The cohomology of extraspecial groups

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

The extraspecial p-groups occupy a distinctive place in the cohomology of and representation theory of finite groups. They are minimal non-abelian in the sense that any non-trivial factor group is elementary abelian. Proving the extraspecial group case was a major hurdle in investigations such as Quillen [25], Chouinard [9], Avrunin and Scott [3] and Alperin and Evens [1], to name just a few. Many of the theorems in these papers could be proved easily by induction for all finite groups one the extraspecial case had been demonstrated Other problems, such as the characterisation of endotrivial modules (Dade [10, 11]), might need only the resolution of the extraspecial case for a complete solution. Many of these questions, even some on representation theory such as in [5], depend on a reasonable calculation of the cohomology of the extraspecial p-groups. The purposes of this paper are to provide a coherent and simplified account of much of the work which has been done in this area and to explain the current state of knowledge.

Briefly the work to date in the area can be summarised as follows. Twenty years ago, Quillen [24] calculated the mod 2 cohomology of the extraspecial 2-groups. The integral cohomology for these groups was calculated by Harada and Kono [15]. The corresponding calculations for p odd still elude us, although strong partial information has been obtained by Tezuka and Yagita [30]. The cohomology of the extraspecial groups of order p^{3} has been completely worked out by Lewis [20] in the integral case and by Diethelm [12] and Leary [19] in characteristic p.

A secondary purpose of our writing is to demonstrate a few of the many techniques which can be used in the field. Although occasionally we have repeated a known argument from the literature, we have tried to present improved or different versions of most proofs. For the sake of brevity, it has been necessary to assume that the reader has some knowledge of methods such as spectral sequences and Steenrod operations. We make one exception by including a section which gives the background on Bézout’s theorem. This material is probably less familiar to specialists in group cohomology and is difficult to find in an easily readable form.

∗Partially supported by a grant from the NSF.

In Sections 2–6 we treat the modular and integral cohomology of the extraspecial 2groups. Sections 2 and 3 are preliminary to the simplified version of Quillen’s calculation of the mod 2 cohomology (Theorem 4.1). The presentation avoids the case analysis of [24]. The main new ingredient is a statement (Lemma 3.2) about the relationship between a quadratic form and the Frobenius map. Next we analyse the Bockstein spectral sequence and use it to calculate the integral cohomology (Theorem 6.2). Our results are slightly more explicit than those given in [15]. An interesting feature of this calculation is that the spectral sequence converges as slowly as possible. In contrast, the Bockstein spectral sequence for the cohomology of the spin groups converges much more quickly, despite the close relationship between the mod 2 calculations (cf. Quillen [24], Section 6, Kono [16]).

In preparation for a treatment of the odd characteristic cases, we present a brief account of some relevant algebraic geometry in Section 7. We apply the geometry to the symplectic forms which come from the odd extraspecial groups in the next section. An extended and slightly simplified version of the calculation by Tezuka and Yagita of the mod p cohomology of extraspecial p-groups for p odd is given next. We complete the computation of the cohomology modulo the Jacobson radical (Theorem 10.1). In Section 11, we complete the calculation of the E_{2p+1} page of the Lyndon–Hochschild–Serre spectral sequence for these groups. The spectral sequence is itself an object of interest, as can be seen from [5]. The calculation of later pages of the spectral sequence seems to be very difficult except in the cases of the group of order p^{3}. This work has been accomplished by Diethelm [12] and Leary [19]. It is reviewed here only to the extent of showing how some of Leary’s calculations can be simplified using different techniques.

Finally, we prove a general result which in many ways sums up a lot of the philosophy of this work. Theorem 14.1 says that if 1 → N → G → Ḡ → 1 is a central extension with N cyclic of order p, then the subvariety of the kernel of the inflation map in the maximal ideal spectrum of H^{∗}(Ḡ, K) is the same as the subvariety of the ideal generated by the extension class in degree two and all of its images under all Steenrod operations.

2 Frobenius stable subspaces

The purpose of this section is to prove a lemma (quoted as “well known” in Quillen [24], Tezuka and Yagita [30]) needed in Sections 3 and 8. Let E be a vector space of dimension n over F_{p}, and let K be an algebraically closed field of characteristic p. Let V = K ⊗_{F_{p}} E, and denote by K[V] the coordinate ring of V; namely the ring of polynomial functions K[x_{1}, …, x_{n}], where x_{1}, …, x_{n} forms an F_{p}-basis for the dual of E. Denote by F : V → V the Frobenius morphism, namely the map of varieties corresponding to the K-linear ring homomorphism F : K[V] → K[V] sending each x_{i} to x_{i}^{p}.

Lemma 2.1 A K-linear subspace V′ ≤ V is F -stable if and only if V′ = K ⊗_{F_{p}} E′ for some F_{p}-linear subspace E′ ≤ E.

Proof This can be proved by a direct inductive argument, or as follows. It suffices to show that the linear span of the images of a vector v under powers of the Frobenius map is a proper subspace of V if and only if v is contained in K ⊗_{F_{p}} E′ for some subspace E′ of codimension one. But this follows immediately from the fact that

det ( \begin{matrix}x_{1} & ⋯ & x_{n} \\ x_{1}^{p} & ⋯ & x_{n}^{p} \\ ⋮ & & ⋮ \\ x_{1}^{p^{n−1}} & ⋯ & x_{n}^{p^{n−1}} \end{matrix} )

is a non-zero multiple of the product ∏(λ_{1}x_{1} + ⋯ + λ_{n}x_{n}), as (λ_{1}, …, λ_{n}) runs over nonzero representatives of the one dimensional subspaces of the F_{p}-dual of E. To prove this fact, notice that the two expressions have the same degree, and that whenever one of the λ_{1}x_{1} + ⋯ + λ_{n}x_{n} vanishes, so does the determinant. □

3 Quadratic forms

In this section, we describe the relationship between quadratic forms and the Frobenius map needed for the calculation of the cohomology of extraspecial 2-groups. The results of this section are true for quadratic forms over any finite field, with the same proofs, but for simplicity we shall work over F_{2}, which is all we need.

Recall that a quadratic form on a vector space V over a field K of characteristic two is a map q : V → K with the property that

q(x + y) = q(x) + q(y) + b(x, y) (3.1)

with b : V × V → K a (symmetric) bilinear form. A linear subspace W of V is said to be isotropic if q(w) = 0 for all w ∈ W. If K′ is an extension field of K, we extend q to a quadratic form on K′ ⊗_{K} V, also denoted q by abuse of notation, by setting

q(∑_{i} λ_{i}v_{i}) = ∑_{i} λ_{i}^{2}q(v_{i}) + ∑_{i<j} λ_{i}λ_{j}b(v_{i}, v_{j}).

Lemma 3.2 Suppose that K is a perfect field of characteristic 2. Let q be a quadratic form defined on a vector space E over F_{2}. Let V = K ⊗_{F_{2}} E, and denote by F : V → V the Frobenius morphism.

Let h be the codimension in E of a maximal isotropic subspace E′. Suppose that v is a vector in V with the property that the linear span of v, F (v), …, F^{r−1}(v) is isotropic, and r ≥ h. Then the linear span of v, F (v), …, F^{r}(v) is also isotropic.

Proof Let W = K ⊗_{F_{2}} E′, so that W is a maximal F -stable isotropic subspace of V. Since r + 1 > h, there is a linear relation of the form

w + μ_{s}F^{s}(v) + μ_{s+1}F^{s+1}(v) + ⋯ + μ_{t}F^{t}(v) = 0

with w ∈ W, 0 ≤ s ≤ t ≤ r and μ_{s}, μ_{t} non-zero. Note that we are not assuming that w ≠ 0. Since K is perfect, F is invertible and so without loss of generality s = 0. Then by applying F^{r−t} and adding if necessary, we obtain such a relation with t = r. Applying q(−) and b(w, −) to the above relation, we obtain

b(w, μ_{0}v + ⋯ + μ_{r}F^{r}(v)) + q(μ_{0}v + ⋯ + μ_{r}F^{r}(v)) = 0
b(w, μ_{0}v + ⋯ + μ_{r}F^{r}(v)) = 0.

Subtracting and using the fact that q(F^{i}(v)) = 0 and b(F^{i}(v), F^{j}(v)) = 0 for |i − j| < r, we obtain

μ_{0}μ_{r}b(v, F^{r}(v)) = 0.

Since μ_{0} and μ_{r} are non-zero, we deduce that b(v, F^{r}(v)) = 0, which completes the proof of the lemma. □

Now, keeping the notation of the lemma, assume that K is algebraically closed, and denote by K[V] the coordinate ring of V; namely the ring of polynomial functions on V. We regard q(v) as an element of K[V] given by a degree two polynomial, and b(v, F^{i}(v)) as an element of K[V] given by a polynomial of degree 2^{i} + 1.

Proposition 3.3 The sequence q(v), b(v, F (v)), b(v, F^{2}(v)), …, b(v, F^{h−1}(v)) is a regular sequence in K[V]; and b(v, F^{j}(v)) is in the radical of the ideal they generate, for j ≥ h. The variety defined by this ideal is the union of the F -stable isotropic subspaces of V; namely the union over the isotropic subspaces E′ ≤ E of K ⊗_{F_{2}} E′.

Proof Let J_{0} = {0}, and for 1 ≤ r ≤ ∞ we denote by J_{r} the ideal in K[V] generated by the sequence of elements q(v), b(v, F (v)), …, b(v, F^{r−1}(v)). Since

q(∑_{i=0}^{r−1} λ_{i}F^{i}(v)) = ∑_{i=0}^{r−1} λ_{i}^{2}q(v)^{2^{i}} + ∑_{0≤i<j≤r−1} b(v, F^{j−i}(v))^{2^{i}},

it follows that a vector w ∈ V lies in the subvariety V (J_{r}) if and only if the linear span of w, F (w), …, F^{r−1}(w) is isotropic. So by the lemma, we have

V (J_{h}) = V (J_{h+1}) = ⋯ = V (J_{∞}),

so that the new generators are in the radical of the ideal generated by the previous ones by Hilbert’s Nullstellensatz. Now V (J_{∞}) is just the union of the F -stable isotropic subspaces of V. By Lemma 2.1, an isotropic subspace of V is F -stable if and only if it is of the form K ⊗_{F_{2}} E′ for some isotropic subspace E′ ≤ E. So V (J_{∞}) is a subvariety of codimension h in V. Since J_{h} is generated by exactly h elements, it follows from a theorem of Macaulay that these elements form a regular sequence (see [33], Appendix 6, Theorem 2). □

In fact, we shall see in Theorem 4.1 that for r ≥ h, b(v, F^{r}(v)) is in the ideal generated by the sequence q(v), b(v, F (v)), b(v, F^{2}(v)), …, b(v, F^{h−1}(v)) and not just in its radical.

4 Mod 2 cohomology of extraspecial 2-groups

In this section, we examine the spectral sequence of a central extension

1 → N → G →^{ψ} E → 1

with N cyclic of order 2 and E an elementary abelian 2-group of order 2^{n}. This was treated in Quillen [24], but we repeat the calculation using the results of the last section, and using the Evens norm map to replace the Stiefel–Whitney class calculation used there. Strictly speaking for G to be extraspecial one requires N = Z(G) in the above situation. However, we shall abuse terminology by using the term even when this condition does not hold. Hence the results of this section apply also to the “almost” extraspecial groups G which satisfy the above conditions but have Z(G) ≅ Z/p^{2}.

