THE COHOMOLOGY OF SPLIT EXTENSIONS OF ELEMENTARY ABELIAN 2-GROUPS AND TOTARO’S EXAMPLE

STEPHEN F. SIEGEL

Abstract. In a previous paper we derived an expression for the differentials in the Lyndon-Hochschild-Serre spectral sequence of a split extension G = H ⋊Q of finite groups with coefficients in a field. Here we apply that result to the case where H and Q are elementary abelian 2-groups and char k = 2. We then work out a special case, in which H has rank 4 and Q has rank 2, from a class of examples constructed by Burt Totaro. Totaro proved that the spectral sequence arising from this extension could not collapse, but using our methods we are able to obtain complete information on the spectral sequence.

1. Introduction

1.1. Background. Most spectral sequences that have been used in group cohomology calculations arise from central group extensions. Recently, however, there has been growing interest in spectral sequences arising from non-central extensions. Benson-Feshbach [3], for example, raised some questions about the vanishing of differentials in split extensions with abelian kernel. (See also [8].) Burt Totaro [11] then constructed an interesting family of counterexamples to answer all of these questions negatively. In particular, for each prime p, Totaro constructs a split group extension in which both kernel and quotient are elementary abelian p-groups, such that the differential d_{p} in the Lyndon-Hochschild-Serre (LHS) spectral sequence with mod-p coefficients is non-zero. To our knowledge, this was the first example of that phenomenon, at any prime.

Totaro’s argument is simple and elegant, and manages to show d_{p} ≠ 0 for very general reasons. Yet it gives almost no other information about the spectral sequence, and in particular it does not show what d_{p}, or any other differential, actually is. It therefore might be interesting to examine one of these examples in greater detail, in the hopes of gaining greater insight into the mystery of p-group cohomology.

We do this for the smallest of Totaro’s examples, in which p = 2 and the split extension is an elementary abelian 2-group of rank 2 acting on one of rank 4. We calculate the E_{2}-page (which has a very rich structure), and the differentials d_{2}. We then show that d_{3} = 0, and, by the position of the generators, it follows that E_{3} = E_{∞}.

The calculation of the differentials is achieved by applying the main theorem of Siegel [9]. In that paper, we gave a general method for calculating differentials in the case of a split extension of finite groups. As we show in Section 2, that method works out particularly elegantly in the case where p = 2 and the quotient group is an elementary abelian 2-group. Section 3 deals with the E_{2}-page of Totaro’s extension, and Section 4 applies the method of Section 2 to this case. That method requires the calculation of a certain twisting cochain, and an algorithm for doing this in the case where both quotient and kernel are elementary abelian 2-groups is explained in Section 5. The output of that algorithm for our example is given in Figures 2 and 3. Though it is on the borderline of something which can be carried out by hand, we used a computer to implement the algorithm.

Date: 9 November 1996.

1991 Mathematics Subject Classification. Primary 20J06.

The author was supported by a National Science Foundation postdoctoral fellowship.

1.2. Preliminaries. For this paper, by a complex of modules over a ring R we will mean a left ℤ-graded R-module C = ⨁_{n∈ℤ} C_{n} together with an endomorphism ∂ satisfying ∂(C_{n}) ⊆ C_{n−1} and ∂^{2} = 0. Hom and tensor products of complexes are defined as in Brown [4, Ch. I §0]. By bigraded complex we will mean an Rmodule E = ⨁_{r,s∈ℤ} E^{r,s} together with an endomorphism δ satisfying δ(E^{r,s}) ⊆ ∑_{i≥r} E^{i,r+s−i+1}. The bigraded complex yields a spectral sequence arising from the filtration F^{p}E = ∑_{r≥p,s∈ℤ} E^{r,s}.

Let k be a field. If V is a k-module then ⟨v_{1}, …, v_{n}⟩ denotes the subspace spanned by v_{1}, …, v_{n} ∈ V. An unlabeled tensor product will always mean “⊗_{k}”. If U and V are graded k-submodules of a graded k-algebra A, then UV denotes the graded k-submodule spanned by all uv, u ∈ U, v ∈ V. If W = UV and the natural map U ⊗ V → W is an isomorphism then we will also write W = U ⊗ V (in analogy with an internal direct sum). If G is a group, σ an automorphism of G, and U and V are kG-modules, then we let Hom_{kG}^{σ}(U, V) denote the set of f in Hom_{k}(U, V) satisfying f(gx) = σ(g)f(x) for all g ∈ G, x ∈ U. Finally, we use the notation common in computer science, where for a proposition P, (P) = 1 if P, else (P) = 0.

2. Elementary abelian 2-groups

2.1. Notation. Let E = ⟨g_{1}, …, g_{r}⟩ be an elementary abelian 2-group of rank r, and k a field of characteristic 2. Let X_{i} = g_{i} + 1 ∈ kE for each i. We construct the minimal resolution P → k as follows. Let P = kE[x_{1}, …, x_{r}], the polynomial algebra in r indeterminates over the commutative ring kE. The grading determined by degree gives P the structure of a graded kE-module which is finitely-generated and free in each degree. Now for any r-tuple α = (α_{1}, …, α_{r}) ∈ ℕ^{r} (ℕ = {0, 1, 2, … }), let |α| = ∑_{i=1}^{r} α_{i}, g^{α} = g_{1}^{α_{1}} ⋯ g_{r}^{α_{r}}, and x^{α} = x_{1}^{α_{1}} ⋯ x_{r}^{α_{r}}. Let ε(i) ∈ ℕ^{r} be the r-tuple which is 1 in position i and 0 elsewhere. Define the differential on P by ∂(x^{α}) = ∑ X_{i}x^{α−ε(i)}, where the sum is taken over all 1 ≤ i ≤ r such that α_{i} ≠ 0, and define the augmentation by ϵ(1) = 1, and ϵ(x^{α}) = 0 if x^{α} ≠ 1.

Now suppose A is a graded commutative k-algebra on which E acts as graded algebra automorphisms. As a graded k-module, the cochains C(E, A) = Hom_{kE}(P, A) may be identified with A[x_{1}, …, x_{r}], where ax^{α} corresponds to the map which takes x^{α} to a and vanishes on x^{β} for β ≠ α. Now the action of kE on A induces an action of kE[x_{1}, …, x_{r}] on A[x_{1}, …, x_{r}], and it is not difficult to see that the differential in C(E, A) is given by multiplication by ∑_{i=1}^{r} X_{i}x_{i}. Moreover,

Lemma 2.1. The cup product in H^{∗}(E, A) is induced by the associative product on A[x_{1}, …, x_{r}] defined by ax^{α} ⌣ bx^{β} = ag^{α}(b)x^{α+β}.

Proof. There is a diagonal approximation map P → P ⊗ P defined by x^{α} ↦ ∑_{β+γ=α} x^{β} ⊗ g^{β}(x^{γ}). One can see this by checking the case r = 1 and then using the fact that the tensor product of chain maps is again a chain map. By applying Hom_{kE}(P, −) one obtains the product described in the lemma.

