{2}^{r,s} for s ≤ n, one really only needs to know α and υ on degrees 0, …, n.

4. The calculation of E_{2}

Let p be an odd prime, k a field of characteristic p, and

G = ⟨a, b, c | c = [a, b], a^{p} = b^{p} = [a, c] = [b, c] = 1⟩,

where [g, h] = g^{−1}h^{−1}gh. G is the (unique, up to isomorphism) extraspecial group of order p^{3} and exponent p. Let H = ⟨a, c⟩ and B = ⟨b⟩. We consider the LHS spectral sequence (E_{r}, d_{r}) arising from the split extension 1 → H → G → B → 1.

Our first task is to analyze E_{2} ≅ H^{∗}(B, H^{∗}(H, k)). Our approach is to determine the kB-module structure of H^{∗}(H, k). Recall that by Jordan canonical form, there are, up to isomorphism, p indecomposable kB-modules (cf. [1]). We denote these by J_{i} (1 ≤ i ≤ p), where J_{i} has dimension i.

Define α_{1}, β_{1} ∈ H^{1}(H, k) ≅ Hom(H, k) by α_{1}(a^{i}c^{j}) = i, and β_{1}(a^{i}c^{j}) = j. Let α_{2} = δ(α_{1}) and β_{2} = δ(β_{1}), where δ, as always, denotes the appropriate Bockstein homomorphism. Then H^{∗}(H, k) is the gradedcommutative ring^{1} k[α_{1}, β_{1}, α_{2}, β_{2}], and the action of B on H^{∗}(H, k) is given by b(α_{i}) = α_{i} and b(β_{i}) = α_{i} + β_{i} (i = 1, 2).

1All graded rings presented with generators and relations will be assumed to be graded-commutative, i.e. xy = (−1)^{deg(x) deg(y)}yx, for all homogeneous elements x, y.

The product on E_{2} can be calculated using the identifications of (4) in the following way: if χ ∈ H^{s}(H, k) represents x ∈ E_{2}^{r,s} and ζ ∈ H^{s′}(H, k) represents y ∈ E_{2}^{r′,s′}, then (−1)^{r′s}xy ∈ E_{2}^{r+r′,s+s′} is represented by

\begin{cases}χζ & if r or r′ is even \\ ∑_{0≤i<j<p} b^{i}(χ)b^{j}(ζ) & if r and r′ are odd. \end{cases}  (7)

Now let W be the graded k-submodule of H^{∗}(H, k) spanned by {α_{2}^{i}β_{2}^{j} | i ≥ 0, 0 ≤ j < p}. W is a kB-submodule of H^{∗}(H, k), and multiplication by α_{2} : W^{n} → W^{n+2} is an injective kB-homomorphism for n ≥ 0 and an isomorphism for n ≥ 2(p − 1). Furthermore, W^{2i} ≅ J_{i+1} (i ≤ p − 1) and rad_{kB}(W) is spanned by {α_{2}^{i}β_{2}^{j} | i ≥ 1, 0 ≤ j < p} (cf. [1, pp. 14–16]).

Set N = k[α_{1}, β_{1}] ⊗ W, and x_{2p} = ∏_{i=0}^{p−1}(β_{2} + iα_{2}). It is easy to see that x_{2p} is B-invariant and that k[α_{2}, β_{2}] = k[x_{2p}] ⊗ W. It follows that

H^{∗}(H, k) = k[α_{1}, β_{1}] ⊗ k[α_{2}, β_{2}] = k[x_{2p}] ⊗ N.

Moreover, since k[x_{2p}] is a trivial kB-module,

soc_{kB}^{i}(H^{∗}(H, k)) = k[x_{2p}] ⊗ soc_{kB}^{i}(N),  i ≥ 0,  (8)

and (8) holds with “soc” replaced by “rad” as well. In particular, the identifications given in (4) yield isomorphisms of graded k-modules

E_{2}^{r,∗} ≅ \begin{cases}k[x_{2p}] ⊗ soc_{kB}(N) & if r = 0 \\ k[x_{2p}] ⊗ soc_{kB}^{p−1}(N)/rad_{kB}(N) & if r is odd \\ k[x_{2p}] ⊗ soc_{kB}(N)/rad_{kB}^{p−1}(N) & if r > 0 is even. \end{cases} (9)

It remains to analyze the structure of N. The referee has pointed out that the structure of soc_{kB}(N) was determined by Minh [12, Theorem 2.4] (though the generating set given there is not minimal) and Mùi [13, Theorem 5.6]; we compute this structure directly, using representation theory.