If we write elements of N and E additively, we can identify N with F_{2} and E with a vector space over F_{2}. We define a quadratic form q : E → N and the associated bilinear form b : E × E → N by

q(x) = x̃^{2}, b(x, y) = [x̃, ỹ] = [x̃, ỹ]^{−1} = [ỹ, x̃]

where x̃ and ỹ are any elements such that ψ(x̃) = x and ψ(ỹ) = y. The condition (3.1) follows from the relation

(x̃ỹ)^{2} = x̃^{2}ỹ(ỹ^{−1}x̃^{−1}ỹx̃)ỹ = x̃^{2}ỹ^{2}[x̃, ỹ].

We have H^{∗}(E, F_{2}) = F_{2}[x_{1}, …, x_{n}] with deg(x_{i}) = 1. Let K be an algebraically closed field of characteristic two, and let V = K ⊗_{F_{2}} E, a vector space of dimension n over K. Then H^{∗}(E, K) = K[x_{1}, …, x_{n}] = K[V], the coordinate ring of V which we regard as an affine variety over K. The element of H^{2}(E, F_{2}) classifying the above central extension is q(v). This is regarded as a polynomial function on V of degree two as follows. Let e_{1}, …, e_{n} be a basis of E over F_{2}. Then if v = (x_{1}, …, x_{n}) = ∑_{i=1}^{n} x_{i}e_{i}, we have

q(v) = q(∑_{i=1}^{n} x_{i}e_{i}) = ∑_{i=1}^{n} q(e_{i})x_{i}^{2} + ∑_{1≤i<j≤n} b(e_{i}, e_{j})x_{i}x_{j}.

Thus the q(e_{i}) and b(e_{i}, e_{j}) are the coefficients of q(v) as a quadratic polynomial in K[V] with coefficients in F_{2}.

Recall that the Frobenius map F on K[V] is defined to be the K-linear ring homomorphism sending x_{i} to x_{i}^{2}. Thus we have

Sq^{1} q(v) = ∑_{1≤i<j≤n} b(e_{i}, e_{j})(x_{i}x_{j}^{2} + x_{j}x_{i}^{2}) = b(v, F (v)),

as a polynomial of degree three in K[V]. Here, Sq^{1} is the first Steenrod square. See Steenrod and Epstein [29] for the properties of these operations. Similarly for r ≥ 1 we have

Sq^{2^{r−1}}Sq^{2^{r−2}} ⋯ Sq^{1} q(v) = ∑_{1≤i<j≤n} b(e_{i}, e_{j})(x_{i}x_{j}^{2^{r}} + x_{j}x_{i}^{2^{r}}) = b(v, F^{r}(v)).

Theorem 4.1 (Quillen [24]) Let G be as above. Then letting v = (x_{1}, …, x_{n}),

H^{∗}(G, F_{2}) = F_{2}[x_{1}, …, x_{n}]/(q(v), b(v, F (v)), …, b(v, F^{h−1}(v))) ⊗ F_{2}[ζ].

Here, x_{1}, …, x_{n} are degree one elements inflated from H^{∗}(E, F_{2}). The integer h is the codimension in V of a maximal F -stable isotropic subspace; in other words, 2^{h} is the index in G of a maximal elementary abelian subgroup. The element ζ is any element of degree 2^{h} which restricts non-trivially to H^{∗}(N, F_{2}).

The elements b(v, F^{j}(v)), for j ≥ h, are in the ideal generated by the regular sequence q(v), b(v, F (v)), …, b(v, F^{h−1}(v)).

The kernel of the partial inflation map

F_{2}[x_{1}, …, x_{n}] = H^{∗}(E, F_{2}) = E_{2}^{∗,0} → E_{2^{r}+1}^{∗,0}

in the Lyndon–Hochschild–Serre spectral sequence for the central extension 1 → N → G → E → 1 is the ideal generated by q(v), b(v, F (v)), …, b(v, F^{r−1}(v)). For r ≤ h, this defines a subvariety of codimension r.

Proof We examine the Lyndon–Hochschild–Serre spectral sequence

H^{∗}(E, H^{∗}(N, F_{2})) ⇒ H^{∗}(G, F_{2}).

The E_{2} page of this spectral sequence is

H^{∗}(E, F_{2}) ⊗ H^{∗}(N, F_{2}) = F_{2}[x_{1}, …, x_{n}] ⊗ F_{2}[z]

with x_{1}, …, x_{n} ∈ E_{2}^{1,0} ≅ H^{1}(E, F_{2}) and z ∈ E_{2}^{0,1} ≅ H^{1}(N, F_{2}). Since q(v) ∈ H^{2}(E, F_{2}) is the element corresponding to the central extension, we have d_{2}(z) = q(v) ∈ E_{2}^{2,0}. Since Steenrod operations commute with transgressions (Serre [27], Section II.9 c), we have, for r ≥ 1,

d_{2^{r}+1}(z^{2^{r}}) = d_{2^{r}+1}(Sq^{2^{r−1}}Sq^{2^{r−2}} ⋯ Sq^{1}(z))
= Sq^{2^{r−1}}Sq^{2^{r−2}} ⋯ Sq^{1} q(v) = b(v, F^{r}(v)).

Now by Proposition 3.3, the elements q(v), b(v, F (v)), …, b(v, F^{h−1}(v)) form a regular sequence in H^{∗}(E, F_{2}), and so by induction on r we have, for 1 ≤ r ≤ h and 2^{r−1} + 1 ≤ j ≤ 2^{r},

E_{j}^{∗∗} = F_{2}[x_{1}, …, x_{n}]/(q(v), b(v, F (v)), …, b(v, F^{r−1}(v))) ⊗ F_{2}[z^{2^{r}}].

To complete the calculation, it suffices to show that the element z^{2^{h}} ∈ E_{2}^{0,2^{h}} is a universal cycle. Let H be a maximal elementary abelian subgroup of G, of index 2^{h}. Then there exists an element ẑ ∈ H^{1}(H, F_{2}) whose restriction to H^{1}(N, F_{2}) gives the element z ∈ H^{0}(E, H^{1}(N, F_{2})). One then easily checks using the Mackey formula for Evens’ norm map [13], that norm_{H,G}(ẑ) is an element of H^{∗}(G, F_{2}) restricting to z^{2^{h}} ∈ H^{∗}(N, F_{2}). Thus z^{2^{h}} is a universal cycle. In particular, for j ≥ h, d_{2^{j}+1}(z^{2^{j}}) = 0, and so b(v, F^{j}(v)) is in the ideal generated by q(v), b(v, F (v)), …, b(v, F^{h−1}(v)). □

5 The Bockstein spectral sequence

The exact sequence 0 → Z →^{p} Z →^{ϕ} Z/p → 0 gives rise to the exact couple

H^{∗}(G, Z) →^{p} H^{∗}(G, Z)

δ ↖ ↙ ϕ_{∗}

H^{∗}(G, Z/p).

The corresponding spectral sequence is known as the Bockstein spectral sequence:

H^{∗}(G, F_{p}) ⇒ (H^{∗}(G, Z)/torsion) ⊗ F_{p} = F_{p}.

Here, “torsion” means the Z-torsion elements of H^{∗}(G, Z) and this includes all elements of positive degree for G finite. The E_{1} term of this spectral sequence is the cochain complex

0 → H^{0}(G, F_{p}) →^{β} H^{1}(G, F_{p}) →^{β} H^{2}(G, F_{p}) → ⋯

where β = ϕ_{∗}δ is the Bockstein homomorphism. We use the term β-cohomology for the cohomology of this complex. It should be noted that, for the case which we wish to consider, where p = 2, we have β = Sq^{1}.

Recall from Theorem 4.1 that if

1 → N → G → E → 1

is a central extension with N cyclic of order two and E an elementary abelian 2-group of order 2^{n}, and with associated quadratic form q, then

H^{∗}(G, F_{2}) = R/(α_{1}, … α_{h}) ⊗ F_{2}[ζ],

where R = H^{∗}(E, F_{2}) = F_{2}[x_{1}, …, x_{n}], α_{1} = q(v), α_{i} = b(v, F^{i−1}(v)) for 2 ≤ i ≤ h, and ζ is an element representing z^{2^{h}} ∈ E_{∞}^{0,2^{h}}. We shall calculate H^{∗}(G, Z) by examining the Bockstein spectral sequence.

Lemma 5.1 The action of the Bockstein homomorphism β on R, given on the x_{i} by β(x_{i}) = x_{i}^{2}, is exact in positive degrees. If h ≥ 2, then we have

β(α_{1}) = α_{2}
β(α_{2}) = 0
β(α_{i}) = α_{i−1}^{2} for 3 ≤ i ≤ h.

There is an element γ_{1} of degree 3 with β(γ_{1}) = α_{1}^{2}. There are elements γ_{j}, 2 ≤ j ≤ h, with deg(γ_{j}) = 2^{j−1}, and satisfying γ_{2} = α_{1} (so that β(γ_{2}) = α_{2}) and for 3 ≤ j ≤ h,

β(γ_{j}) = α_{j} + ∑_{i=2}^{j−1} γ_{i}^{2^{j−i}−1}β(γ_{i}).

The elements γ_{j} are well defined modulo Im(β) and the ideal generated by the α_{i} with i < j.

Proof For a cyclic group of order p, it is clear that the Bockstein homomorphism is exact in positive degrees. Applying the Künneth theorem to the Bockstein regarded as a derivation, it follows that the same is true for all elementary abelian p-groups.

Since β = Sq^{1}, we have β(α_{1}) = α_{2}, and since β ∘ β = 0 we have β(α_{2}) = 0. For 3 ≤ i ≤ h we have

β(α_{i}) = Sq^{1}Sq^{2^{i−1}}α_{i−1} = Sq^{2^{i−1}+1}α_{i−1} = α_{i−1}^{2}.

Since β is exact in positive degrees, and β(α_{1}^{2}) = 2α_{1}β(α_{1}) = 0, there is an element γ_{1} of degree 3 with β(γ_{1}) = α_{1}^{2}. We set γ_{2} = α_{1}, so that β(γ_{2}) = α_{2}. Now let 3 ≤ j ≤ h, and suppose by induction on j that we have elements γ_{2}, …, γ_{j−1} with the given properties. Then

β(α_{j} + ∑_{i=2}^{j−1} γ_{i}^{2^{j−i}−1}β(γ_{i})) = α_{j−1}^{2} + ∑_{i=2}^{j−1} γ_{i}^{2^{j−i}−2}β(γ_{i})^{2}

= α_{j−1}^{2} + β(γ_{j−1})^{2} + (∑_{i=2}^{j−2} γ_{i}^{2^{j−i−1}−1}β(γ_{i}))^{2}
= β(γ_{j−1})^{2} + β(γ_{j−1})^{2} = 0.

So by exactness of β in positive degrees, there exists an element γ_{j} with β(γ_{j}) as given, and this element is unique modulo Im(β). It is easy to check also that altering the γ_{i} (i < j) by an element of Im(β) will alter γ_{j} by an element of the ideal generated by the α_{i} (i < j). □

Proposition 5.2 If h ≥ 2 then the β-cohomology of R/(α_{1}, …, α_{h}) is an exterior algebra

Λ(γ_{1}, γ_{3}, γ_{4}, …, γ_{h}),

where the γ_{j} are the elements defined in Lemma 5.1 with deg(γ_{1}) = 3 and deg(γ_{j}) = 2^{j−1}, 3 ≤ j ≤ h.

Proof For 2 ≤ j ≤ h, the ideal (α_{1}, …, α_{j}) is closed under the action of β. So we calculate inductively the β-cohomology of R/(α_{1}, …, α_{j}).

We start with R/(α_{1}, α_{2}). By the lemma, the β-cohomology of R is spanned by 1 in degree zero. Since β(α_{2}) = 0, and α_{2} is a non zero divisor, multiplication by α_{2} is an isomorphism of β-modules R → (α_{2}), so the β-cohomology of the ideal (α_{2}) is spanned by α_{2} in degree 3. Now examine the long exact sequence in β-cohomology associated to the short exact sequence of cochain complexes

0 → (α_{2}) → R → R/(α_{2}) → 0.