2.2. Differentials. Let G = H ⋊ Q be a split group extension, where for now H is any finite group, and Q = ⟨g_{1}, …, g_{s}⟩ is an elementary abelian 2-group of rank s. Let Y = kQ[y_{1}, …, y_{s}] be the minimal kQ-resolution, and P the minimal kHresolution. Our method for calculating differentials will require us to find a twisting system for the extension, which is a collection of maps f_{α} ∈ Hom_{kH}^{σ}(P, P)_{|α|−1}, where σ is the automorphism of kH induced by g^{α}, (0 ≠ α ∈ ℕ^{s}) satisfying

(i) ϵ ∘ f_{α} = ϵ if |α| = 1
(ii) ∂ ∘ f_{α} + f_{α} ∘ ∂ = 1(α = 2ε(i) for some i) + ∑_{β+γ=α} f_{β} ∘ f_{γ}.

(2.1)

Now in the LHS spectral sequence {E_{r}, d_{r}} we have

E_{1} = Hom_{kQ}(Y, Hom_{kH}(P, k)) ≈ H^{∗}(H, k)[y_{1}, …, y_{s}].

For γ ∈ ℕ^{s} with r = |γ| ≥ 1, let f̂_{γ} = Hom_{kH}(f_{γ}, k) ∈ End_{k}(H^{∗}(H, k))_{1−r}, and set χ_{r} = ∑_{|γ|=r} f̂_{γ}y^{γ} ∈ End_{k}(H^{∗}(H, k))[y_{1}, …, y_{s}]. This last ring acts on E_{1}, and we let δ_{r}: E_{1}^{p,q} → E_{1}^{p+r,q−r+1} denote multiplication by χ_{r}.

Theorem 2.2. There exists a twisting system for the extension G = H ⋊Q. Moreover, given a twisting system, set δ = ∑_{r≥1} δ_{r}; we then have δ^{2} = 0, and the spectral sequence arising from the bigraded complex (E_{1}, δ) equals the LHS spectral sequence.

Proof. Define t: k ⊗_{kQ} Y → Hom_{kG}(P ↑_{H}^{G}, P ↑_{H}^{G}) ≈ ⨁_{σ∈Q} σ ⊗ Hom_{kH}^{σ}(P, P) by

t(y^{α}) = 1(|α| = 1) + g^{α} ⊗ f_{α}.

The conditions on the f_{α} are seen to be exactly equivalent to the condition that t be a twisting cochain for the group extension 1 → H → G → Q → 1, as defined in [9, Section 5]. So existence follows from [9, Theorem 5.1] and the spectral sequence statement follows from [9, Theorem 8.1].

3. Totaro’s example: E_{2}-page

Let Q = ⟨g_{1}, g_{2}⟩, H = ⟨h_{1}, …, h_{4}⟩ be elementary abelian 2-groups of ranks 2 and 4, respectively. Let Q act on H according to

T_{1} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}) , T_{2} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \end{matrix}) ,

where T_{j} is the matrix of the linear transformation induced by g_{j} on the 𝔽_{2}-vector-space H with respect to the basis h_{1}, …, h_{4}. We let G be the semidirect product of H and Q defined by this action; this is the smallest of the class of examples considered by Totaro in [11]. We continue with the notation of the previous section; in particular P = 𝔽_{2}H[x_{1}, …, x_{4}] and Y = 𝔽_{2}Q[y_{1}, y_{2}] are the minimal resolutions of 𝔽_{2} over 𝔽_{2}H and 𝔽_{2}Q, respectively. Let x_{1}, …, x_{4} ∈ H^{1}(H, 𝔽_{2}) = Hom(H, 𝔽_{2}) denote the dual basis to h_{1}, …, h_{4}. Let A = H^{∗}(H, 𝔽_{2}) = 𝔽_{2}[x_{1}, …, x_{4}]. As a graded kQ-module, A = Sym(M), where M is spanned by the x_{i} and the action of Q is given by the transposes of T_{1} and T_{2}. Let {E_{r}, d_{r}} denote the LHS spectral sequence of the split extension.

We seek to describe E_{2}. But first we note that, though M is an indecomposable 𝔽_{2}Q-module, if we let k = 𝔽_{4} then k ⊗_{𝔽_{2}} M = M_{1} ⊕ M_{2}, where M_{1} = ⟨v_{1}, v_{2}⟩, M_{2} = ⟨v_{3}, v_{4}⟩, and

v_{1} = ωx_{1} + x_{2}, v_{2} = ω̄x_{3} + x_{4}, v_{3} = ω̄x_{1} + x_{2}, v_{4} = ωx_{3} + x_{4}.

Q fixes v_{1} and v_{3}, and g_{1}v_{2} = ωv_{1}+v_{2}, g_{1}v_{4} = ω̄v_{3}+v_{4}, g_{2}v_{2} = v_{1}+v_{2}, g_{2}v_{4} = v_{3}+v_{4}. The Frobenius automorphism λ ↦ \overline{λ} of k over 𝔽_{2} extends to a ring-automorphism of A, defined by \overline{λx^{α}} = \overline{λ}x^{α}, which is of order 2 and commutes with the action of Q. This automorphism exchanges M_{1} and M_{2}; specifically, \overline{v_{1}} = v_{3} and \overline{v_{2}} = v_{4}. Setting S = k[v_{1}, v_{2}] = Sym(M_{1}), we therefore have k ⊗ A = S ⊗ \overline{S}. Hence we will deal primarily with k ⊗_{𝔽_{2}} E_{2} instead of E_{2}. Let γ_{i} = [y_{i}] ∈ E_{2}^{1,0} (i = 1, 2).

Proposition 3.1. The following hold:

(i) For each indecomposable 𝔽_{2}Q-module U let p_{U}(t) = ∑_{n≥0} a_{n}t^{n}, where a_{n} is the multiplicity of U as a summand of H^{n}(H, 𝔽_{2}). Then

p_{𝔽_{2}}(t) = \frac{1}{(1 − t^{4})^{2}} , p_{M}(t) = \frac{t + t^{3}}{(1 − t^{4})^{2}} , p_{Ω𝔽_{2}}(t) = \frac{2t^{2}}{(1 − t^{4})^{2}} ,

p_{Ω^{2}𝔽_{2}}(t) = \frac{t^{4}}{(1 − t^{4})^{2}} , p_{𝔽_{2}Q}(t) = \frac{t^{2}(1 + 2t + 2t^{3} − t^{4})}{(1 − t)^{2}(1 − t^{4})^{2}},

and p_{U} = 0 for U not isomorphic to one of the five modules above. (ii) H^{∗}(H, 𝔽_{2})^{Q} is generated by x_{1}, x_{2},

q_{1} = x_{1}^{2}x_{3} + x_{1}x_{3}^{2} + x_{1}^{2}x_{4} + x_{1}x_{4}^{2} + x_{2}^{2}x_{4} + x_{2}x_{4}^{2},
q_{2} = x_{1}^{2}x_{3} + x_{1}x_{3}^{2} + x_{2}^{2}x_{3} + x_{2}x_{3}^{2} + x_{2}^{2}x_{4} + x_{2}x_{4}^{2},
z_{1} = x_{4}(x_{1}^{2}x_{2} + x_{1}^{2}x_{4} + x_{1}x_{2}^{2} + x_{1}x_{2}x_{4} + x_{2}^{2}x_{4} + x_{4}^{3}) = ∏_{σ∈Q} σ(x_{4}),
z_{2} = x_{3}(x_{1}^{2}x_{2} + x_{1}^{2}x_{3} + x_{1}x_{2}^{2} + x_{1}x_{2}x_{3} + x_{2}^{2}x_{3} + x_{3}^{3}) = ∏_{σ∈Q} σ(x_{3}),