Proposition 3. (i) soc_{kB}(N) = k[x_{1}, x_{2}, y_{2}, x_{3}]/(y_{2}^{2}, x_{1}y_{2}, y_{2}x_{3}, x_{1}x_{3}+ x_{2}y_{2}), where x_{1} = α_{1}, x_{2} = α_{2}, y_{2} = α_{1}β_{1}, and x_{3} = α_{1}β_{2} − β_{1}α_{2}.

(ii) rad_{kB}(N^{1}) = kα_{1}, and for 1 ≤ i < p,

{α_{2}^{j}β_{2}^{i−j} | 1 ≤ j ≤ i} ∪ {α_{1}β_{1}α_{2}^{j}β_{2}^{i−j−1} | 1 ≤ j < i}

is a basis for rad_{kB}(N^{2i}), and

{α_{1}α_{2}^{j}β_{2}^{i−j} | 1 ≤ j ≤ i} ∪ {β_{1}α_{2}^{j}β_{2}^{i−j} | 2 ≤ j ≤ i} ∪ {α_{1}β_{2}^{i} + iβ_{1}α_{2}β_{2}^{i−1}}

is a basis for rad_{kB}(N^{2i+1}).

(iii) For each s ≥ 0, the following chart gives the kB-module structure of N^{s}, a basis for soc_{kB}^{p−1}(N^{s})/rad_{kB}(N^{s}), and a basis for soc_{kB}(N^{s})/rad_{kB}^{p−1}(N^{s}):

N^{s}  soc^{p−1}/rad  soc/rad^{p−1}
s ≥ 2p − 1 J_{p} ⊕ J_{p}  ∅  ∅
s = 2p − 2 J_{p−1} ⊕ J_{p} [α_{1}β_{1}β_{2}^{p−2}]  [x_{2}^{p−2}y_{2}]
s = 2p − 3 J_{p−2} ⊕ J_{p} [α_{1}β_{2}^{p−2}] [x_{2}^{p−3}x_{3}]
s = 2i + 1^{∗} J_{i} ⊕ J_{i+2} [α_{1}β_{2}^{i}], [β_{1}β_{2}^{i}] x_{1}x_{2}^{i}, x_{2}^{i−1}x_{3}
s = 2i^{∗∗}  J_{i} ⊕ J_{i+1} [α_{1}β_{1}β_{2}^{i−1}], [β_{2}^{i}] x_{2}^{i}, x_{2}^{i−1}y_{2}
s = 1  J_{2}  [β_{1}]  x_{1}
s = 0  J_{1}  1  1
∗ 1 ≤ i ≤ p − 3 ^{∗∗} 1 ≤ i ≤ p − 2

Proof. For 1 < i < p, J_{2} ⊗ J_{i} ≅ J_{i−1} ⊕ J_{i+1} (cf. [1, p. 50]) and J_{2} ⊗ J_{p} ≅ J_{p} ⊕ J_{p} since J_{p} is projective. This, together with the known structure of W, gives the kB-module structure of N.

Next, we prove (i). One checks that x_{1}, x_{2}, y_{2}, and x_{3} are fixed by b, satisfy the relations, and that the subring they generate is contained in N. On the other hand, the graded-commutative ring S described abstractly by these generators and relations satisfies S^{1} = kx_{1}, and, for i ≥ 1,

\begin{aligned}S^{2i} & = & kx_{2}^{i} ⊕ kx_{2}^{i−1}y_{2} \\ S^{2i+1} & = & kx_{1}x_{2}^{i} ⊕ kx_{2}^{i−1}x_{3}. \end{aligned}  (10)

By the known structure of N, S has the same dimension as soc_{kB}(N) in each degree. Hence S ≅ soc_{kB}(N), and we have established (i).

The expression for rad_{kB}(N) in even degress follows from the basis for rad_{kB}(W), since B fixes α_{1}β_{1}. To obtain the expression for odd degrees, one computes (b − 1)N explicitly; the last term comes from (b − 1)β_{1}β_{2}^{i}.

To fill in the chart for s = 2p − 3, observe that N(b)α_{1}β_{2}^{p−2} = 0 and x_{1}x_{2}^{p−2} = −N(b)β_{1}β_{2}^{p−2}. For s = 2p − 2, N(b)α_{1}β_{1}β_{2}^{p−2} = 0 and x_{2}^{p−1} = −N(b)β_{2}^{p−1}. The proof of the rest of (iii) is routine.