The connecting homomorphism is just β, and since β(α_{1}) = α_{2}, the β-cohomology of R/(α_{2}) is spanned by 1 and α_{1}.

Since α_{1} is a non zero divisor modulo (α_{2}) (Proposition 3.3) and β(α_{1}) = α_{2}, multiplication by α_{1} is an isomorphism of β-modules R/(α_{2}) → (α_{1}, α_{2})/(α_{2}), and so the β-cohomology of (α_{1}, α_{2})/(α_{2}) is spanned by α_{1} and α_{1}^{2}.

Finally, by examining the long exact sequence in β-cohomology of

0 → (α_{1}, α_{2})/(α_{2}) → R/(α_{2}) → R/(α_{1}, α_{2}) → 0

we see that since β(γ_{1}) = α_{1}^{2} (Lemma 5.1), the β-cohomology of R/(α_{1}, α_{2}) is spanned by 1 and γ_{1}.

Now assume by induction on j that the β-cohomology of R/(α_{1}, …, α_{j−1}) is an exterior algebra Λ(γ_{1}, γ_{3}, …, γ_{j−1}) (where the γ_{i} are as in Lemma 5.1). Then since α_{j} is a non zero divisor modulo (α_{1}, …, α_{j−1}) (Proposition 3.3) and β(α_{j}) = α_{j−1}^{2}, we see that the β-cohomology of (α_{1}, …, α_{j})/(α_{1}, …, α_{j−1}) is α_{j}.Λ(γ_{1}, γ_{3}, …, γ_{j−1}). By examining the long exact sequence in β-cohomology of

0 → (α_{1}, …, α_{j})/(α_{1}, …, α_{j−1}) → R/(α_{1}, …, α_{j−1}) → R/(α_{1}, …, α_{j}) → 0

we see that since β(γ_{j}) ≡ α_{j} mod (α_{1}, …, α_{j−1}) (Lemma 5.1), the β-cohomology of R/(α_{1}, …, α_{j}) is Λ(γ_{1}, γ_{3}, …, γ_{j}). Taking j = h, this completes the proof. □

Theorem 5.3 (A) If h ≥ 2, then the β-cohomology of H^{∗}(G, F_{2}) is

Λ(γ_{1}, γ_{3}, γ_{4}, …, γ_{h}) ⊗ F_{2}[ζ],

with deg(γ_{1}) = 3, deg(γ_{j}) = 2^{j−1} (j ≥ 3) and deg(ζ) = 2^{h}. Numbering so that this is the E_{2} page of the Bockstein spectral sequence, we have for r ≤ h

E_{r}^{∗} = Λ(γ_{1}γ_{3} … γ_{r}, γ_{r+1}, …, γ_{h}) ⊗ F_{2}[ζ],

E_{h+1}^{∗} = E_{h}^{∗}, and E_{r}^{∗} = F_{2} for r ≥ h + 2. The differentials are given by β_{2}(γ_{1}) = γ_{3}, β_{r}(γ_{1}γ_{3} … γ_{r}) = γ_{r+1}, β_{r}(γ_{i}) = 0 for r < i, β_{r}(ζ) = 0 for r < h+1 and β_{h+1}(γ_{1}γ_{3} … γ_{h}) = ζ.

(B) If h = 1, then either

(i) G ≅ Z/4 × elementary abelian, in which case after a suitable change of basis, q(v) = x_{1}^{2}, and the β-cohomology of H^{∗}(G, F_{2}) is

H^{∗}(Z/4, F_{2}) ≅ F_{2}[x_{1}]/(x_{1}^{2}) ⊗ F_{2}[ζ]

with deg(ζ) = 2 and β_{2}(x_{1}) = ζ, or

(ii) G ≅ D_{8} × elementary abelian, in which case after a suitable change of basis, q(v) = x_{1}x_{2}, and β(ζ) = (x_{1} + x_{2})ζ. The β-cohomology of H^{∗}(G, F_{2}) is

F_{2}[x_{1}ζ, ζ^{2}]/((x_{1}ζ)^{2})

and β_{2}(x_{1}ζ) = ζ^{2}.

(C) If h = 0 then G is elementary abelian and the β-cohomology of H^{∗}(G, F_{2}) is F_{2}.

Proof The cases with h ≤ 1 are easy to calculate explicitly, so we shall assume that h ≥ 2.

To calculate the β-cohomology of H^{∗}(G, F_{2}), by Proposition 5.2 it suffices to show that the representative ζ ∈ H^{∗}(G, F_{2}) of z^{2^{h}} ∈ E_{∞}^{0,2^{h}} may be chosen so that β(ζ) = 0. Now the subgroup Out(G, N) of Out(G) stabilising N is the orthogonal group of the quadratic form O(q), and acts on the spectral sequence of the central extension. If β(ζ) has nonzero image in E_{∞}^{1,2^{h}} (≅ E) then this element is invariant under O(q). As long as h ≥ 2, there is no such O(q)-invariant element. So β(ζ) is some element of E_{∞}^{2^{h}+1,0}. But by Proposition 5.2, β is exact in degree 2^{h} + 1 on E_{∞}^{∗,0} ≅ R/(α_{1}, …, α_{h}). So by rechoosing the representative ζ of z^{2^{h}} if necessary, we have β(ζ) = 0. This completes the calculation of the E_{2} page of the Bockstein spectral sequence. Note that we are numbering the spectral sequence in such a way that β_{n} is the differential on E_{n}^{∗}.

Now the differential β_{2} must vanish on γ_{3}, γ_{4}, …, γ_{h} and ζ, because there is nothing in the appropriate degrees for them to hit. It follows that the only possibility for a non-zero differential is γ_{1} ↦ γ_{3}. So for some value of j_{2} ≥ 2 we have β_{j_{2}}(γ_{1}) = γ_{3}, β_{j_{2}}(γ_{i}) = 0 (i ≥ 3) and β_{j_{2}}(ζ) = 0. It follows that

E_{j_{2}+1}^{∗} = Λ(γ_{1}γ_{3}, γ_{4}, …, γ_{h}) ⊗ F_{2}[ζ].

Now we have deg(γ_{1}γ_{3}) = 7, and so the only possibility for a non-zero differential is γ_{1}γ_{3} ↦ γ_{4}. So for some value of j_{3} > j_{2} we have β_{j_{3}}(γ_{1}γ_{3}) = γ_{4}, β_{j_{3}}(γ_{i}) = 0 (i ≥ 4), β_{j_{3}}(ζ) = 0.

Continuing in this way, we see that there are integers

2 ≤ j_{2} < j_{3} < ⋯ < j_{h}

such that for 3 ≤ r < h, β_{j_{r}}(γ_{1}γ_{3} … γ_{r}) = γ_{r+1}, β_{j_{r}}(γ_{i}) = 0 (i ≥ r + 1), β_{j_{r}}(ζ) = 0 and β_{j_{h}}(γ_{1}γ_{3} … γ_{h}) = ζ.

It remains to prove that j_{r} = r for r < h and j_{h} = h+1. If H is a maximal elementary abelian subgroup of G, of index 2^{h}, and w is an element of H^{∗}(G, Z) whose reduction modulo two is ζ, then 2 res_{G,H}(w) = 0, and so

2^{h+1}w = tr_{H,G}(2 res_{G,H}(w)) = 0

so that j_{h} ≤ h + 1.

If the values of j_{r} are not as stated, then j_{h} − j_{h−1} = 1. Choose a subgroup G_{0} of G of index 2, containing N, and for which the value of h is one less than the value of h for G. The differentials in the Bockstein spectral sequence commute with restriction to G_{0}. So if h = 2, then ζ restricts to an element of H^{∗}(G_{0}, F_{2}) which survives to E_{2}^{∗} but not E_{3}^{∗}, whereas γ_{1} restricts to zero. Therefore β_{2}(γ_{1}) = 0 and β_{3}(γ_{1}) = ζ.

If h ≥ 3 we work by induction on h. The element res_{G,G_{0}}(γ_{1}γ_{3} … γ_{h−1}) survives to E_{j_{h−1}}^{∗}, and

β_{j_{h−1}}res_{G,G_{0}}(γ_{1}γ_{3} … γ_{h−1}) = res_{G,G_{0}}(β_{j_{h−1}}(γ_{1}γ_{3} … γ_{h−1}))
= res_{G,G_{0}}(γ_{h}) = 0,

so res_{G,G_{0}}(γ_{1}γ_{3} … γ_{h−1}) survives to E_{j_{h−1}+1}^{∗}, and hence by the inductive hypothesis applied to G_{0} we see that j_{h−1} = h − 1. Moreover,

res_{G,G_{0}}β_{h}(γ_{1}γ_{3} … γ_{h}) = β_{h}res_{G,G_{0}}(γ_{1}γ_{3} … γ_{h}) = β_{h}(0) = 0

and it follows that j_{h} = h + 1. □

6 Integral cohomology of extraspecial 2-groups

We first describe the integral cohomology of an elementary abelian 2-group. If E is an elementary abelian 2-group of order 2^{n}, then every element in positive degree in H^{∗}(E, Z) is killed by multiplication by two. It follows that the task of calculating H^{∗}(E, Z) is equivalent to calculating the image = kernel of the Bockstein β on H^{∗}(E, F_{2}) = F_{2}[x_{1}, …, x_{n}], since in positive degrees, reduction modulo two is an isomorphism between H^{∗}(G, Z) and this image.

If I is a non-empty subset of {1, …, n}, we denote by u_{I} the element of H^{∗}(E, Z) whose reduction modulo two is given by

ū_{I} = β (∏_{i∈I} x_{i}) = (∏_{i∈I} x_{i}) (∑_{i∈I} x_{i}) .

We write u_{i} for u_{{i}}, so that ū_{i} = x_{i}^{2}.

Proposition 6.1 H^{∗}(E, Z) is generated by the (graded commuting) elements u_{I}, ∅ ≠ I ⊆ {1, …, n}, subject to the following relations:

(i) 2 u_{I} = 0,

(ii) u_{I}u_{J} = ∑_{i∈I} u_{i}u_{(I∪J)\{i}} if I ∩ J = ∅,

(iii) u_{I}u_{J} = (∏_{j∈J} u_{j}) (∑_{j∈J} u_{(I\J)∪{j}}) if I ⊇ J,

(iv) u_{I}u_{J} = u_{I∪J}u_{I∩J} + u_{I\J}u_{J\I} ( ∏_{j∈I∩J} u_{j}) otherwise,

(v) ∑_{i∈I} u_{i}u_{I\{i}} = 0.

In the presence of relation (i), the relations (ii)–(v) are equivalent to the single relation

(vi) u_{I}u_{J} = ( ∏_{j∈I∩J} u_{j}) ( ∑_{j∈I∩J} u_{(I△J)∪{j}} + ∑_{j∈J\I} u_{j}u_{(I△J)\{j}}) with no restriction on

I and J (non-empty). Here, I△J denotes the symmetric difference (I \ J) ∪ (J \ I).

Proof The elements ū_{I} generate the image of β on H^{∗}(E, F_{2}) since the Bockstein of any monomial a^{2} ∏_{i∈I} x_{i} is a^{2}β (∏_{i∈I} x_{i}), and any square is in the subring generated by the ū_{i}.

It is easy to check that the relations (i)–(v) hold between the ū_{I}, and hence between the u_{I}, and that in the presence of (i), the relations (ii)–(v) are equivalent to (vi). Modulo these relations, every element of positive degree can be expressed as a linear combination of elements of the form

(monomial in the u_{i}).u_{I}

with I a non-empty subset (not necessarily of size bigger than one) whose largest element is at least as large as the largest of the i appearing in the monomial. Elements of this form are linearly independent modulo two, because the last monomials in the x_{i} appearing in their reductions are distinct. □

To avoid spending excessive time on trivialities, we shall only consider extraspecial 2-groups for which the codimension h of a maximal isotropic subspace is at least two. The cases with h = 1 are easy to deal with using the information given in Theorem 5.3. We continue with the notation of Section 5.