subject to the relations

0 = q_{1}^{2} + x_{1}^{2}z_{1} + x_{1}^{2}z_{2} + x_{2}^{2}z_{1} + x_{1}^{2}x_{2}q_{1} + x_{1}x_{2}^{2}q_{1},
0 = q_{2}^{2} + x_{1}^{2}z_{2} + x_{2}^{2}z_{1} + x_{2}^{2}z_{2} + x_{1}^{2}x_{2}q_{2} + x_{1}x_{2}^{2}q_{2},

and has Poincaré series (1 + t^{3})^{2}/(1 − t)^{2}(1 − t^{4})^{2}. (iii) H^{∗}(H, k)^{Q} is generated by v, v̄, u, ū, s, s̄, where v = v_{1}, u = v_{1}v_{4}^{2} + v_{2}v_{3}^{2},

and s = v_{1}^{3}v_{2} + v_{2}^{4} = ∏_{σ∈Q} σ(v_{2}), subject to the relations

0 = ωu^{2} + ū^{2} + ωv^{2}s̄ + v̄^{2}s + v^{3}u + ωv̄^{3}ū,
0 = u^{2} + ωū^{2} + v^{2}s̄ + ωv̄^{2}s + ωv^{3}u + v̄^{3}ū.

(iv) As a bigraded k-module we have k ⊗_{𝔽_{2}} E_{2} = k[s, s̄] ⊗ Ẽ_{2}, where

Ẽ_{2} = k[v, v̄] ⊗ ⟨1, u, ū, uū⟩ ⊕ k[γ_{1}, γ_{2}] ⊗ ⟨γ_{2}, ζ, ζ̄, ζζ̄⟩ ⊕

k[γ_{1}] ⊗ ⟨γ_{1}, γ_{1}v, γ_{1}v̄, γ_{1}u, γ_{1}ū⟩ ⊕ ⟨γ_{1}vu⟩,

where ζ = [(ω̄v_{1}v_{2} + ωv_{2}^{2})y_{1} + (v_{1}v_{2} + v_{2}^{2})y_{2}].

Note. Ẽ_{2} is depicted in Figure 1. In the leftmost column we give the kQ-module structure of (W \overline{W})^{q}, and at each point (p, q) we give the dimension of and a basis for Ẽ_{2}^{p,q}.

[figure: 16kQ; •; 16; v^{6} ⋯ v̄^{6} ; v^{3}u ⋯ v̄^{3}u ; v^{3}ū ⋯ v̄^{3}ū ; uū; •; 0; •; 0; •; 0; •; 0; •; 0; 12kQ; •; 12; v^{5} ⋯ v̄^{5} ; v^{2}u vv̄u ; v̄^{2}u v^{2}ū ; vv̄ū v̄^{2}ū; •; 0; •; 0; •; 0; •; 0; •; 0; Ω^{2} ⊕ 7kQ; •; 9; v^{4} ⋯ v̄^{4} ; vu vū ; v̄u v̄ū; •; 1; γ_{1}vu; •; 1; η = ζζ̄; •; 2; {γ_{1}η \atop γ_{2}η}; •; 3; γ_{1}^{2}η ; γ_{1}γ_{2}η ; γ_{2}^{2}η; •; 4; {M_{1} ⊕ \atop M_{2} ⊕ 4kQ}; •; 6; {v^{3} ⋯ v̄^{3} \atop u ū}; •; 2; {γ_{1}u \atop γ_{1}ū}; •; 2; {γ_{1}^{2}u \atop γ_{1}^{2}ū}; •; 2; {γ_{1}^{3}u \atop γ_{1}^{3}ū}; •; 2; {γ_{1}^{4}u \atop γ_{1}^{4}ū}; •; 2; 2Ω ⊕ kQ; •; 3; \begin{matrix}v^{2} \\ vv̄ \\  & v̄^{2} \end{matrix}; •; 2; ζ ; ζ̄; •; 4; {γ_{1}ζ γ_{2}ζ \atop γ_{1}ζ̄ γ_{2}ζ̄}; •; 6; {γ_{1}^{2}ζ ⋯ γ_{2}^{2}ζ \atop γ_{1}^{2}ζ̄⋯ γ_{2}^{2}ζ̄}; •; 8; {γ_{1}^{3}ζ ⋯ γ_{2}^{3}ζ \atop γ_{1}^{3}ζ̄⋯ γ_{2}^{3}ζ̄}; •; 10; M_{1} ⊕ M_{2}; •; 2; v v̄; •; 2; γ_{1}v γ_{1}v̄; •; 2; γ_{1}^{2}v γ_{1}^{2}v̄; •; 2; γ_{1}^{3}v γ_{1}^{3}v̄; •; 2; γ_{1}^{4}v γ_{1}^{4}v̄; •; 2; k; •; 1; 1; •; 2; γ_{1} γ_{2}; •; 3; γ_{1}^{2} ; γ_{1}γ_{2} ; γ_{2}^{2}; •; 4; γ_{1}^{3} ⋯ γ_{2}^{2}; •; 5; γ_{1}^{4} ⋯ γ_{2}^{4}; •; 6]

Figure 1. Ẽ_{2}^{p,q} = H^{p}(Q, (W \overline{W})^{q})

The remainder of this section will consist of a proof of this proposition. We begin by analyzing the structure of S. The following is well-known:

Lemma 3.2. Let k be a field of characteristic 2 and let σ be the algebra automorphisms of k[x, y] defined by σ(x) = x, σ(y) = x + y. Then k[x, y]^{σ} is a polynomial ring in the two generators x and y(x + y).

By Lemma 3.2, S^{g_{2}} = k[v_{1}, w], where w = v_{2}(v_{1} + v_{2}). We have g_{1}w = v_{1}^{2} + w. Now multiplication by v_{1} : (S^{2n})^{g_{2}} → (S^{2n+1})^{g_{2}} is an isomorphism of k⟨g_{1}⟩- modules, so S^{g_{2}} = (S^{2∗})^{g_{2}} ⊕v_{1}(S^{2∗})^{g_{2}}. Again, Lemma 3.2 implies (S^{2∗})^{Q} = k[v_{1}^{2}, s], where s = w(v_{1}^{2} + w), so has Poincaré series 1/(1 − t^{2})(1 − t^{4}). By multiplying this by 1 + t we see that the Poincaré series of S^{Q} is 1/(1 − t)(1 − t^{4}), and hence S^{Q} = k[v_{1}, s].

We have S^{0} ≅ k, S^{1} ≅ M_{1}, S^{2} ≅ Ωk as it is 3-dimensional with 1-dimensional socle, and S^{3} ≅ kQ as it is 4-dimensional with 1-dimensional socle. Let W = k[v] ∑_{i=0}^{3} S^{i}. We claim that S = k[s] ⊗ W. To prove this it suffices to show that s, v is a regular sequence in S. For then multiplication by v_{1} induces an injective kQ-homomorphism from S^{n}/sS^{n−4} to S^{n+1}/sS^{n−3} for all n ≥ 4. Since both domain and range have dimension 4 it is therefore an isomorphism. It follows that S = k[s]W. Hence the natural map k[s] ⊗ W → S is surjective, and by comparing Poincaré series we see it is an isomorphism. Hence the claim follows from the following, with R = k[v_{1}] and x = v_{2}:

Lemma 3.3. Let k be a field and R = ⊕_{n≥0}R_{n} a graded commutative k-algebra with R_{0} = k, and assume that R is an integral domain. Let 𝔪 = ⊕_{n>0}R_{n}. Suppose f is a monic polynomial of positive degree in R[x], and let 0 ≠ a ∈ 𝔪. Then f, a is a regular sequence in R[x].