Now define γ_{1} ∈ H^{1}(B, k) ≅ E_{2}^{1,0} by γ_{1}(b^{i}) = i, and let γ_{2} = δ(γ_{1}). Let

z_{2j} = [β_{1}β_{2}^{j−1}] ∈ E_{2}^{1,2j−1}, z_{2j+1} = [β_{2}^{j}] ∈ E_{2}^{1,2j}, 1 ≤ j ≤ p − 2,

set w_{n} = −x_{1}z_{n−1} (3 ≤ n ≤ 2p − 2) and w_{2p−1} = y_{2}z_{2p−3}. Notice that w_{2i} is represented by α_{1}β_{2}^{i−1} and w_{2i+1} by α_{1}β_{1}β_{2}^{i−1}.

Corollary 4. (i) Multiplication by x_{2p} : E_{2}^{r,s} → E_{2}^{r,s+2p} is an isomorphism for r ≥ 1, s ≥ 0.

(ii) Multiplication by γ_{2} : E_{2}^{r,s} → E_{2}^{r+2,s} is surjective for all r, s ≥ 0

and is an isomorphism if r ≥ 1 or s ≤ 2p − 4. (iii) E_{2}^{0,∗} ≅ k[x_{2p}] ⊗ k[x_{1}, x_{2}, y_{2}, x_{3},]/(y_{2}^{2}, x_{1}y_{2}, y_{2}x_{3}, x_{1}x_{3} + x_{2}y_{2}) (iv) We have the following bases for E_{2}^{r,s}:

\begin{matrix}s & 0 & 1 & 2i & 2i + 1 \\ E_{2}^{0,s} & 1 & x_{1} & x_{2}^{i}, x_{2}^{i−1}y_{2} & x_{1}x_{2}^{i}, x_{2}^{i−1}x_{3} \end{matrix} (1 ≤ i ≤ p − 1)

\begin{matrix}s & 0 & 1 & 2 ≤ s ≤ 2p − 4 & 2p − 3 & 2p − 2 & 2p − 1 \\ E_{2}^{1,s} & γ_{1} & z_{2} & z_{s+1}, w_{s+1} & w_{2p−2} & w_{2p−1} & ∅ \end{matrix}

\begin{matrix}s & 2p − 3 & 2p − 2 & 2p − 1 \\ E_{2}^{2,s} & γ_{2}x_{2}^{p−3}x_{3} & γ_{2}x_{2}^{p−2}y_{2} & ∅ \end{matrix}

(v) E_{2} is generated by x_{1}, γ_{1}, x_{2}, y_{2}, γ_{2}, x_{3}, x_{2p}, z_{2}, z_{3}, …, z_{2p−3}.

Proof. As in (9), we may write E_{2}^{r,∗} ≅ k[x_{2p}] ⊗ C, for a graded kBmodule C. By Proposition 3(iii), if r ≥ 1 then C is concentrated in degrees 0 through 2p − 1, proving (i). Now for any kB-module V, multiplication by γ_{2} : H^{r}(B, V) → H^{r+2}(B, V) is just the natural map under the identifications in (4). This is a surjection for r ≥ 0 and an isomorphism for r ≥ 1. If r = 0 and s ≤ 2p − 4 then Proposition 3(iii) shows that rad_{kB}^{p−1}(H^{s}(H, k)) = 0, hence the natural map is an isomorphism in this case too, proving (ii). Statements (iii) and (iv) follow directly from Proposition 3 and (9). Now (v) is clear since one can get a basis for E_{2}^{r,s} for any r, s by starting with one of the bases in (iv) and multiplying by appropriate powers of x_{2p} and γ_{2p}, according to (i) and (ii).

5. Differentials

We now determine all the differentials in the spectral sequence. The data is presented diagrammatically in Figure 1; the integer to the upper right of each point there indicates k-dimension.