Theorem 6.2 Suppose that 1 → N → G → E → 1 is a central extension with N cyclic of order two and E an elementary abelian 2-group of order 2^{n}, with associated quadratic form q, and h ≥ 2. Then H^{∗}(G, Z) is generated by elements u_{I}, ∅ ≠ I ⊆ {1, …, n} (inflated from E), v_{j}, 3 ≤ j ≤ h, of degree 2^{j−1}, and w of degree 2^{h}, subject to the relations (i) and (vi) of Proposition 6.1, together with the following relations:

(vii) v_{j}^{2} = 0 for 3 ≤ j ≤ h,

(viii) the polynomials in the u_{I} reducing modulo two to β(γ_{j}), 1 ≤ j ≤ h, are equal to zero,

(ix) the polynomials in the u_{I} reducing modulo two to γ_{j} are equal to 2^{j−2}v_{j} for 3 ≤ j ≤ h,

(x) 2^{h+1}w = 0.

Proof This follows immediately from the structure of H^{∗}(E, Z) given in Proposition 6.1 and the behaviour of the Bockstein spectral sequence for H^{∗}(G, F_{2}) described in Theorem 5.3. For 3 ≤ j ≤ h, the elements v_{j} correspond to the fact that γ_{j} is in the image of β_{j−1}. Thus 2^{j−1}v_{j} = 0 and 2^{j−2}v_{j} reduces modulo two to γ_{j}. The element w corresponds to the fact that ζ is in the image of β_{h+1}, so that 2^{h+1}w = 0 and 2^{h}w reduces modulo two to ζ. Note also that the intersection of the ideal (α_{1}, …, α_{h}) with the image of β is generated by the β(γ_{j}), 1 ≤ j ≤ h. □

7 Bézout’s Theorem

Next, we wish to prove the analogue of Proposition 3.3 for symplectic forms. In contrast to the case of quadratic forms, it turns out that the corresponding ideal is always a radical ideal. To prove this, as in Quillen [24] and Tezuka and Yagita [30], we require the special case of Bézout’s theorem for complete intersections. It is hard to extract a short proof of this from the literature, so we consider it worthwhile including a discussion of this theorem here. Other discussions may be found in for example Section 27 of Krull [17], Section 8 of Fulton [14], Boda and Vogel [6]. For the sake of completeness, we end this section with a discussion of Serre’s version [28] of Bézout’s theorem for more general intersections; we shall not need this version in this paper, but we hope the reader will find the discussion enlightening. We advise the reader not familiar with primary decomposition to begin by reading for example Section 6 of Matsumura [21].

Suppose that R = K[x_{1}, …, x_{n}] is a polynomial ring, regarded as a graded ring with deg(x_{i}) = 1, and suppose that M is a finitely generated graded R-module. Then the Poincaré series

p(M, t) = ∑ t^{i} dim_{K} M_{i}

is a rational function of t of the form f(t)/(1−t)^{d}, where f(t) is a polynomial with integer coefficients with f(1) > 0, and the integer d is the Krull dimension dim(M) (see Atiyah and Macdonald [2], Chapter 11 and Zariski and Samuel [33], Chapter 8, §§8 and 10). The positive integer f(1) is called the degree deg(M). It is equal to (d − 1)! times the leading coefficient of the Hilbert polynomial of M. In other words, as n gets large, we have

dim_{K} M_{n} = \frac{f(1)n^{d−1}}{(d − 1)!} + O(n^{d−2}).

If I is an ideal in R, we define the degree deg(I) to be the degree of the R-module R/I.

Lemma 7.1 If I is an ideal in R = K[x_{1}, …, x_{n}] generated by a regular sequence of homogeneous elements α_{1}, …, α_{h} then

deg(I) = deg(α_{1}) … deg(α_{h}).

Proof First notice that the subring S = K[α_{1}, …, α_{h}] is a polynomial subring. Because α_{1}, …, α_{h} is a regular sequence, R is a free S-module. The Poincaré series for S is 1/ ∏_{i=1}^{h}(1 − t^{n_{i}}) where n_{i} = deg(α_{i}). As an S-module, R has a free basis consisting of any homogeneous set of representatives of a homogeneous K-basis for the graded ring R/I. So if p(R/I, t) is the Poincaré series for R/I, then the Poincaré series for R is given by

\frac{1}{(1 − t)^{n}} = \frac{p(R/I, t)}{(1 − t^{n_{1}}) ⋯ (1 − t^{n_{h}})},

where n_{i} = deg(α_{i}). Thus

p(R/I, t) = \frac{(1 + t + ⋯ + t^{n_{1}−1}) … (1 + t + ⋯ + t^{n_{h}−1})}{(1 − t)^{n−h}}.

Thus the Krull dimension is n − h, and substituting t = 1 in (1 − t)^{n−h}p(R/I, t) we obtain the required formula for the degree. □

In particular, we remark that according to this lemma the notation deg(α) is unambiguous for the degree of the principal ideal (α).

Lemma 7.2 Suppose that 0 → M → M′ → M′′ → 0 is a short exact sequence of finitely generated R-modules. Then one of the following holds:

(i) The Krull dimensions of M, M′ and M′′ are equal, and

deg(M′) = deg(M) + deg(M′′).

(ii) The Krull dimensions of M and M′ are equal, that of M′′ is strictly smaller, and deg(M) = deg(M′).

(iii) The Krull dimensions of M′ and M′′ are equal, that of M is strictly smaller, and deg(M′) = deg(M′′).

Proof This is clear by looking at the behaviour of Poincaré series. □

Lemma 7.3 If p is a prime ideal in R = K[x_{1}, …, x_{n}], and M is a finitely generated pprimary R-module, then the localisation M_{p} has finite length l_{p}(M) ≠ 0 as an R_{p}-module, and

deg(M) = l_{p}(M) deg(p).

Proof According to Matsumura [21], Theorem 6.4, there exists a chain of submodules

0 = M^{0} ⊆ M^{1} ⊆ ⋯ ⊆ M^{s} = M

such that for each i we have M^{i}/M^{i−1} ≅ A/p_{i} with p_{i} ∈ Spec(A). Since the localisation M_{p_{i}} is non-zero, p_{i} ∈ Supp(M) ([21], §4) and so by [21] Theorem 6.5, p_{i} ⊇ p. Since localisation is exact, l_{p}(M) is the number of ideals p_{i} which are equal to p. Since for p_{i} properly containing p, we have dim A/p_{i} < dim A/p, it follows from Lemma 7.2 that deg(M) = l_{p}(M) deg(p). □

Lemma 7.4 Suppose that the finitely generated graded R-module M, has shortest primary decomposition

⋂_{i=1}^{s} M_{(i)} = {0}

with M/M_{(i)} a p_{i}-primary module. Then we have a short exact sequence

0 → M → ⨁_{i=1}^{s} M/M_{(i)} → M′ → 0

with dim M′ < dim M, and

deg(M) = ∑_{dim(A/p_{i})=dim(M)} deg(M/M_{(i)}).

Proof To show that dim M′ < dim M we examine the support Supp(M′). We have

Supp(M′) ⊆ ⋃_{i=1}^{s} Supp(M/M_{(i)}) = Supp(M)

by [21] Theorem 6.5. Moreover, if p_{i} is a minimal prime in Ass(M), or equivalently a minimal prime in Supp(M), then by [21] Theorem 6.8 (iii) the map M_{p_{i}} → (M/M_{(i)})_{p_{i}} is an isomorphism and so p_{i} ∉ Supp(M′). Thus dim M′ < dim M, and so by Lemma 7.2 we have

deg(M) = deg(⨁_{i=1}^{s} M/M_{(i)}) = ∑_{dim(A/p_{i})=dim(M)} deg(M/M_{(i)}). □

Lemma 7.5 If q is a p-primary ideal then l_{p}(R/q) = 1 if and only if q = p.

Proof If l_{p}(R/q) = 1 then the exact sequence

0 → p/q → R/q → R/p → 0

shows that l_{p}(p/q) = 0, so that p is not an associated prime of p/q. But p/q is a submodule of a p-primary module, so Ass_{R}(p/q) = ∅ and hence by [21] Theorem 6.1, p/q = {0}. □

Proposition 7.6 (Bézout’s Theorem for complete intersections)

Suppose that I is an ideal in R = K[x_{1}, …, x_{n}] generated by a regular sequence α_{1}, …, α_{h}, with α_{j} homogeneous of degree deg(α_{j}). If I has primary decomposition I = ⋂_{i=1}^{s} q_{i}, where q_{i} is p_{i}-primary, then

∏_{j=1}^{h} deg(α_{j}) = ∑_{i=1}^{s} l_{p_{i}}(R/q_{i}) deg(p_{i}). (7.7)

We have I = \sqrt{I} if and only if each l_{p_{i}}(q_{i}) = 1.

Proof Since R is a Cohen–Macaulay ring, it follows from Zariski and Samuel [33], Theorem 2 of Appendix 6, that the Krull dimension of each R/q_{i} is equal to n − h. In other words, I is equidimensional (unmixed) with no embedded primes. Thus we have, by Lemmas 7.1, 7.3 and 7.4,

∏_{j=1}^{h} deg(α_{j}) = deg(I) = ∑_{i=1}^{s} deg(q_{i}) = ∑_{i=1}^{s} l_{p_{i}}(R/q_{i}) deg(p_{i}).

We have I = \sqrt{I} if and only if each q_{i} is equal to p_{i}, which by Lemma 7.5 happens if and only if each l_{p_{i}}(R/q_{i}) = 1. □

In the above version of Bézout’s theorem for complete intersections, the local length is taken as the definition of intersection multiplicity. More generally, this is the wrong definition, as was shown in an example by van der Waerden [31], §11, using Macaulay’s space curve. There are (at least) two good algebraic definitions of intersection multiplicity in the general case. Samuel’s definition [26] uses Hilbert polynomials, while Serre’s [28] is homological. A proof of the equivalence of these definitions may be found in Section 4.3 of Serre [28], and a proof of Bézout’s theorem using Samuel’s definition may be found in Boda and Vogel [6].

If I_{1} and I_{2} are homogeneous ideals in R = K[x_{1}, …, x_{n}], then Serre defines the intersection multiplicity as follows. If p is a minimal associated prime of the ideal (I_{1}, I_{2}) generated by I_{1} and I_{2} then the intersection multiplicity is given by

i(I_{1}, I_{2}; p) = ∑_{j≥0}(−1)^{j}l_{p}(Tor_{j}^{R}(R/I_{1}, R/I_{2}))

so that the term with j = 0 is l_{p}(R/I_{1} ⊗_{R} R/I_{2}) = l_{p}(R/(I_{1}, I_{2})), and the remaining terms may be thought of as correction terms for the fact that the varieties defined are not complete intersections.

Theorem 7.8 (Bézout’s Theorem)

If I_{1} and I_{2} are homogeneous ideals in R = K[x_{1}, …, x_{n}] satisfying n+dim R/(I_{1}, I_{2}) = dim R/I_{1} + dim R/I_{2} then

deg(I_{1}) deg(I_{2}) = ∑_{p} i(I_{1}, I_{2}; p) deg(p)

where the sum runs over those associated primes p of (I_{1}, I_{2}) for which R/p has the largest Krull dimension.