Proof. Given an element of R[x]/R[x]f we may choose g ∈ R[x] representing that element with deg(g) < deg(f). Thus deg(ag) = deg(g) < deg(f), while every non-zero element of R[x]f has degree at least deg(f). Hence a + R[x]f is not a zero-divisor in R[x]/R[x]f.

We thus have the following Poincaré series for the multiplicities of the 4 indecomposables which occur as summands of S:

p_{k}(t) = \frac{1}{1 − t^{4}} , p_{M_{1}}(t) = \frac{t}{1 − t^{4}} , p_{Ωk}(t) = \frac{t^{2}}{1 − t^{4}} , p_{kQ}(t) = \frac{t^{3}}{(1 − t)(1 − t^{4})}.

The same statement holds replacing M_{1} with M_{2} and S with \overline{S}. Since k ⊗_{𝔽_{2}} A ≅ S ⊗ \overline{S}, we need to know the tensor products of these indecomposables. But the tensor product of a free module with any module is free, and furthermore we claim

M_{1} ⊗ M_{2} ≅ kQ, M_{1} ⊗ Ωk ≅ M_{1} ⊕ kQ, Ωk ⊗ Ωk ≅ Ω^{2}k ⊕ kQ.

The first isomorphism can be seen using varieties (cf. Evens [7, Theorem 10.1.1]), as

V_{Q}(k̃ ⊗ M_{1} ⊗ M_{2}) = V_{Q}(k̃ ⊗ M_{1}) ∩ V_{Q}(k̃ ⊗ M_{2})
= V ((y_{1} + ωy_{2})) ∩ V ((y_{1} + ω̄y_{2})) = {0},

where k̃ is an algebraic closure of k, so k̃ ⊗ M_{1} ⊗ M_{2}, and therefore M_{1} ⊗ M_{2}, is free. The other two follow from the facts that for any f.g. kQ-module U, Ωk ⊗ U is isomorphic to the direct sum of ΩU and a projective module (cf. [1, Corollary 3.1.6]), and ΩM_{i} ≅ M_{i} because M_{i} is periodic and hence has period 1, by Benson-Carlson [2, Proposition 2.2]. From this we can determine the kQ-module structure of k⊗_{𝔽_{2}}A. To obtain the 𝔽_{2}Q-module structure of A, use the Noether-Deuring Theorem [6, Theorem 29.7], which says that for any f.g. 𝔽_{2}Q-modules U, V, U ≅ V ⇔ k ⊗_{𝔽_{2}} U ≅ k ⊗_{𝔽_{2}} V. So the multiplicities of 𝔽_{2}, Ω𝔽_{2}, Ω^{2}𝔽_{2}, 𝔽_{2}Q, and M in A equal the respective multiplicities of k, Ωk, Ω^{2}k, kQ, and M_{1} in k ⊗_{𝔽_{2}} A. Hence

p_{𝔽_{2}} = p_{k}^{2}, p_{M} = p_{k}p_{M_{1}} + p_{Ωk}p_{M_{1}}, p_{Ω𝔽_{2}} = 2p_{k}p_{Ωk}, p_{Ω^{2}𝔽_{2}} = p_{Ωk}^{2},
p_{𝔽_{2}Q} = 2p_{k}p_{kQ} + p_{M_{1}}^{2} + 2p_{Ωk}p_{M_{1}} + 4p_{M_{1}}p_{kQ} + p_{Ω^{2}k} + 6p_{Ωk}p_{k} + 4p_{kQ}^{2},

from which we obtain part (i) of the proposition.

We now turn to (ii). The Poincaré series for A^{Q} is as claimed since it equals ∑_{U} dim_{k}(U^{Q})p_{U}(t), the sum taken over representatives U of the isomorphism classes of indecomposable kQ-modules. We must next check the elements are invariant and satisfy the relations. One is helped here by the automorphism ϕ of M of order 3, defined by x_{1} ↦ x_{2}, x_{2} ↦ x_{1} + x_{2}, x_{3} ↦ x_{3} + x_{4}, x_{4} ↦ x_{3}. We have ϕ(q_{1}) = q_{2}, ϕ(q_{2}) = q_{1} + q_{2}, and similarly for z_{i} replacing q_{i}. So if q_{1} is invariant, so is q_{2}. Since z_{1} is a norm, it is invariant, and therefore z_{2} is as well, since the norm commutes with ϕ. Finally ϕ takes the first relation to the second, so only the first must be checked.

Let B be the subring of A^{Q} generated by the x_{i}, q_{i}, z_{i}. We will show that B = A^{Q}. First we claim that the subring T of B generated by x_{1}, x_{2}, z_{1}, z_{2} is a polynomial ring in those 4 generators. That follows from the following lemma (with R = k[x_{1}, x_{2}], S = R[x_{4}, x_{3}]), the proof of which is an easy exercise:

Lemma 3.4. Let R be an integral domain, n and d positive integers, and S = R[w_{1}, …, w_{n}], the polynomial ring in n variables. Suppose z_{1}, …, z_{n} ∈ S and for each i there exists 0 ≠ r_{i} ∈ R such that deg(z_{i} − r_{i}w_{i}^{d}) < d. Then z_{1}, …, z_{n} are algebraically independent over R.

In particular, T has Poincaré series 1/(1−t)^{2}(1−t^{4})^{2}. Now consider the subspace T + q_{1}T + q_{2}T + q_{1}q_{2}T of B. We claim that this sum is direct. To see this, suppose

f_{0} + q_{1}f_{1} + q_{2}f_{2} + q_{1}q_{2}f_{3} = 0 f_{i} ∈ T (3.1)

Consider these as polynomials in x_{4} with coefficients in 𝔽_{2}[x_{1}, x_{2}, x_{3}]. Write q_{i} = ∑_{j=0}^{2} a_{i,j}x_{4}^{j}, q_{1}q_{2} = ∑_{j=0}^{4} b_{j}x_{4}^{j}, with a_{i,j}, b_{j} of degree 0. We have a_{1,1} = x_{1}^{2} + x_{2}^{2}, a_{1,2} = x_{1} + x_{2}, a_{2,1} = x_{2}^{2}, a_{2,2} = x_{2}, b_{3} ≠ 0, b_{4} ≠ 0. For each i such that f_{i} ≠ 0, write f_{i} = ∑_{j=0}^{m_{i}} c_{i,j}z_{1}^{j} = c_{i,m_{i}}x_{4}^{4m_{i}} + g_{i}, where deg(f_{i}) = 4m_{i}, deg(c_{i,j}) = 0 and deg(g_{i}) ≤ 4m_{i}−2. We may do this since the coefficient of x_{4}^{3} in z_{4} is 0. Assuming not all the f_{i} are 0, let n be the maximum degree of f_{0}, q_{1}f_{1}, q_{2}f_{2}, q_{1}q_{2}f_{3}. There are two possibilities: either n = deg(f_{0}) = deg(q_{1}q_{2}f_{3}) ≡ 0 (mod 4), or n = deg(q_{1}f_{1}) = deg(q_{2}f_{2}) ≡ 2 (mod 4). In the first case, say n = 4m. Consideration of the x_{4}^{n}- and x_{4}^{n−1}-terms of (3.1) yields

c_{0,m}x_{4}^{n} + 0x_{4}^{n−1} = b_{4}c_{3,m−1}x_{4}^{n} + b_{3}c_{3,m−1}x_{4}^{n−1},

which implies c_{3,m−1} = 0, contradicting n = deg(q_{1}q_{2}f_{3}). In the second case, say n = 4m + 2. A similar consideration shows (x_{1} + x_{2})c_{1,m} = x_{2}c_{2,m} and (x_{1}^{2} + x_{2}^{2})c_{1,m} = x_{2}^{2}c_{2,m}, from which it is easy to see c_{2,m} = 0, contradicting n = deg(q_{2}f_{2}). Hence all the f_{i} are 0, so the sum is direct and therefore has Poincaré series (1 + 2t^{3} + t^{6})/(1 − t)^{2}(1 − t^{4})^{2}, which equals the Poincaré series of A^{Q}. Hence B = A^{Q}.