Theorem 5. (i) x_{1}, γ_{1}, x_{2}, y_{2}, z_{2}, γ_{2}, x_{3}, z_{3}, x_{2p} all live to E_{∞}. In par-

ticular, if p = 3 then E_{2} = E_{∞}. (ii) If p > 3 then in addition

\begin{aligned}d_{2}(z_{2i+2}) & = & iγ_{2}w_{2i+1} & = & −ix_{1}γ_{2}z_{2i}, & 1 ≤ i ≤ p − 3, \\ d_{2}(z_{2i+1}) & = & iγ_{2}w_{2i} & = & −ix_{1}γ_{2}z_{2i−1}, & 2 ≤ i ≤ p − 2. \end{aligned}

Proof. Let Q′ → k be the minimal k⟨a⟩-resolution, where Q′_{n} = k⟨a⟩e′_{n}, and Q′′ → k the minimal k⟨c⟩-resolution, with Q′′_{n} = k⟨c⟩e′′_{n}. Let P = Q′ ⊗ Q′′, so P → k is the minimal kH-resolution. For n, k ∈ ℤ, set e_{k}^{n} = e′_{n−k} ⊗ e′′_{k} if 0 ≤ k ≤ n and 0 otherwise. Hence e_{0}^{n}, e_{1}^{n}, …, e_{n}^{n} is a kH-basis for P_{n} (n ≥ 0). Also, write ē_{k}^{n} for the image of e_{k}^{n}

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14  1 γ_{1}1 γ_{2}1 1 1 1 x_{1} 1 z_{2} 1 1 1 1 1 2 z_{3} 2 2 2 2 2 y_{2} x_{2} x_{3} 2 z_{4} 2 2 2 2 2 2 z_{5} 2 2 2 2 2 2 z_{6} 2 2 2 2 2 2 z_{7} 2 2 2 2 2 2 1 1 1 1 1 2 1 1 1 1 1 2 0 0 0 0 0 x_{10} 3 1 1 1 1 1 3 1 1 1 1 1 4 2 2 2 2 2 4 2 2 2 2 2 4 2 2 2 2 2  1 γ_{1}1 γ_{2}1 1 1 1 x_{1} 1 z_{2} 1 1 1 1 1 2 z_{3} 2 2 1 2 1 y_{2} x_{2} x_{3} 2 1 2 0 2 0 2 1 2 0 2 0 2 1 2 0 2 0 2 w_{7} 1 2 1 2 1 2 w_{8} 1 1 1 1 1 2 w_{9} 1 1 1 1 1 2 0 0 0 0 0 x_{10} 3 1 1 1 1 1 3 1 1 1 1 1 4 2 2 1 2 1 4 1 2 0 2 0 4 1 2 0 2 0

Figure 1. E_{2} and E_{3} = E_{∞} for extraspecial 5^{3}, exponent 5.

in P_{n}/rad_{kH}(P_{n}) ≅ H_{n}(H, k). It is not hard to see that the duality between H^{∗}(H, k) and H_{∗}(H, k) is given by

(α_{1}^{i_{1}}β_{1}^{j_{1}}α_{2}^{i_{2}}β_{2}^{j_{2}})^{∗} = ē_{j_{1}+2j_{2}}^{i_{1}+j_{1}+2i_{2}+2j_{2}},  0 ≤ i_{1}, j_{1} ≤ 1, 0 ≤ i_{2}, j_{2}. (11)

Define ρ, κ ∈ kH by ρ = ∑_{0≤j≤i<p} a^{i}c^{j}, and κ = ∑_{i=0}^{p−1} ia^{i}. Define maps α ∈ Hom_{kH}(P, P^{b^{−1}})_{0} and υ ∈ Hom_{kH}(P, P)_{1} by

\begin{aligned}αe_{2j}^{2i} & = & ∑_{k≥j} \binom{k}{j}(e_{2k}^{2i} − ρe_{2k+1}^{2i}), & υe_{2j}^{2i} & = & −(j + 1)κe_{2j+2}^{2i+1}, \\ αe_{2j+1}^{2i} & = & ∑_{k≥j} \binom{k}{j}ce_{2k+1}^{2i}, & υe_{2j+1}^{2i} & = & −(j + 1)e_{2j+3}^{2i+1}, \\ αe_{2j}^{2i+1} & = & ∑_{k≥j} \binom{k}{j}(ce_{2k}^{2i+1} + e_{2k+1}^{2i+1}), & υe_{2j}^{2i+1} & = & −(j + 1)e_{2j+2}^{2i+2}, \\ αe_{2j+1}^{2i+1} & = & ∑_{k≥j} \binom{k}{j}e_{2k+1}^{2i+1}, & υe_{2j+1}^{2i+1} & = & −(j + 1)κe_{2j+3}^{2i+2}, \end{aligned} (12)

for 0 ≤ j ≤ i < p (of course the sums are finite since e_{m}^{n} = 0 for m > n). It is straightforward, though a bit tedious, to check that α is a chain map, and that ∂υ + υ∂ = 1 − α^{p}. (The reader who checks this will find the following relations in kH useful: ρ(c − 1) = N(a) − cN(ac), ρ(a − 1) = N(ac) − N(c), and κ(1 − a) = N(a).) Hence we can apply Corollary 2.