Proof We proceed by reduction to the diagonal. Geometrically, this consists of the observation that V_{1} ∩ V_{2} is isomorphic to the intersection of V_{1} × V_{2} ⊆ A^{2n} with the diagonal. Algebraically, we proceed as follows. We regard R as an R ⊗_{K} R-module via multiplication R⊗_{K} R → R. If P is a projective resolution of R/I_{1} as an R-module and Q is a projective resolution of R/I_{2} as an R-module, then P ⊗_{K} Q is a projective resolution of R/I_{1} ⊗_{K} R/I_{2} as an R ⊗_{K} R-module, and P ⊗_{R} Q = (P ⊗_{K} Q) ⊗_{R⊗_{K}R} R, so

Tor_{j}^{R}(R/I_{1}, R/I_{2}) ≅ Tor_{j}^{R⊗_{K}R}(R/I_{1} ⊗_{K} R/I_{2}, R)

So we have reduced to the case where the second ideal is generated by a regular sequence of degree one elements I_{2} = (α_{1}, …, α_{h}). In this case, we can calculate Tor by using the Koszul complex. Namely, let K(α_{1}, …, α_{h}) = K(α_{1}) ⊗_{K} ⋯ ⊗_{K} K(α_{h}) with K(α_{i}) equal to the two-term complex ⋯ → 0 → R →^{α_{i}} R → 0 → ⋯ with the non-zero terms in degrees zero and one. Since K(α_{1}, …, α_{h}) is a projective resolution of R/(α_{1}, …, α_{h}) as an Rmodule, Tor_{j}^{R}(R/I_{1}, R/I_{2}) is the homology of the chain complex R/I_{1} ⊗_{R} K(α_{1}, …, α_{h}). Now the term of this complex of homological degree j consists of \binom{h}{j} copies of R/I_{1} shifted in internal degree by j, and so the Poincaré series is given by

p(R/I_{1} ⊗_{R} K_{j}(α_{1}, …, α_{h}), t) = \binom{h}{j}t^{j}p(R/I_{1}, t).

Since the alternating sum of the Poincaré series of the terms in a complex of graded modules is the same as the alternating sum of the Poincaré series of the homology, we have

∑_{j=0}^{h}(−1)^{j}p(Tor_{j}^{R}(R/I_{1}, R/I_{2}), t) = ∑_{j=0}^{h}(−1)^{j}p(R/I_{1} ⊗ K_{j}(α_{1}, …, α_{h}), t)

= ∑_{j=0}^{h}(−1)^{j}\binom{h}{j}t^{j}p(R/I_{1}, t) = p(R/I_{1}, t)(1 − t)^{h}.

So the degrees on the left and right agree, and so by Lemma 7.4 we have

deg(I_{1}) = ∑_{j=0}^{h}(−1)^{j} deg Tor_{j}^{R}(R/I_{1}, R/I_{2})

= ∑_{p} ∑_{j=0}^{h}(−1)^{j}l_{p}Tor_{j}^{R}(R/I_{1}, R/I_{2}) deg p

where the sum runs over the primes p for which R/p has the largest Krull dimension. Since deg(I_{2}) = 1, this proves the theorem. □

8 Symplectic forms

For p odd, we need to consider symplectic forms instead of quadratic forms. Recall that a symplectic form on a vector space V over a field K is a bilinear form b : V × V → K with the property that b(x, x) = 0 and b(x, y) = −b(y, x). A linear subspace W of V is said to be isotropic if b(w, w′) = 0 for all w and w′ in W. Again, b extends in an obvious way to K′ ⊗_{K} V for K′ an extension field of K.

Lemma 8.1 Suppose that K is a field of characteristic p. Let b be a symplectic form defined on a vector space E over F_{p}. Let V = K ⊗_{F_{p}} E, and denote by F : V → V the Frobenius morphism.

Let h be the codimension in E of a maximal isotropic subspace E′. If v is a vector in V with the property that the linear span of v, F (v), …, F^{r}(v) is isotropic, with r ≥ h, then the linear span of v, F (v), …, F^{r+1}(v) is also isotropic.

Proof Let W = K ⊗_{F_{p}} E′, so that W is a maximal F -stable isotropic subspace of V. As in the proof of Lemma 3.2, we obtain a linear relation of the form

w + μ_{0}v + μ_{1}F (v) + ⋯ + μ_{r}F^{r}(v) = 0

with w ∈ W and μ_{0}, μ_{r} non-zero. Again, we are not assuming that w ≠ 0. Applying F, we obtain

F (w) + μ_{0}^{p}F (v) + ⋯ + μ_{r}^{p}F^{r+1}(v) = 0.

Since w and F (w) lie in W, which is isotropic, we obtain from these two equations

0 = b(w, F (w)) = μ_{0}μ_{r}^{p}b(v, F^{r+1}(v)).

Since μ_{0} and μ_{r} are non-zero, we deduce that b(v, F^{r+1}(v)) = 0, which completes the proof of the lemma. □

Now assume that K is algebraically closed, and denote by K[V] the coordinate ring of V. Using the same argument as in Proposition 3.3, together with Bézout’s theorem, we obtain the following.

Proposition 8.2 (Tezuka and Yagita [30])

The sequence b(v, F (v)), b(v, F^{2}(v)), …, b(v, F^{h}(v)) is a regular sequence in K[V]. The ideal generated by these elements is a radical ideal, and contains b(v, F^{r}(v)) for r ≥ h + 1. The variety defined by this ideal is the union of the F -stable isotropic subspaces of V; namely the union over the isotropic subspaces E′ ≤ E of K ⊗_{F_{p}} E′.

Proof For r = 0 set J_{r} = {0}, and for 1 ≤ r ≤ ∞ we denote by J_{r} the ideal in K[V] generated by the elements b(v, F (v)), ⋯, b(v, F^{r}(v)). Since

b(∑_{i=0}^{r} λ_{i}F^{i}(v), ∑_{j=0}^{r} μ_{j}F^{j}(v)) = ∑_{0≤i<j≤r}(λ_{i}μ_{j} − λ_{j}μ_{i})b(v, F^{j−i}(v))^{p^{i}},

it follows that a vector v ∈ V lies in the subvariety V (J_{r}) if and only if the linear span of v, F (v), …, F^{r}(v) is isotropic. So by the lemma, we have

V (J_{h}) = V (J_{h+1}) = ⋯ = V (J_{∞})

so that the new generators are in the radical of the ideal generated by the previous ones by Hilbert’s Nullstellensatz. Now V (J_{∞}) is just the union of the F -stable isotropic subspaces of V. By Lemma 2.1, an isotropic subspace is F -stable if and only if it is of the form K ⊗_{F_{p}} E′ for some isotropic subspace E′ ≤ E. So V (J_{∞}) is a subvariety of codimension h in V. Since J_{h} is generated by exactly h elements, it follows that these elements form a regular sequence.

It remains to prove that J_{h} is its own radical. For this purpose, we proceed as in Quillen [24], Tezuka and Yagita [30], by applying Bézout’s Theorem (Proposition 7.6) to the ideal J_{h}. The left-hand side of Equation (7.7) is equal to ∏_{i=1}^{h}(p^{i} +1). The right-hand side has one term for each component of the variety V (J_{h}). The number of these is equal to the number of maximal isotropic subspaces. Counting in two ways the number of pairs consisting of a maximal isotropic subspace and a basis, we see that the number of maximal isotropic subspaces is also equal to ∏_{i=1}^{h}(p^{i} + 1). So each l_{p_{i}}(q_{i}) has to be one, and hence J_{h} = \sqrt{J_{h}}. □

9 Invariant theory

The remaining ingredient in the case of p odd is an invariant theoretic calculation. Let G be an elementary abelian p-group of order p^{h} with basis g_{1}, …, g_{h}, acting on F_{p}[x_{1}, …, x_{m}, z], m ≥ h as follows:

g_{i}(x_{j}) = x_{j} (j ≤ m)
g_{i}(z) = x_{i} + z.

Let ζ denote the product

ζ = ∏_{(λ_{1},...,λ_{h})}(z − λ_{1}x_{1} − ⋯ − λ_{h}x_{h}),

where the product runs over the p^{h} possible values of λ_{1}, …, λ_{h} ∈ F_{p}, so that ζ has degree p^{h} as a function of z.

Theorem 9.1 The invariants of the action of G on F_{p}[x_{1}, …, x_{m}, z] are precisely the elements of the subring F_{p}[x_{1}, …, x_{m}, ζ].

Proof Clearly x_{1}, …, x_{m} and ζ are invariants of the action. Conversely, suppose that f(x_{1}, …, x_{m}, z) is a polynomial which is invariant, not in F_{p}[x_{1}, …, x_{m}, ζ], and of smallest possible degree subject to these restraints. Then by subtracting a suitable polynomial in x_{1}, …, x_{m}, we may assume that every term in f involves z, so that f is divisible by z. Since f is invariant under the action of G, it is also divisible by every image of z under the action of G, and hence by their product, since F_{p}[x_{1}, …, x_{m}, z] is a unique factorisation domain. The product of all the images of z is ζ, so f is divisible by ζ. But then f/ζ is a polynomial of smaller degree, not in F_{p}[x_{1}, …, x_{m}, ζ], and invariant under G. This contradiction proves the theorem. □

Remark It is shown in Wilkerson [32] that the non-zero coefficients of powers of z appearing in ζ are the Dickson invariants, which are the invariants for the action of GL(h, p) on F_{p}[x_{1}, …, x_{h}]. Thus the ring of invariants of this action is a polynomial ring in these coefficients. We shall not need this fact.

10 Mod p cohomology of extraspecial p-groups, p odd

Let p be an odd prime. In this section, we examine the spectral sequence of a central extension

1 → N → G →^{ψ} E → 1

with N cyclic of order p and E an elementary abelian p-group of order p^{n}. Our calculations repeat and extend the calculations given in Tezuka and Yagita [30].

If we write elements of N and E additively, we can identify N with F_{p} and E with a vector space over F_{p}. We define a linear form λ : E → N by taking λ(x) = x̃^{p}, where tildex is any element with ψ(x̃) = x. Since commutators are central and of order p, we have

(x̃ỹ)^{p} = x̃^{p}ỹ^{p}[ỹ, x̃]^{p(p−1)/2} = x̃^{p}ỹ^{p}

so that λ is indeed linear. We denote by G_{1} the subset of G consisting of the identity and the elements of order p. The above shows that either G_{1} = G or G_{1} is a subgroup of index p in G.

We also define a symplectic form b : E × E → N by b(x, y) = [x̃, ỹ] for x̃, ỹ ∈ G with ψ(x̃) = x, ψ(ỹ) = y. Let s denote the corresponding element of Λ^{2}Hom(E, N) = Hom(Λ^{2}E, N). That is, s(v ∧ w) = b(v, w).

We have H^{∗}(E, F_{p}) = Λ(y_{1}, …, y_{n}) ⊗ F_{p}[x_{1}, …, x_{n}] with deg(y_{i}) = 1, deg(x_{i}) = 2 and with the Bockstein map given by β(y_{i}) = x_{i}, β(x_{i}) = 0. Thus

H^{2}(E, F_{p}) ≅ Λ^{2}Hom(E, F_{p}) ⊕ Hom(E, F_{p})

with basis y_{ij} = y_{i}y_{j}, i < j, and x_{i}. The elements s ∈ Λ^{2}Hom(E, F_{p}) and λ ∈ Hom(E, F_{p}) determine the element of H^{2}(E, F_{p}) corresponding to the given central extension. Let e_{1}, …, e_{n} be a basis for E over F_{p}. Then if v = ∑_{i=1}^{n} x_{i}e_{i}, we have λ(v) = ∑_{i=1}^{n} λ(e_{i})x_{i}, while if w ∧ w′ = ∑_{i<j} y_{ij}e_{i} ∧ e_{j} then

s(w ∧ w′) = ∑_{i<j} b(e_{i}, e_{j})y_{ij}.