We now turn to the relations. Let B̃ = 𝔽_{2}[x̃_{1}, x̃_{2}, q̃_{1}, q̃_{2}, z̃_{1}, z̃_{2}]. We have a map of 𝔽_{2}-algebras B̃ ↠ B which has kernel containing (r_{1}, r_{2}), where the r_{i} are defined as in (ii), and we wish to show this is exactly the kernel. But Lemma 3.3 (with R = 𝔽_{2}[x̃_{1}, x̃_{2}, q̃_{2}, z̃_{1}, z̃_{2}], x = q̃_{1}) shows that r_{1}, r_{2} is a regular sequence, so the Poincaré series of B̃/(r_{1}, r_{2}) is (1 − t^{6})^{2}/(1 − t)^{2}(1 − t^{3})^{2}(1 − t^{4})^{2}, which equals the Poincaré series of B. So (r_{1}, r_{2}) is the kernel, proving (ii).

Now for (iii). We have x_{1} = v + v̄, x_{2} = ω̄v + ωv̄, q_{1} = ωu + ω̄ū, q_{2} = ω̄u + ωū, z_{1} = ωs + ω̄s̄+ ω̄ūv + ωuv̄, z_{2} = s + s̄+ ūv + uv̄. So by (ii), the alleged generators do generate k ⊗_{𝔽_{2}} B. The relations are the same as those in (ii) multiplied by ω̄, proving (iii).

To prove (iv), recall that S = k[s] ⊗ W. Applying the Frobenius, we have \overline{S} = k[s̄]⊗\overline{W}, so k⊗_{𝔽_{2}} A = S ⊗\overline{S} = k[s, s̄]⊗W ⊗\overline{W}, and therefore k⊗_{𝔽_{2}} E_{2} = k[s, s̄]⊗Ẽ_{2}, where Ẽ_{2} = H^{∗}(Q, W \overline{W}). Consider the natural map from k[v, v̄] ⊗ ⟨1, u, ū, uū⟩ to (W \overline{W})^{Q}. As v, v̄, u, ū are algebraically independent (by Lemma 3.4, with R = k[v, v̄], S = R[v_{4}, v_{2}]), the map is certainly injective, and a comparison of the Poincaré series shows it is an isomorphism. That describes Ẽ_{2}^{0∗}. To get the rest of Ẽ_{2} we must describe the cohomology of the relevant indecomposables:

Lemma 3.5. For the following kQ-modules U, we give a presentation for U in diagram form, B(Q, U) = B Hom_{kQ}(Y, U), and a complement H̃(Q, U) of B(Q, U) in Z(Q, U).

U diagram B(Q, U)  H̃(Q, U)
M_{1} [figure: a_{2}; a_{1}; ωa_{1}] k[y_{1}, y_{2}]⟨ωa_{1}y_{1} + a_{1}y_{2}⟩  k[y_{1}]a_{1}
Ωk  [figure: b; c; a] k[y_{1}, y_{2}]⟨ay_{1}, ay_{2}⟩  ⟨a⟩ ⊕ k[y_{1}, y_{2}]⟨cy_{1} + by_{2}⟩
Ω^{2}k [figure: c_{3}; c_{4}; c_{5}; c_{1}; c_{2}] \begin{matrix}⟨c_{2}y_{1} + c_{1}y_{2}⟩⊕ \\ k[y_{1}, y_{2}]⟨c_{1}y_{1}, c_{1}y_{2}^{2}, c_{2}y_{1}^{2}, c_{2}y_{2}⟩ \end{matrix} \begin{matrix}⟨c_{1}, c_{2}, c_{2}y_{1}⟩⊕ \\ k[y_{1}, y_{2}]⟨c_{5}y_{1}^{2} + c_{4}y_{1}y_{2} + c_{3}y_{2}^{2}⟩ \end{matrix}

In the diagrams, [figure](resp. [figure] ) signifies the action of g_{1} + 1 (resp. g_{2} + 1). The proofs are routine. Next, we have the following explicit decompositions of the W^{i}\overline{W}^{j}, where the action of Q on the basis elements corresponds to the labeling given in Lemma 3.5:

module summand basis
W^{2}\overline{W}^{0} Ωk  a = v_{1}^{2}, b = v_{1}v_{2} + v_{2}^{2}, c = ω̄v_{1}v_{2} + ωv_{2}^{2}
W^{1}\overline{W}^{2} M_{1}  a_{1} = v_{1}v_{4}^{2} + v_{2}v_{3}^{2}, a_{2} = v_{1}v_{3}v_{4} + v_{2}v_{4}^{2},
kQ  v_{1}v_{3}^{2}, v_{1}v_{3}v_{4}, v_{2}v_{3}^{2}, v_{2}v_{3}v_{4}
W^{2}\overline{W}^{2} Ω^{2}k  c_{1} = ab̄ + bā, c_{2} = aā + cā + ac̄, c_{3} = bā + bb̄,
c_{4} = bā + cb̄ + bc̄, c_{5} = cc̄
kQ  aā, cā, ab̄, cb̄

Using this and the Frobenius, we have explicit decompositions of (W \overline{W})^{q} for 0 ≤ q ≤ 4. Hence by Lemma 3.5, Ẽ_{2}^{∗,1} = H^{∗}(Q, M_{1} ⊕M_{2}) = k[γ_{1}]⊗⟨v, v̄⟩, and therefore Ẽ_{2}^{+,1} = k[γ_{1}] ⊗ ⟨γ_{1}v, γ_{1}v̄⟩. Similarly,

Ẽ_{2}^{+,2} = H^{+}(Q, W^{2}) ⊕ H^{+}(Q, \overline{W}^{2}) = k[γ_{1}, γ_{2}] ⊗ ⟨ζ, ζ̄⟩,
Ẽ_{2}^{+,3} = H^{+}(Q, ⟨a_{1}, a_{2}⟩) ⊕ H^{+}(Q, ⟨ā_{1}, ā_{2}⟩) = k[γ_{1}] ⊗ ⟨γ_{1}u, γ_{1}ū⟩,
Ẽ_{2}^{+,4} = H^{+}(Q, ⟨c_{1}, …, c_{5}⟩) = ⟨γ_{1}c_{2}⟩ ⊕ k[γ_{1}, γ_{2}]η,

where η = [c_{5}y_{1}^{2} + c_{4}y_{1}y_{2} + c_{3}y_{2}^{2}]. It remains to show [γ_{1}c_{2}] = [γ_{1}vu] and η = ζζ̄. The first is straightforward, and Lemma 2.1 implies ζζ̄ is represented by

(cy_{1} + by_{2}) ⌣ (c̄y_{1} + b̄y_{2}) = c\overline{g_{1}(c)}y_{1}^{2} + (c\overline{g_{1}(b)} + b\overline{g_{2}(c)})y_{1}y_{2} + b\overline{g_{2}(b)}y_{2}^{2}
= cc̄y_{1}^{2} + (c(ā + b̄) + b(ā + c̄))y_{1}y_{2} + bb̄y_{2}^{2}
= c_{5}y_{1}^{2} + (c_{4} + cā)y_{1}y_{2} + (c_{1} + c_{3} + ab̄)y_{2}^{2}
∼ c_{5}y_{1}^{2} + c_{4}y_{1}y_{2} + c_{3}y_{2}^{2},

as cā and ab̄ are in the kQ-summand and c_{1}y_{2}^{2} is a coboundary by Lemma 3.5. This completes the proof of Proposition 3.1.