We first calculate d_{2} on z_{2i+2} = [β_{1}β_{2}^{i}] (1 ≤ i ≤ p − 3). According to (11), the map f : P_{2i+1} → k, defined by f(e_{n}^{2i+1}) = 1 if n = 2i+1 and 0 otherwise, corresponds to β_{1}β_{2}^{i}. Using (12) we see that f ∘ υ : P_{2i} → k sends e_{n}^{2i} to −i if n = 2i−1 and 0 otherwise. Hence −f ∘υ corresponds to i(ē_{2i−1}^{2i})^{∗} = iα_{1}β_{1}β_{2}^{i−1}, which represents iγ_{2}w_{2i+1}.

Similarly, z_{2i+1} (1 ≤ i ≤ p − 2) is represented by f = (ē_{2i}^{2i})^{∗}, and −f ∘ υ = i(ē_{2i−2}^{2i−1})^{∗} = iα_{1}β_{2}^{i−1}. If i ≥ 2 this shows d_{2}(z_{2i+1}) = iγ_{2}w_{2i}. If i = 1, Proposition 3(ii) says α_{1} ∈ rad_{kB}(H^{1}(H, k)), so d_{2}(z_{3}) = 0. Finally y_{2} = [α_{1}β_{1}] is represented by f = (ē_{1}^{2})^{∗}, and f ∘ υ = 0, so d_{2}(y_{2}) = 0.

Now for a split extension, all differentials into the horizontal edge vanish (cf. [6, Proposition 7.3.2]), so x_{1}, y_{2}, z_{2}, and z_{3} live to E_{∞}. Therefore δ(x_{1}) = x_{2} and

δ(y_{2}) = δ(α_{1}β_{1}) = δ(α_{1})β_{1} − α_{1}δ(β_{1}) = α_{2}β_{1} − α_{1}β_{2} = −x_{3}

live to E_{∞} as well. Since

x_{2p} = ∏_{σ∈G/H} σ^{∗}(β_{2}) = res_{G→H}N_{H→G}(β_{2}),

where N_{H→G} denotes the Evens norm map (cf. Theorem 6.1.1 of [6]), x_{2p} lives to E_{∞}. Of course we could have used the Charlap-Vasquez theorem for these cases as well, but then we would not get the additional information about the vanishing of the higher differentials.

Corollary 6. If p > 3 then:

(i) Multiplication by x_{2p} : E_{3}^{r,s} → E_{3}^{r,s+2p} is an isomorphism for r ≥ 1, s ≥ 0.

(ii) Multiplication by γ_{2} : E_{3}^{r,s} → E_{3}^{r+2,s} is surjective for r, s ≥ 0, and

is an isomorphism for r ≥ 2, s ≥ 0. (iii) For r even, E_{3}^{r,∗} = E_{2}^{r,∗}. For r = 1, 3 we have the following k-bases

for E_{3}^{r,s} (0 ≤ s ≤ 2p − 1):

\begin{matrix}s & 0 & 1 & 2 & 3 & 4 & 5 & 6 ≤ s ≤ 2p − 2 & 2p − 1 \\ E_{3}^{1,s} & γ_{1} & z_{2} & z_{3}, γ_{1}y_{2} & x_{1}z_{3} & z_{2}x_{3} & x_{3}z_{3} & w_{s+1} & ∅ \end{matrix} ; \begin{matrix}s & 0 & 1 & 2 & 3 ≤ s ≤ 2p − 5 & 2p − 4 ≤ s ≤ 2p − 2 & 2p − 1 \\ E_{3}^{3,s} & γ_{1}γ_{2} & γ_{2}z_{2} & γ_{2}z_{3} & ∅ & γ_{2}w_{s+1} & ∅ \end{matrix}

(iv) E_{3} is generated by x_{1}, γ_{1}, x_{2}, y_{2}, z_{2}, γ_{2}, x_{3}, z_{3}, x_{2p}, and w_{7}, …, w_{2p−1}.