Let K be an algebraically closed field of characteristic p, and let V = K ⊗_{F_{p}} E, a vector space of dimension n over K. Then modulo the Jacobson radical, the cohomology ring of E is the coordinate ring of V regarded as an affine variety over K,

H^{∗}(E, K)/J = K[x_{1}, …, x_{n}] = K[V].

The Frobenius map F on K[V] is the K-linear ring homomorphism sending each x_{i} to x_{i}^{p}.

We also write W for another copy of K ⊗_{F_{p}} E, and regard the y_{i} as linear forms on W. We have H^{3}(E, F_{p}) ⊆ H^{3}(E, K) ≅ Hom_{K}(W ⊗ V, K) ⊕ Λ^{3}Hom_{K}(W, K), and

β s (w, v) = ∑_{i<j} b(e_{i}, e_{j})(x_{i}y_{j} − y_{i}x_{j}) = −b(w, v).

Applying the Steenrod operation P^{1}, we have

P^{1}β s (w, v) = ∑_{i<j} b(e_{i}, e_{j})(x_{i}^{p}y_{j} − y_{i}x_{j}^{p}) = −b(w, F (v))
βP^{1}β s (w, v) = ∑_{i<j} b(e_{i}, e_{j})(x_{i}^{p}x_{j} − x_{i}x_{j}^{p}) = −b(v, F (v))

as an element of H^{2p+1}(E, K) in the first case and H^{2p+2}(E, K) in the second case. Similarly for r ≥ 1 we have

P^{p^{r−1}}P^{p^{r−2}} ⋯ P^{1}β s (w, v) = ∑_{i<j} b(e_{i}, e_{j})(x_{i}^{p^{r}}y_{j} − y_{i}x_{j}^{p^{r}}) = −b(w, F^{r}(v))
βP^{p^{r−1}}P^{p^{r−2}} ⋯ P^{1}β s (w, v) = ∑_{i<j} b(e_{i}, e_{j})(x_{i}^{p^{r}}x_{j} − x_{i}x_{j}^{p^{r}}) = −b(v, F^{r}(v)).

The following is a slight extension of the result of Tezuka and Yagita [30], who prove the theorem only up to inseparable isogeny.

Theorem 10.1 Let G, E, V, … be as above. Denote by V_{1} the kernel on V of the linear form λ. Denote by h (resp. h_{1}) the codimension in V (resp. V_{1}) of a maximal F -stable isotropic subspace, so that p^{h} is the index in G of a maximal abelian subgroup. Then writing J for the Jacobson radical, we have

H^{∗}(G, F_{p})/J = F_{p}[x_{1}, …, x_{n}]/(λ(v), b(v, F (v)), …, b(v, F^{h_{1}}(v))) ⊗ F_{p}[ζ].

Here, x_{1}, …, x_{n} are degree two elements inflated from elements of the same name in H^{∗}(E, F_{p}). The element ζ is any element of degree 2p^{h} which restricts non-trivially to H^{∗}(N, F_{p}).

The ideal generated by the elements λ(v), b(v, F (v)), …, b(v, F^{h_{1}}(v)) contains the elements b(v, F^{j}(v)), for j > h_{1}. If λ ≠ 0, then these elements form a regular sequence; if λ = 0, the remaining elements form a regular sequence.

Proof We examine the Lyndon–Hochschild–Serre spectral sequence

H^{∗}(E, H^{∗}(N, F_{p})) ⇒ H^{∗}(G, F_{p}).

The E_{2} page of this spectral sequence is

H^{∗}(E, F_{p}) ⊗ H^{∗}(N, F_{p}) = Λ(y_{1}, …, y_{n}, t) ⊗ F_{p}[x_{1}, …, x_{n}, z]

with y_{1}, …, y_{n} ∈ E_{2}^{1,0}, t ∈ E_{2}^{0,1}, x_{1}, …, x_{n} ∈ E_{2}^{2,0} and z ∈ E_{2}^{0,2}. Since s(w, w′) + λ(v) ∈ H^{2}(E, F_{p}) is the element corresponding to the central extension, we have

d_{2}(t) = s(w ∧ w′) + λ(v) ∈ E_{2}^{2,0}.

Since Steenrod operations commute with transgressions (Serre [27], Section II.9 c) and β(λ) = 0 (since λ ∈ Im(β)), z survives to E_{3} and

d_{3}(z) = d_{3}(β(t)) = β(d_{2}(t)) = βs (w, v) = −b(w, v) ∈ E_{3}^{3,0},

and similarly for r ≥ 1, z^{p^{r}} survives to E_{2p^{r}+1} and

d_{2p^{r}+1}(z^{p^{r}}) = d_{2p^{r}+1}(P^{p^{r−1}}P^{p^{r−2}} ⋯ P^{1}β(t))
= P^{p^{r−1}}P^{p^{r−2}} ⋯ P^{1}β d_{2}(t)
= P^{p^{r−1}}P^{p^{r−2}} ⋯ P^{1}β s(w, v) = −b(w, F^{r}(v)).

Also, by Kudo’s transgression theorem (Kudo [18]; see also May [22] p. 169, and Mc-Cleary [23], §6.1.1), z^{p^{r−1}(p−1)}b(w, F^{r}(v)) is not in the image of any differential, and in E_{2p^{r−1}(p−1)+1} we have

d_{2p^{r−1}(p−1)+1}(z^{p^{r−1}(p−1)}b(w, F^{r−1}(v)) ≡ −βP^{p^{r−1}}P^{p^{r−2}} ⋯ P^{1}β s(v) ≡ b(v, F^{r}(v)),

the congruence being modulo the images of the previous differentials. We should remark at this stage that this element is not guaranteed not to have been hit by a previous differential, so we should beware of deducing too much (see the remark following the end of the proof). But we may certainly deduce that b(v, F^{r}(v)) is in the kernel of inflation, Ker(inf_{E,G}).

Applying Proposition 8.2 to the subspace V_{1} of V, we see that the elements λ(v) (if this is non-zero), b(v, F (v)), …, b(v, F^{h_{1}}(v)) form a regular sequence which generates a radical ideal J_{∞} in K[x_{1}, …, x_{n}] containing b(v, F^{j}(v)) for j > h_{1}. The above discussion shows that J_{∞} ≤ Ker(inf_{E,G}).

Now the inverse image of a subgroup E′ of E is elementary abelian if and only if, as a vector space, it is in the kernel of λ and isotropic for the form b. Any element of H^{∗}(E, K) which is in the kernel of inf_{E,G} must restrict to zero on such a subgroup E′. So we have

K[x_{1}, …, x_{n}] ∩ Ker(inf_{E,G}) ⊇ J_{∞} = \sqrt{J_{∞}}
= I (⋃_{E′}(K ⊗_{F_{p}} E′))
= K[x_{1}, …, x_{n}] ∩ ⋂_{E′} Ker(res_{E,E′})
⊇ K[x_{1}, …, x_{n}] ∩ Ker(inf_{E,G}),

where E′ runs over the isotropic subspaces of E in the kernel of λ. It follows that the inequalities are equalities, so that K[x_{1}, …, x_{n}] ∩ Ker(inf_{E,G}) is the ideal J_{∞} generated by λ(v), b(v, F (v)), …, b(v, F^{h_{1}}(v)). Since these elements are in F_{p}[x_{1}, …, x_{n}] it follows that F_{p}[x_{1}, …, x_{n}] ∩ Ker(inf_{E,G}) is also the ideal generated by these elements.

Next, we show that z^{p^{h}} ∈ E_{2}^{0,2p^{h}} is a universal cycle. Let H_{1} be a maximal elementary abelian subgroup of G, and let H be a maximal abelian subgroup containing it, so that |G : H| = p^{h}. Then z is a restriction of an element which we shall also call z in H^{2}(H, F_{p}). One then easily checks using the Mackey formula for Evens’ norm map [13], that norm_{H,G}(z) is an element of H^{∗}(G, F_{p}) restricting to z^{p^{h}} ∈ H^{∗}(N, F_{p}). In fact, the same formula shows that the res_{G,H_{1}}(ζ) is the product of the images of z under the action of G/H. Thus z^{p^{h}} is a universal cycle.

For any such maximal abelian subgroup H of G, G/H is elementary abelian and acts on H^{∗}(H_{1}, F_{p})/J. After rechoosing the basis of E suitably, the latter is isomorphic to the polynomial subring F_{p}[x_{1}, …, x_{m}, z] of H^{∗}(H_{1}, F_{p}), and G/H acts in the manner described in Section 9. The image of res_{G,H_{1}} lies in the ring of invariants, which by Theorem 9.1 is F_{p}[x_{1}, …, x_{m}, ζ].

Now by a theorem of Quillen [25], an element of H^{∗}(G, F_{p}) is nilpotent if and only if its restriction to every elementary abelian subgroup is nilpotent. So any non-nilpotent element restricts to some non-zero element of F_{p}[x_{1}, …, x_{m}, ζ] as above, for some choice of H_{1}. It follows that in E_{∞}^{∗∗}, every non-nilpotent element has vertical grading divisible by 2p^{h}.

To complete the proof of the structure of H^{∗}(G, F_{p})/J, it remains to show that ζ is a non zero-divisor in H^{∗}(G, F_{p}). To prove this, one shows by induction on l ≥ 2 that multiplication by ζ is an isomorphism from E_{l}^{r,s} to E_{l}^{r,s+2p^{h}} commuting with the differential d_{l}. □

Remark In the case h = h_{1} of the above theorem, there remains something of a mystery as to what element in the spectral sequence actually kills b(w, F^{h}(v)). The point is that z^{p^{h}} survives to E_{∞}, so that b(w, F^{h}(v)) must have been killed by some previous differential.

11 E_{2p+1}^{∗,∗}

We now consider the special case in which n = 2h + 1, h_{1} = h and

d_{2}(t) = s(w ∧ w′) + λ(v) = y_{1}y_{2} + y_{3}y_{4} + ⋯ + y_{2h−1}y_{2h} + x_{2h+1}.

In this case, we completely determine the E_{2p+1} page of the Lyndon–Hochschild–Serre spectral sequence considered in the last section.

Since d_{2}(t) is a non zero divisor in E_{2}^{∗,0} = H^{∗}(E, F_{p}), we have

E_{3}^{∗,∗} = H^{∗}(E, F_{p})/(d_{2}(t)) ⊗ F_{p}[z] = Λ(y_{1}, …, y_{2h+1}) ⊗ F_{p}[x_{1}, …, x_{2h}] ⊗ F_{p}[z].

Note that the odd numbered rows are zero in E_{3}. We now have

d_{3}(z) = −b(w, v) = x_{1}y_{2} − y_{1}x_{2} + x_{3}y_{4} − y_{3}x_{4} + ⋯ + x_{2h−1}y_{2h} − y_{2h−1}x_{2h}.