4. Totaro’s example: differentials

We now apply Theorem 2.2 to find the differentials in this spectral sequence. The Theorem requires us to find a twisting system {f_{α}} for the group extension, and the algorithm used for doing this is described in the following section. The values f̂_{α}(v^{β}) (for |α| ∈ {2, 3}, |β| ≤ 4) appear in Figures 2 and 3. For readability we have abbreviated v^{γ} by γ in the tables.

Proposition 4.1. We have d_{2}(ζ) = γ_{1}^{3}v̄, d_{2}(ζ̄) = γ_{1}^{3}v, and d_{2} vanishes on γ_{1}, γ_{2}, v, v̄, u, ū, s, s̄. Hence k ⊗_{𝔽_{2}} E_{3} = k[s, s̄] ⊗ Ẽ_{3}, where

Ẽ_{3} = k[v, v̄] ⊗ ⟨1, u, ū, uū⟩ ⊕ k[γ_{1}, γ_{2}] ⊗ ⟨γ_{2}, ξ, ξ̄, η⟩ ⊕

k[γ_{1}] ⊗ ⟨γ_{1}, γ_{1}u, γ_{1}ū⟩ ⊕ ⟨γ_{1}v, γ_{1}v̄, γ_{1}^{2}v, γ_{1}^{2}v̄, γ_{1}vu⟩,

η = ζζ̄, and ξ = (γ_{1} + ωγ_{2})ζ. Moreover, E_{3} = E_{∞}, so H^{∗}(G, 𝔽_{2}) has Poincaré series

\frac{1 + 2t + t^{2} + 2t^{4} + t^{5} + t^{7}}{(1 − t^{4})^{2}(1 − t)^{2}} .

Proof. Let b_{1} = ω̄v_{1}v_{2} + ωv_{2}^{2}, b_{2} = v_{1}v_{2} + v_{2}^{2}. By Theorem 2.2, d_{2}(ζ) is represented by

(f̂_{20}y_{1}^{2} + f̂_{11}y_{1}y_{2} + f̂_{02}y_{2}^{2})(b_{1}y_{1} + b_{2}y_{2}) = v_{3}y_{1}^{3} + ω̄v_{3}y_{1}^{2}y_{2} + ωv_{3}y_{1}y_{2}^{2} + v_{3}y_{2}^{3}
∼ v_{3}y_{1}^{3},

which represents γ_{1}^{3}v̄. Here we also used Lemma 3.5, which implies v_{3}y_{2} ∼ ω̄v_{3}y_{1}. Similarly, d_{2}(u) is represented by

f̂_{20}(u)y_{1}^{2} + f̂_{11}(u)y_{1}y_{2} + f̂_{02}(u)y_{2}^{2} = ωay_{1}^{2} + ay_{2}^{2} ∼ 0.

Applying the Frobenius gives the desired values of d_{2} on ζ̄ and ū. We also know all differentials vanish on s and s̄ as s = res_{H}^{G} norm_{H}^{G}(v_{2}) by [7, Theorem 6.1.1 (N4)], where norm_{H}^{G} denotes the Evens norm map. To show that d_{2}(η) = 0 we could use the fact that d_{2} is a derivation, but we will do this using Theorem 2.2 as follows. First, we may choose a representative x^{2,4} ∈ Ẽ_{1}^{2,4} of η such that the coefficients of x^{2,4} lie in W^{2}\overline{W}^{2}. From Figure 2, we see that f̂_{α}(W^{2}\overline{W}^{2}) ⊆ W^{3} ⊕ \overline{W}^{3} ⊕ ⟨v_{1}^{2}v_{3}⟩ ⊕ ⟨v_{1}v_{3}^{2}⟩ whenever |α| = 2. In particular, the coefficients of δ_{2}(x^{2,4}) lie in the sum of the four kQ-summands of (W \overline{W})^{3}, so d_{2}(η) = 0. Finally, d_{2} vanishes on the other generators because of their positions and the fact that all differentials into the horizontal edge vanish in the case of a split extension (cf. [7, Proposition 7.3.2]).