Proof. To see that multiplication by γ_{2} is surjective, argue in E_{2} as follows: suppose r, s ≥ 0, y ∈ E_{2}^{r+2,s} and d_{2}(y) = 0. By Corollary 4(ii), there exists x ∈ E_{2}^{r,s} such that γ_{2}x = y. Hence γ_{2}d_{2}(x) = 0. Since γ_{2} : E_{2}^{r+2,s−1} → E_{2}^{r+4,s−1} is injective, d_{2}(x) = 0, i.e., x represents an element of E_{3}. To get injectivity on E_{3}, suppose x ∈ E_{2}^{r,s} (r ≥ 2), d_{2}(x) = 0, and γ_{2}x = d_{2}(w), for some w ∈ E_{2}^{r,s+1}. Then w = γ_{2}v, for some v ∈ E_{2}^{r−2,s+1}. Hence γ_{2}x = γ_{2}d_{2}(v), and since γ_{2} : E_{2}^{r,s} → E_{2}^{r+2,s} is injective, x = d_{2}(v), i.e., x represents 0 in E_{3}, and we have proved (ii). The proof of (i) is entirely similar.

It follows from Theorem 5 that E_{3}^{1,s} = kw_{s+1} for 3 ≤ s ≤ 2p − 2. But w_{4} = −x_{1}z_{3} and

z_{2}x_{3} = [β_{1}][α_{1}β_{2} − β_{1}α_{2}] = −[α_{1}β_{1}β_{2}] = −w_{5}.

For s = 5, z_{3}x_{3} = [β_{2}][α_{1}β_{2} − β_{1}α_{2}] = [α_{1}β_{2}^{2} − β_{1}α_{2}β_{2}]. The cosets are in M/rad_{kB}(M), where M = H^{5}(H, k). But according to Proposition 3(ii), α_{1}β_{2}^{2} + 2β_{1}α_{2}β_{2} lies in rad_{kB}(M), hence z_{2}x_{3} = \frac{3}{2}[α_{1}β_{2}^{2}] = \frac{3}{2}w_{6}, and this is non-zero since p > 3. Now the rest of (iii) is obvious, and (iv) follows from (i), (ii), and (iii).

Theorem 7. If p > 3 then E_{3} = E_{∞}. Hence for p ≥ 3, the Poincaré series for H^{∗}(G, k) is

p_{G}(t) = \frac{1 + t + 2t^{2} + 2t^{3} + ∑_{i=4}^{2p−1} t^{i}}{(1 − t)(1 − t^{2p})} .

Proof. We need to show that w_{7}, …, w_{2p−1} live to E_{∞}. Let m be a generator of (ℤ/pℤ)^{×}. Hence m^{p−1} = 1 but m^{i} ≠ 1 for 1 ≤ i ≤ p − 2. We may also consider m to be in k through the embedding ℤ/p ↪ k. Define automorphisms ϕ and ψ of G by ϕ(a^{i}b^{j}c^{k}) = a^{i}b^{mj}c^{mk} and ψ(a^{i}b^{j}c^{k}) = a^{mi}b^{j}c^{mk}. Since these automorphisms preserve H, they induce automorphisms of H^{∗}(H, k) and E_{n} (n ≥ 2), and one can easily check that these induced automorphisms simply multiply each generator by the scalar given in the following chart:

α_{ϵ} β_{ϵ} x_{1} γ_{ϵ} x_{2} y_{2} x_{3} z_{2i} z_{2i+1} w_{2i} w_{2i+1}
ϕ 1 m 1 m 1 m m m^{i+1} m^{i+1} m^{i} m^{i+1}
ψ m m m 1 m m^{2} m^{2} m^{i} m^{i} m^{i} m^{i+1}

Now let n ≥ 3 and assume d_{r} = 0 for 3 ≤ r < n; we show d_{n} = 0. We first consider the case where n = 2j + 1 is odd (j ≥ 1).

Consider d_{n}(w_{2i}) ∈ E_{3}^{2(j+1),2(i−j)−1} (4 ≤ i ≤ p − 1). If i − j ≥ 2 then employing the basis for E_{3} given in Corollary 6, we have

d_{n}(w_{2i}) = λ_{1}γ_{2}^{j+1}x_{1}x_{2}^{i−j−1} + λ_{2}γ_{2}^{j+1}x_{2}^{i−j−2}x_{3},  (13)

for unique λ_{1}, λ_{2} ∈ k. Applying ϕ to (13) yields

λ_{1}(1 − m^{i−j−1}) = 0 = λ_{2}(1 − m^{i−j−2}).