Since b(w, v) has odd degree, we have b(w, v)^{2} = 0, and so we can regard the even numbered rows as a complex with differential given by multiplication by b(w, v). Notice that each of the even numbered rows is the same and is isomorphic to E_{3}^{∗,0} = H^{∗}(E, F_{p})/(d_{2}(t)), namely Λ(y_{2h+1}) tensored with F_{p}[x_{1}, …, x_{2h}] ⊗ Λ(y_{1}, …, y_{2h}). Moreover, the element b(w, v) is homogeneous of degree one with respect to the grading on the exterior algebra Λ. So the rows are copies of Λ(y_{2h+1}) tensored with the complex

⋯ → F_{p}[x_{1}, …, x_{2h}]⊗Λ^{j}(y_{1}, …, y_{2h}) ⟶^{x_{1}y_{2}−y_{1}x_{2}+···} F_{p}[x_{1}, …, x_{2h}]⊗Λ^{j+1}(y_{1}, …, y_{2h}) → ⋯

The latter is the Koszul complex for the regular sequence −x_{2}, x_{1}, −x_{4}, x_{3}, …, −x_{2h}, x_{2h−1}, and so it is exact except for the right hand end, where the homology consists of the one dimensional space spanned by the single element y_{1} … y_{2h}. So after tensoring with Λ(y_{2h+1}), the homology is spanned by y_{1} … y_{2h} and y_{1} … y_{2h+1}. So we have

E_{4}^{∗,0} = E_{2}^{∗,0}/(s(w ∧ w′) + λ(v), b(w, v)) = E_{3}^{∗,0}/(b(w, v))
= E_{3}^{∗,0}/Ker(b(w, v)) ⊕ ⟨y_{1} … y_{2h}⟩ ⊕ ⟨y_{1} … y_{2h+1}⟩
E_{4}^{∗,2j} = \{ \begin{matrix}z^{j}E_{4}^{∗,0} & j ≡ 0 (mod p) \\ ⟨z^{j}y_{1} … y_{2h}⟩ ⊕ ⟨z^{j}y_{1} … y_{2h+1}⟩ & j ≢ 0, p − 1 (mod p) \\ z^{j}(b(w, v)) ⊕ ⟨z^{j}y_{1} … y_{2h}⟩ ⊕ ⟨z^{j}y_{1} … y_{2h+1}⟩ & j ≡ p − 1 (mod p). \end{matrix}

Here, ⟨y_{1} … y_{2h}⟩ denotes the space of dimension one spanned by the class of the element y_{1} … y_{2h}, and so on.

We now observe that as a module for F_{p}[x_{1}, …, x_{2h}], E_{3}^{∗,0}/Ker(b(w, v)) ≅ (b(w, v)), and so the only elements of E_{4}^{∗,0} killed by multiplication by the elements x_{1}, …, x_{2h} are y_{1} … y_{2h} and y_{1} … y_{2h+1}. It follows that

d_{2j+1}(z^{j}y_{1} … y_{2h}) = 0, d_{2j+1}(z^{j}y_{1} … y_{2h+1}) = 0 1 ≤ j ≤ p − 2,

and so E_{2p−1}^{∗,∗} = E_{4}^{∗,∗}.

Next, by Kudo’s transgression theorem [18] we have

d_{2p−1}(z^{p−1}b(w, v)) = b(v, F (v)) = −x_{1}^{p}x_{2} + x_{1}x_{2}^{p} − ⋯ − x_{2h−1}^{p}x_{2h} + x_{2h−1}x_{2h}^{p}

and this is still a non zero divisor in E_{2p}^{∗,0}. If h ≥ 2, we also have d_{2p−1}(z^{p−1}y_{1} … y_{2h}) = 0 and d_{2p−1}(z^{p−1}y_{1} … y_{2h+1}) = 0 for the same reason as before, and so

E_{2p}^{∗,2j} = E_{2p+1}^{∗,2j} = \{ \begin{matrix}z^{j}.E_{3}^{∗,0}/(b(w, v), b(v, F (v))) & j ≡ 0 (mod p) \\ ⟨z^{j}y_{1} … y_{2h}⟩ ⊕ ⟨z^{j}y_{1} … y_{2h+1}⟩ & j ≢ 0 (mod p). \end{matrix}

12 A group of order p^{4}

To get any further with the spectral sequence in the case h ≥ 2, the next task is to understand the kernel and image of multiplication by d_{2p+1}(z^{p}) = −b(w, F (v)) on E_{2p+1}^{∗,0}, and this appears to be hard. Instead, we complete the calculation in case h = 1.

The following result can be found in Leary [19]. The difference of our approach is that we do not first construct the integral cohomology, and we obtain one of the relations by using a construction developed in [4]. The proof depends heavily on the results of the previous section.

Theorem 12.1 Suppose that

1 → N → G → E → 1

is a central extension with |E| = p^{3} and |N| = p (p odd), corresponding to the element y_{1}y_{2} + x_{3} ∈ H^{2}(E, F_{p}). Then H^{∗}(G, F_{p}) is generated by elements y_{1}, y_{2}, y_{3} in degree one, x_{1}, x_{2} in degree two, c_{j} in degree 2j (2 ≤ j ≤ p − 1), and ζ in degree 2p, subject to the following relations:

x_{1}y_{2} = y_{1}x_{2} x_{1}^{p}y_{2} = y_{1}x_{2}^{p} x_{1}^{p}x_{2} = x_{1}x_{2}^{p}

c_{j}y_{1} = \{ \begin{matrix}0 & 2 ≤ j < p − 1 \\ −x_{1}^{p−1}y_{1} & j = p − 1 \end{matrix} c_{j}y_{2} = \{ \begin{matrix}0 & 2 ≤ j < p − 1 \\ −x_{2}^{p−1}y_{2} & j = p − 1 \end{matrix}

c_{j}x_{1} = \{ \begin{matrix}0 & 2 ≤ j < p − 1 \\ −x_{1}^{p} & j = p − 1 \end{matrix} c_{j}x_{2} = \{ \begin{matrix}0 & 2 ≤ j < p − 1 \\ −x_{2}^{p} & j = p − 1 \end{matrix}

c_{i}c_{j} = \{ \begin{matrix}0 & i + j < 2p − 2 \\ x_{1}^{2p−2} − x_{1}^{p−1}x_{2}^{p−1} + x_{2}^{2p−2} & i = j = p − 1 \end{matrix}

Proof As remarked in the last section, the element z^{p} of E_{2}^{0,2p} is represented by the Evens norm from an elementary abelian subgroup of index p, and so it survives to E_{∞}. It follows that −b(w, F (v)) = x_{1}^{p}y_{2} − y_{1}x_{2}^{p} is zero in E_{2p+1}^{∗,0}, and the only element that could hit it is z^{p−1}y_{1}y_{2}, so d_{2p−1}(z^{p−1}y_{1}y_{2}) is some non-zero multiple of x_{1}^{p}y_{2} − y_{1}x_{2}^{p}. Thus the E_{∞} page of the spectral sequence is given by

E_{∞}^{∗,0} = H^{∗}(E, F_{p})/(y_{1}y_{2} + x_{3}, x_{1}y_{2} − y_{1}x_{2}, x_{1}^{p}y_{2} − y_{1}x_{2}^{p}, x_{1}^{p}x_{2} − x_{1}x_{2}^{p})

E_{∞}^{∗,2j} = \{  \begin{matrix}z^{j}.E_{∞}^{∗,0} & j ≡ 0 (mod p) \\ ⟨z^{j}y_{1}y_{2}⟩ ⊕ ⟨z^{j}y_{1}y_{2}y_{3}⟩ & j ≢ 0, p − 1 (mod p) \\ 0 & j ≡ p − 1 (mod p) \end{matrix}

Note that E_{∞}^{∗,0} has a basis consisting of the monomials in x_{1}, x_{2}, y_{1} and y_{2} with the following properties: the degree in each y_{i} is at most one; if y_{1} is present then x_{2} is not; and if x_{1} is present then the total degree in x_{2} and y_{2} is strictly less than p.

We write ζ for a representative in H^{∗}(G, F_{p}) of z^{p}, so that deg(ζ) = 2p and ζ is a non zero divisor in H^{∗}(G, F_{p}). We choose elements c_{j} ∈ H^{2j}(G, F_{p}), 2 ≤ j ≤ p − 1, representing z^{j−1}y_{1}y_{2} as follows. First, let a_{j} be any representative, so that it is well defined only modulo inflations from H^{∗}(E, F_{p}) and up to scalar multiplication. That is, the E_{∞} page of the spectral sequence is only a graded version of H^{∗}(G, F_{p}). Hence a_{j} + γ also represents z^{j}y_{1}y_{2} if γ ∈ E_{∞}^{2j,0} = inf_{E,G}(H^{2j}(E, F_{p})). For 2 ≤ j < p − 1, since a_{j}y_{1} is an element of the image of inflation from E, annihilated by y_{1}, it is divisible by y_{1}, so a_{j}y_{1} = f_{j}(x_{1}, x_{2})y_{1}. Setting c_{j} = a_{j} −f_{j}(x_{1}, x_{2}), we have c_{j}y_{1} = 0. By the same argument, c_{j}y_{2} = f′_{j}(x_{1}, x_{2})y_{2}. Then f′_{j}(x_{1}, x_{2})x_{1}y_{2} = c_{j}x_{1}y_{2} = c_{j}y_{1}x_{2} = 0, and so f′_{j}(x_{1}, x_{2}) = 0. The equation c_{j}x_{1}y_{2} = c_{j}y_{1}x_{2} = 0 also implies that c_{j}x_{1} = c_{j}x_{2} = 0.

For j = p−1, the above argument does not work, because there are elements annihilated by x_{1}, so that we may not deduce that f′_{j}(x_{1}, x_{2}) = 0. Instead, we argue as follows. There is an automorphism ρ of G, inverting elements of N, swapping x_{1} and x_{2} and swapping y_{1} and y_{2}, while negating y_{3}. We have ρ^{2} = 1 and ρ(a_{p−1}) = a_{p−1}+α for some α inflated from E. Then ρ(α) = −α and so ρ(a_{p−1} + α/2) = a_{p−1} + α/2. We replace a_{p−1} by a_{p−1} + α/2 to ensure that a_{p−1} is fixed by ρ. Now a_{p−1}y_{1} = f(x_{1}, x_{2})y_{1} and a_{p−1}y_{2} = f(x_{2}, x_{1})y_{2}. Since x_{1}y_{2} = y_{1}x_{2}, we have f(x_{2}, x_{1})x_{1}y_{2} = f(x_{1}, x_{2})y_{1}x_{2} and so f(x_{2}, x_{1}) − f(x_{1}, x_{2}) is a multiple of x_{2}^{p−1} − x_{1}^{p−1}. So we may add to a_{p−1} a symmetric polynomial in x_{1} and x_{2} to obtain an element c_{p−1} satisfying c_{p−1}y_{1} = μx_{1}^{p−1}y_{1} and c_{p−1}y_{2} = μx_{2}^{p−1}y_{2} for some choice of scalar μ.

We now wish to show that μ ≠ 0, so that by replacing c_{p−1} by a suitable scalar multiple we can make μ = −1. To prove that μ ≠ 0, we argue by contradiction. If μ = 0 then c_{p−1}x_{1}y_{2} = 0, so c_{p−1}x_{1} is a multiple of x_{1}^{p} − x_{1}x_{2}^{p−1}; then since c_{p−1}x_{1}y_{1} = 0 we have c_{p−1}x_{1} = 0, and similarly c_{p−1}x_{2} = 0.

To derive a contradiction from the equations c_{p−1}x_{1} = c_{p−1}x_{2} = 0, we use a construction from [4]. Let K be a field of characteristic p. In [4] it is shown that if ζ_{1}, …, ζ_{r} (r = p-rank of G) are homogeneous elements of degree n_{1}, …, n_{r} generating a polynomial subring over which H^{∗}(G, K) is finitely generated as a module, then there is a spectral sequence with

E_{2}^{∗,∗} = H^{∗}(G, K) ⊗ Λ^{∗}(ζ̃_{1}, …, ζ̃_{r}),

with H^{∗}(G, K) along the horizontal axis, exterior generators ζ̃_{i} of degree n_{i} − 1 on the vertical axis, and converging to a finite Poincaré duality ring of formal dimension ∑_{i=1}^{r}(n_{i}− 1). The differentials are explicitly described in [4], and in particular d_{n_{i}}(αζ̃_{i}) = αζ_{i}.

In our case, we may choose ζ_{1} = ζ (the representative of z^{p}) and ζ_{2} = x_{2} − γx_{1}, where γ is a scalar in K but not in F_{p} (so we must choose K strictly bigger than F_{p}). The formal dimension is (2p − 1) + 1 = 2p.