10 STEPHEN F. SIEGEL

\begin{matrix}β\α & 20 & 11 & 02 \\ 1^{2} & 0 & 0 & 0 \\ 12 & 0 & 0 & 0 \\ 13 & 0 & 0 & 0 \\ 14 & 0 & 0 & 0 \\ 2^{2} & ω̄3 & 0 & 3 \\ 23 & 0 & 0 & 0 \\ 24 & ω1 + ω̄3 & ω1 + ω̄3 & ω1 + ω̄3 \\ 3^{2} & 0 & 0 & 0 \\ 34 & 0 & 0 & 0 \\ 4^{2} & ω1 & 0 & 1 \\ 1^{3} & 0 & 0 & 0 \\ 1^{2}2 & 0 & 0 & 0 \\ 1^{2}3 & 0 & 0 & 0 \\ 1^{2}4 & 0 & 0 & 0 \\ 12^{2} & ω̄13 & 0 & 13 \\ 123 & 0 & 0 & 0 \\ 124 & ω1^{2} + ω̄13 & ω1^{2} + ω̄13 & ω1^{2} + ω̄13 \\ 13^{2} & 0 & 0 & 0 \\ 134 & 0 & 0 & 0 \\ 14^{2} & ω1^{2} & 0 & 1^{2} \\ 2^{3} & ω̄23 & 0 & 23 \\ 2^{2}3 & ω̄3^{2} & 0 & 3^{2} \\ 2^{2}4 & ω̄34 & 0 & 34 \\ 23^{2} & 0 & 0 & 0 \\ 234 & ω13 + ω̄3^{2} & ω13 + ω̄3^{2} & ω13 + ω̄3^{2} \\ 24^{2} & ω12 & 0 & 12 \\ 3^{3} & 0 & 0 & 0 \\ 3^{2}4 & 0 & 0 & 0 \\ 34^{2} & ω13 & 0 & 13 \\ 4^{3} & ω14 & 0 & 14 \\ 1^{4} & 0 & 0 & 0 \\ 1^{3}2 & 0 & 0 & 0 \\ 1^{3}3 & 0 & 0 & 0 \\ 1^{3}4 & 0 & 0 & 0 \\ 1^{2}2^{2} & ω̄1^{2}3 & 0 & 1^{2}3 \\ 1^{2}23 & 0 & 0 & 0 \\ 1^{2}24 & ω1^{3} + ω̄1^{2}3 & ω1^{3} + ω̄1^{2}3 & ω1^{3} + ω̄1^{2}3 \\ 1^{2}3^{2} & 0 & 0 & 0 \\ 1^{2}34 & 0 & 0 & 0 \\ 1^{2}4^{2} & ω1^{3} & 0 & 1^{3} \\ 12^{3} & ω̄123 & 0 & 123 \\ 12^{2}3 & ω̄13^{2} & 0 & 13^{2} \\ 12^{2}4 & ω̄134 & 0 & 134 \\ 123^{2} & 0 & 0 & 0 \\ 1234 & ω1^{2}3 + ω̄13^{2} & ω1^{2}3 + ω̄13^{2} & ω1^{2}3 + ω̄13^{2} \\ 124^{2} & ω1^{2}2 & 0 & 1^{2}2 \\ 13^{3} & 0 & 0 & 0 \\ 13^{2}4 & 0 & 0 & 0 \\ 134^{2} & ω1^{2}3 & 0 & 1^{2}3 \\ 14^{3} & ω1^{2}4 & 0 & 1^{2}4 \\ 2^{4} & 0 & 0 & 0 \\ 2^{3}3 & ω̄23^{2} & 0 & 23^{2} \\ 2^{3}4 & ω12^{2} + ω̄2^{2}3 + ω̄234 & ω̄1^{3} + 1^{2}3 + ω12^{2} + ω̄2^{2}3 & ω12^{2} + ω̄2^{2}3 + 234 \\ 2^{2}3^{2} & ω̄3^{3} & 0 & 3^{3} \\ 2^{2}34 & ω̄3^{2}4 & 0 & 3^{2}4 \\ 2^{2}4^{2} & ω12^{2} + ω̄34^{2} & 0 & 12^{2} + 34^{2} \\ 23^{3} & 0 & 0 & 0 \\ 23^{2}4 & ω13^{2} + ω̄3^{3} & ω13^{2} + ω̄3^{3} & ω13^{2} + ω̄3^{3} \\ 234^{2} & ω123 & 0 & 123 \\ 24^{3} & ω124 + ω14^{2} + ω̄34^{2} & 13^{2} + ω14^{2} + ω3^{3} + ω̄34^{2} & 124 + ω14^{2} + ω̄34^{2} \\ 3^{4} & 0 & 0 & 0 \\ 3^{3}4 & 0 & 0 & 0 \\ 3^{2}4^{2} & ω13^{2} & 0 & 13^{2} \\ 34^{3} & ω134 & 0 & 134 \\ 4^{4} & 0 & 0 & 0 \end{matrix}

Figure 2. f̂_{α}(v^{β}) for |α| = 2

COHOMOLOGY OF SPLIT EXTENSIONS OF ELEMENTARY ABELIAN 2-GROUPS 11

\begin{matrix}β\α & 30 & 21 & 12 & 03 \\ 1^{3} & 0 & 0 & 0 & 0 \\ 1^{2}2 & 0 & 0 & 0 & 0 \\ 1^{2}3 & 0 & 0 & 0 & 0 \\ 1^{2}4 & 0 & 0 & 0 & 0 \\ 12^{2} & 0 & 0 & 0 & 0 \\ 123 & 0 & 0 & 0 & 0 \\ 124 & 0 & 0 & 0 & 0 \\ 13^{2} & 0 & 0 & 0 & 0 \\ 134 & 0 & 0 & 0 & 0 \\ 14^{2} & 0 & 0 & 0 & 0 \\ 2^{3} & ω1 + ω̄3 & 1 + ω3 & ω̄1 + 3 & ω1 + ω̄3 \\ 2^{2}3 & 0 & 0 & 0 & 0 \\ 2^{2}4 & ω1 & ω̄1 & ω̄1 & 1 \\ 23^{2} & 0 & 0 & 0 & 0 \\ 234 & 0 & 0 & 0 & 0 \\ 24^{2} & ω̄3 & ω3 & ω3 & 3 \\ 3^{3} & 0 & 0 & 0 & 0 \\ 3^{2}4 & 0 & 0 & 0 & 0 \\ 34^{2} & 0 & 0 & 0 & 0 \\ 4^{3} & ω1 + ω̄3 & ω̄1 + 3 & 1 + ω3 & ω1 + ω̄3 \\ 1^{4} & 0 & 0 & 0 & 0 \\ 1^{3}2 & 0 & 0 & 0 & 0 \\ 1^{3}3 & 0 & 0 & 0 & 0 \\ 1^{3}4 & 0 & 0 & 0 & 0 \\ 1^{2}2^{2} & 0 & 0 & 0 & 0 \\ 1^{2}23 & 0 & 0 & 0 & 0 \\ 1^{2}24 & 0 & 0 & 0 & 0 \\ 1^{2}3^{2} & 0 & 0 & 0 & 0 \\ 1^{2}34 & 0 & 0 & 0 & 0 \\ 1^{2}4^{2} & 0 & 0 & 0 & 0 \\ 12^{3} & ω1^{2} + ω̄13 & 1^{2} + ω13 & ω̄1^{2} + 13 & ω1^{2} + ω̄13 \\ 12^{2}3 & 0 & 0 & 0 & 0 \\ 12^{2}4 & ω1^{2} & ω̄1^{2} & ω̄1^{2} & 1^{2} \\ 123^{2} & 0 & 0 & 0 & 0 \\ 1234 & 0 & 0 & 0 & 0 \\ 124^{2} & ω̄13 & ω13 & ω13 & 13 \\ 13^{3} & 0 & 0 & 0 & 0 \\ 13^{2}4 & 0 & 0 & 0 & 0 \\ 134^{2} & 0 & 0 & 0 & 0 \\ 14^{3} & ω1^{2} + ω̄13 & ω̄1^{2} + 13 & 1^{2} + ω13 & ω1^{2} + ω̄13 \\ 2^{4} & 0 & 0 & 0 & 0 \\ 2^{3}3 & ω13 + ω̄3^{2} & 13 + ω3^{2} & ω̄13 + 3^{2} & ω13 + ω̄3^{2} \\ 2^{3}4 & \begin{matrix}ω̄1^{2} + ω12 + 13 \\ +ω14 + ω3^{2} + ω̄34 \end{matrix} & \begin{matrix}ω̄1^{2} + ω̄12 + 13 \\ +14 + ω3^{2} + ω34 \end{matrix} & \begin{matrix}1^{2} + ω̄12 + ω13 \\ +ω̄14 + ω̄3^{2} + 34 \end{matrix} & \begin{matrix}1^{2} + 12 + ω13 \\ +ω14 + ω̄3^{2} + ω̄34 \end{matrix} \\ 2^{2}3^{2} & 0 & 0 & 0 & 0 \\ 2^{2}34 & ω13 & ω̄13 & ω̄13 & 13 \\ 2^{2}4^{2} & 0 & 0 & 0 & 0 \\ 23^{3} & 0 & 0 & 0 & 0 \\ 23^{2}4 & 0 & 0 & 0 & 0 \\ 234^{2} & ω̄3^{2} & ω3^{2} & ω3^{2} & 3^{2} \\ 24^{3} & \begin{matrix}ω̄1^{2} + ω12 + 13 \\ +ω̄23 + ω3^{2} + ω̄34 \end{matrix} & \begin{matrix}ω̄1^{2} + ω̄12 + 13 \\ +23 + ω3^{2} + ω34 \end{matrix} & \begin{matrix}ω1^{2} + 12 + ω̄13 \\ +ω23 + 3^{2} + ω34 \end{matrix} & \begin{matrix}ω1^{2} + ω12 + ω̄13 \\ +ω̄23 + 3^{2} + 34 \end{matrix} \\ 3^{4} & 0 & 0 & 0 & 0 \\ 3^{3}4 & 0 & 0 & 0 & 0 \\ 3^{2}4^{2} & 0 & 0 & 0 & 0 \\ 34^{3} & ω13 + ω̄3^{2} & ω̄13 + 3^{2} & 13 + ω3^{2} & ω13 + ω̄3^{2} \\ 4^{4} & 0 & 0 & 0 & 0 \end{matrix}