Since 1 ≤ i − j − 1 ≤ p − 2, m^{i−j−1} ≠ 1; hence λ_{1} = 0. If i − j > 2 then λ_{2} = 0 as well. If i − j = 2, we apply ψ to (13) and obtain λ_{2}(1 − m^{i−2}) = 0. Since 2 ≤ i − 2 ≤ p − 3, we have λ_{2} = 0. If i−j = 1 then d_{n}(w_{2i}) = λγ_{2}^{j+1}x_{1} for a unique λ ∈ k. Applying ψ yields λ(1 − m^{i−1}) = 0, forcing λ = 0.

Now consider d_{n}(w_{2i+1}) ∈ E_{3}^{2(j+1),2(i−j)} (3 ≤ i ≤ p − 1). Since all differentials into the horizontal edge vanish, we can assume i − j ≥ 1, so that

d_{n}(w_{2i+1}) = λ_{1}γ_{2}^{j+1}x_{2}^{i−j} + λ_{2}γ_{2}^{j+1}x_{2}^{i−j−1}y_{2},  (14)

for some λ_{1}, λ_{2} ∈ k. Applying ψ yields λ_{1}(1 − m^{j+1}) = 0 = λ_{2}(1 − m^{j}). Since 1 ≤ j ≤ i − 1 ≤ p − 2, we have λ_{2} = 0. If j < p − 2 then λ_{1} = 0 as well. If j = i − 1 = p − 2 then we apply ϕ to (14) and obtain λ_{1}(1 − m) = 0.

We now consider the case where n = 2j is even (j ≥ 2). Since E_{3}^{2i+1,s} = 0 for i ≥ 1 and 3 ≤ s ≤ 2p − 5, the only possible nonzero differentials in this case are those that map into E_{3}^{∗,2} or E_{3}^{∗,1}. In the former case, we must consider d_{n}(w_{2(j+1)}) = λγ_{2}^{j}z_{3} ∈ E_{3}^{2j+1,2}, in the latter, d_{n}(w_{2j+1}) = λγ_{2}^{j}z_{2} ∈ E_{3}^{2j+1,1}. In either case, applying ϕ yields λ(1 − m) = 0, hence λ = 0, completing the inductive step.

To get p_{G}(t), let L be the subring of E_{∞} generated by all of the generators other than x_{2p}. One then has L^{r,s} = 0 for r ≥ 1, s ≥ 2p, L^{r,s} = E_{∞}^{r,s} for s ≤ 2p − 1, and E_{∞} = k[x_{2p}] ⊗ L. It follows that

∑_{r+s=n} dim_{k}(L^{r,s}) = \begin{cases}n + 1 & n = 0, 1 \\ 4 & n = 2 \\ n + 3 & 3 ≤ n ≤ 2p − 1 \\ 2p + 2 & n ≥ 2p, \end{cases}

so 1−t times the Poincaré series for L is 1+t+2t^{2} +2t^{3} +∑_{i=4}^{2p−1} t^{i}.

6. Remarks

(1) One could probably go on from this point to deduce the full ring structure of H^{∗}(G, k), much as in [8], but it is not clear that this yields any new information or insight into the calculation. However, it is reassuring to note that the Poincaré series, as well as the number and the degrees of the generators, agree with the results in [8]!

(2) It would be interesting to know if there is a different way to calculate d_{2} in this spectral sequence. We know of no way other than the Charlap-Vasquez technique outlined here.

(3) It is natural to ask if the Charlap-Vasquez formula can be generalized to a formula for the higher differentials d_{r} (r ≥ 2). Indeed, this can be done (cf. [15]), and will be the subject of a future paper.

Acknowledgements

This paper is based on part of the author’s Ph.D. dissertation. Funding for this work was provided through an Office of Naval Research Graduate Fellowship and a Sloan Foundation Dissertation Fellowship. I am grateful to my advisor, Jon Alperin, for the advice and encouragement he provided during my time at the University of Chicago. Thanks are also due to Len Evens for many helpful conversations related to this field, and to the referee for several helpful comments.

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[15] S. F. Siegel, Cohomology and group extensions, Ph.D. Thesis, University of Chicago, 1993.

Mathematics Department, Northwestern University, Evanston, Illinois 60208-2730

E-mail address: siegel@math.nwu.edu