Since ζ_{1} is a non zero divisor, the E_{∞} page of this spectral sequence has two nonzero rows, namely E_{∞}^{∗,0} = H^{∗}(G, K)/(ζ_{1}, ζ_{2}) and E_{∞}^{∗,1} = kernel of multiplication by ζ_{2} on H^{∗}(G, K)/(ζ_{1}).

− − y_{1}y_{2}ζ̃_{1} y_{1}y_{2}y_{3}ζ̃_{1} c_{2}ζ̃_{1} c_{2}y_{3}ζ̃_{1} ⋯ (c_{p−1}ζ̃_{1})  −  −
1 y_{1} x_{1}  x_{1}y_{2}  x_{1}^{2}  x_{1}^{2}y_{2}  x_{1}^{p−1} x_{1}^{p−1}y_{2} (x_{1}^{p})
y_{2} y_{1}y_{2}  x_{1}y_{3} x_{1}y_{2}y_{3} x_{1}^{2}y_{3} ⋯ x_{1}^{p−2}y_{2}y_{3} x_{1}^{p−1}y_{3} x_{1}^{p−1}y_{2}y_{3}
y_{3} y_{1}y_{3} y_{1}y_{2}y_{3} c_{2} c_{2}y_{3} c_{p−1} c_{p−1}y_{3}
y_{2}y_{3}

If c_{p−1}x_{1} = c_{p−1}x_{2} = 0 then both x_{1}^{p} and x_{1}^{p−1}y_{2}y_{3} are linearly independent elements of degree 2p in E_{∞}, contradicting the fact that the unique non-zero element in this degree is the Poincaré dual of the identity element. This completes the proof that μ ≠ 0, so that without loss of generality μ = −1.

Finally, we evaluate the products c_{i}c_{j}. As long as i + j < 2p − 2, c_{i}c_{j} is annihilated by multiplication by both x_{1} and x_{2}, and so c_{i}c_{j} = 0. On the other hand, if i = j = p − 1 we have c_{p−1}^{2}x_{1} = −c_{p−1}x_{1}^{p} = x_{1}^{2p−1} and c_{p−1}^{2}x_{2} = −c_{p−1}x_{2}^{p} = x_{2}^{2p−1}, and it follows that c_{p−1}^{2} = x_{1}^{2p−2} − x_{1}^{p−1}x_{2}^{p−1} + x_{2}^{2p−2}. □

13 Extraspecial groups of order p^{3}

The mod p cohomology rings of the extraspecial p-groups of order p^{3} for p odd have been completely determined. The exponent p^{2} case was calculated by Diethelm [12], and the results are relatively straightforward. The case of exponent p case was recently calculated by Leary [19], and the answer is very complicated, with the description of the generators and relations taking up more than a page. We shall not reproduce his calculations here, but we shall briefly describe how the calculations can be simplified by using the fact that GL_{2}(F_{p}) acts as automorphisms on the relevant spectral sequence.

The group G described in the previous section is a central product of an extraspecial group P of order p^{3} and exponent p with a cyclic group of order p^{2}. It thus has as a central extension the group P × Z/p^{2}. The spectral sequence of this central extension has E_{3} = E_{∞}, and hence (using the Künneth theorem for P ×Z/p^{2}) the quotient of this spectral sequence by y_{3} and by the generator in degree (0, 2) gives a two row spectral sequence converging to H^{∗}(P, F_{p}) (this spectral sequence is isomorphic to the one considered in Leary [19] using the central product of P with a circle group S^{1}). We write ξ for the element in degree (0, 1), so that the differential in this spectral sequence is given by d_{2}(ξζ^{j}) = y_{1}y_{2}ζ^{j}, and d_{2} vanishes on all other monomials in the generators.

The outer automorphism group Out(P) ≅ GL_{2}(F_{p}) acts on the above two row spectral sequence, and this action commutes with the differentials. We begin with a discussion of the module theory for GL_{2}(F_{p}) (see for example Brauer and Nesbitt [7] Section 30).

Let M be the natural module of dimension two for GL_{2}(F_{p}). Then for 0 ≤ j ≤ p − 1, the symmetric powers S^{j}(M) are simple. Writing S^{j,i}(M) for S^{j}(M) ⊗ det^{i}, the modules S^{j,i}(M) run over a complete set of representatives for the isomorphism classes of simple F_{p}GL_{2}(F_{p})-modules, for 0 ≤ i, j ≤ p − 1.

It is not hard, from the information given in the last section, to identify the first few terms in the E_{2} page of the above two row spectral sequence as modules for GL_{2}(F_{p}). For 1 ≤ j ≤ p − 1 we have

E_{2}^{∗,1} ≅ E_{2}^{∗,0} ⊗ det
E_{2}^{2j−1,0} ≅ S^{j}(M)  E_{2}^{2j,0} ≅ S^{j}(M) ⊕ det^{j}
E_{2}^{2p−1,0} ≅ X_{0}  E_{2}^{2p,0} ≅ X_{0} ⊕ det
E_{2}^{2(p+j)−1,0} ≅ X_{j} ⊕ S^{j,1}(M) E_{2}^{2(p+j),0} ≅ X_{j} ⊕ S^{j,1}(M) ⊕ det

Here, for 0 ≤ j ≤ p − 1, the module X_{j} is a (non-split) extension

0 → S^{j+1}(M) → X_{j} → S^{p−j−2,1}(M) → 0

of dimension p + 1. The E_{∞} page agrees with the E_{2} page except for the disappearance of the copies of det^{j+1} in degrees (2jp, 1) and (2jp + 2, 0) corresponding to the elements ξζ^{j} and y_{1}y_{2}ζ^{j}.

Lemma 13.1 As long as p > 3 we have

Ext_{F_{p}GL_{2}(F_{p})}^{1}(E_{∞}^{1,1}, E_{∞}^{2,0}) = 0.

Proof The scalar matrix ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ) ∈ Z(GL_{2}(F_{p})) acts as scalar multiplication by a on E_{∞}^{2,0} and multiplication by a^{3} on E_{∞}^{1,1}. As long as p > 3, there are scalars a with a^{3} ≢ a

(mod p), so any extension splits as a direct sum of eigenspaces. □

This lemma allows us to choose canonical representatives Y_{1} of y_{1}ξ and Y_{2} of −y_{2}ξ in H^{2}(P, F_{p}) (the minus sign is to take account of the fact that swapping y_{1} and y_{2} negates ξ). We then choose representatives X_{1} of x_{1}ξ and X_{2} of x_{2}ξ in H^{3}(P, F_{p}) by setting X_{j} = β(Y_{j}).

In a similar fashion, we see that for 2 ≤ j ≤ p − 1, the copy of det^{j+1} in E_{∞}^{2j,1} does not extend E_{∞}^{2j,0} and so we have a canonical representative b_{j} of c_{j}ξ in H^{2j+1}(P, F_{p}).

Theorem 13.2 (Leary [19]) Suppose that p > 3, and P is an extraspecial group of order p^{3} and exponent p. Then H^{∗}(P, F_{p}) is generated by elements y_{1}, y_{2} in degree one, x_{1}, x_{2}, Y_{1}, Y_{2} in degree two, X_{1}, X_{2} in degree three, c_{j} in degree 2j (4 ≤ j ≤ p − 1), b_{j} in degree 2j +1 (3 ≤ j ≤ p−1) and ζ in degree 2p, subject to some relations described in Leary [19].

The action of GL_{2}(F_{p}) can be used to derive many of the relations in the cohomology ring. For example, products between y_{i} and Y_{j} in H^{∗}(P, F_{p}) give rise to a map of F_{p}GL_{2}(F_{p})- modules E_{∞}^{1,0} ⊗ E_{∞}^{1,1} → E_{∞}^{2,1} ⊕ E_{∞}^{3,0} which, taken modulo E_{∞}^{3,0}, is the product in E_{∞}^{∗,∗}. We have E_{∞}^{1,0} ⊗ E_{∞}^{1,1} ≅ S^{2,1}(M) ⊕ det^{2}, and so there is no non-zero map to E_{∞}^{3,0}. Hence we have y_{1}Y_{1} = y_{2}Y_{2} = 0 and y_{1}Y_{2} = y_{2}Y_{1}. The same argument can be used to show such relations as y_{1}X_{1} = x_{1}Y_{1}, etc. Other relations can be obtained from the fact that the GL_{2}(F_{p})-action commutes with the Bockstein homomorphism.

Remark In case p = 3, the extension of E_{∞}^{1,1} by E_{∞}^{2,0} as F_{3}GL_{2}(F_{3})-modules really does not split. This may be seen by examining the effect of change of basis on the Massey products ⟨y_{1}, y_{1}, y_{2}⟩ and ⟨y_{2}, y_{2}, y_{1}⟩ (which may be taken as the elements Y_{1} and Y_{2} in the above theorem). Note that ⟨y_{j}, y_{j}, y_{j}⟩ = x_{j} if p = 3 and zero if p > 3.

14 The variety of the kernel of inflation

We end the paper with a couple of observations about the rôle of the extension class in the cohomology of the group extension. The first should be viewed as a point of philosophy which has been illustrated several times in this article. Certainly the proof is no more than the easy demonstration that two things which we know how to measure are the same.

Theorem 14.1 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 the subvariety of VḠ determined by the kernel of the inflation map H^{∗}(Ḡ, K) → H^{∗}(G, K) is equal to the subvariety VḠ(I_{η}) of VḠ determined by the ideal I_{η} in H^{∗}(Ḡ, K) generated by the element η in H^{2}(Ḡ, K) corresponding to the extension, together with its images under all Steenrod operations.

Proof We begin with the case where Ḡ = E is elementary abelian. In this case, the theorem follows from the description of the kernel of inflation map given in Theorem 4.1 in case p = 2 and Theorem 10.1 in case p is odd.

In the general case, we note that by a theorem of Quillen [25], we have

VḠ = ⋃_{E≤Ḡ} res_{Ḡ,E}^{∗}V_{E}

as E runs over the elementary abelian subgroups E of Ḡ. So for any ideal I in H^{∗}(Ḡ, K) we have

VḠ(I) = ⋃_{E≤Ḡ} res_{Ḡ,E}^{∗}V_{E}(res_{Ḡ,E}(I)).

Restriction from Ḡ to E of the kernel of inflation and of I_{η} give the corresponding ideals for the extension of E by N. □

It is interesting to note that if p = 2 then the cohomology ring of an extraspecial p-group is Cohen–Macaulay. However, this happens only for a few small cases when p is odd. This may be partially accounted for by the following observation.

Theorem 14.2 Suppose that

1 → N → G → Ḡ → 1

is a central extension with N cyclic of order 2, and corresponding to an element η ∈ H^{2}(Ḡ, F_{2}). Then the following are equivalent:

(i) η is nilpotent.

(ii) Every image of η under an element of the Steenrod algebra is nilpotent.

(iii) Every elementary abelian 2-subgroup of Ḡ lifts to an elementary abelian subgroup of G.

Proof The cohomology of an elementary abelian 2-group contains no non-zero nilpotent elements, so by a theorem of Quillen [25] an element of cohomology is nilpotent if and only if its restriction to every elementary abelian subgroup is zero, so (i) is equivalent to (iii). Since restriction commutes with Steenrod operations this also shows that (i) and (ii) are equivalent. □

Remark The corresponding statement is false for p odd, because the cohomology of an elementary abelian p-group has non-zero nilpotent elements. However, the same argument shows that an element η ∈ H^{2}(Ḡ, F_{p}) (p odd) is nilpotent if and only if every cyclic subgroup of order p in Ḡ lifts to a cyclic subgroup of order p in G.

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