Figure 3. f̂_{α}(v^{β}) for |α| = 3

It is straightforward to obtain the description of E_{3}. Since E_{3}^{p,1} = 0 for p ≥ 3, it remains only to show that d_{3}(η) = 0. Let N denote the kQ summand of W^{1}\overline{W}^{2} specified in Section 3, and let J = rad(kQ). Since J^{2}N = ⟨v_{1}v_{3}^{2}⟩, it is not difficult to see we may choose x^{3,3} ∈ Ẽ_{1}^{3,3} such that δ_{1}(x^{3,3}) = δ_{2}(x^{2,4}) and the coefficients of x^{3,3} are in the space W^{3} ⊕ \overline{W}^{3} ⊕ JN ⊕ J\overline{N} (note JN = ⟨v_{1}v_{3}^{2}, v_{1}v_{3}v_{4}, v_{2}v_{3}^{2}⟩). Using Figure 2, we observe that each f̂_{α} with |α| = 2 takes this space into W^{1}\overline{W}^{1}. Similarly, Figure 3 shows that f̂_{β}(W^{2}\overline{W}^{2}) ⊆ W^{1}\overline{W}^{1} whenever |β| = 3. By Theorem 2.2, d_{3}(η) is represented by δ_{2}(x^{3,3}) + δ_{3}(x^{2,4}), which has coefficients in W^{1}\overline{W}^{1} ≅ kQ; hence d_{3}(η) = 0.

5. How to calculate a twisting cochain

5.1. An algorithm. We return to the notation of Section 2. Viewing (2.1), one sees that the problem of calculating a twisting system for the extension G = H ⋊ Q reduces to the following two problems: (1) given σ ∈ Aut(H), construct a map f ∈ Z_{0} Hom_{kH}^{σ}(P, P) satisfying ϵ∘f = ϵ, and (2) given d ≥ 0 and f ∈ B_{d} Hom_{kH}^{σ}(P, P), construct a map g ∈ Hom_{kH}^{σ}(P, P)_{d+1} satisfying ∂g + g∂ = f.

To solve these problems we now define the standard contracting homotopy s ∈ Hom_{k}(P, P)_{1}. For any S ⊆ {1, 2, …, r}, let X_{S} = ∏_{i∈S} X_{i}. Define m(S) to be min(S) or ∞ if S = ∅, and define M(S) to be max(S) or −∞ if S = ∅. (Here ∞ is just a fixed integer greater than r.) Given α ∈ ℕ^{r}, define m(α) to be the least i such that α_{i} ≠ 0, or ∞ if α = 0, and define M(α) to be the greatest i such that α_{i} ≠ 0, or −∞ if α = 0. Now set

s(X_{S}x^{α}) = X_{S−{m(S)}}x_{m(S)}x^{α}(S ≠ ∅ and m(S) ≤ m(α))

Then ∂s + s∂ = 1 + ηϵ, where η: k → P is the unit map, as required. So if n ≥ 1, x ∈ P_{n}, and ∂(x) = 0, then ∂s(x) = x.

Problem (1) can now be solved as follows. Define f(1) = 1, and extend this in the unique way on P_{0}. Suppose f has been defined through degree n. Then define f(x^{α}) = sf∂(x^{α}) for |α| = n + 1, and extend to P_{n+1}. Then ∂f(x^{α}) = f∂(x^{α}), and since the x^{α} form a kH-basis of P_{n+1} we have ∂f = f∂ on all of P_{n+1}, as required.

The solution for Problem (2) is similar. First define g(1) = sf(1), and extend on P_{0}. If d = 0 then since f ∼ 0 we have ϵ ∘ f = 0, so ∂g(1) = ∂sf(1) = f(1) + ηϵ(f(1)) = f(1), as required. If d > 0 then ∂f(1) = f∂(1) = 0 so ∂g(1) = f(1) in this case as well. Now suppose g has been constructed through degree n to satisfy ∂g +g∂ = f. Then ∂f −∂g∂ vanishes on P_{n+1}. So define g(x^{β}) = s(f(x^{β})−g∂(x^{β})) for |β| = n + 1, and extend to P_{n+1}, and we have ∂g + g∂ = f on P_{n+1}.

We call the result of this algorithm the standard twisting system. Of course, it depends on the choice of generators for H and Q, but after fixing those it is welldefined. We implemented this algorithm in MIT Scheme, and executed it on a Sun SparcStation 20 (see [10] for source code and output). For the Burt Group of order 64, the program also converted from the x^{β}-basis of H^{∗}(H, k) to the v^{β}-basis, and produced f̂_{α} from f_{α}; those algorithms are completely standard.

5.2. Properties. It appears that much (if not all) of the data produced by the algorithm described above can be expressed by explicit, closed formulas. We will show how to do this for the f_{α} where |α| = 1 here. Finding similar formulas for |α| > 1 is one of the important open problems in the field, and would greatly reduce the amount of calculation required.

We will need two additional kH-algebra structures on the graded kH-module kH[x_{1}, …, x_{r}]. The first of these we denote by u ⊗ v ↦ u ∗ v and is defined by

x^{α} ∗ x^{β} = \binom{α + β}{β}x^{α+β} = \binom{α_{1} + β_{1}}{β_{1}} … \binom{α_{r} + β_{r}}{β_{r}}x^{α+β},

where α, β ∈ ℕ^{r}. This is just the “divided polynomial algebra” structure, and is associative and commutative. It also satisfies ∂(u ∗ v) = ∂(u) ∗ v + u ∗ ∂(v), which means (P, ∂, ∗) is a differential graded kH-algebra (cf. [5, Ch. XII §7]). The second we denote by u ⊗ v ↦ u ∘ v and is defined by

x^{α} ∘ x^{β} = x^{α+β}(M(α) ≤ m(β))

It is associative, but not commutative (unless r = 1). Both algebras have unit 1.

Lemma 5.1. Let s be the standard contracting homotopy for P.

(i) Ker s = Im s ∪ k · 1 = ∑ kX_{S}x^{α}, where the sum is taken over all subsets S ⊆ {1, 2, …, r} and α ∈ ℕ^{r} such that m(S) > m(α).

(ii) s(u ∘ v) = s(u) ∘ v for u ∈ P, v ∈ Im s (iii) u, v ∈ Im s ⇒ u ∗ v ∈ Im s

Proof. Each basis element X_{S}x^{α} of P_{n} such that S ≠ ∅ and m(S) ≤ m(α) is taken under s to a distinct basis element X_{S−{m(S)}}x_{m(S)}x^{α} of P_{n+1}. That is all that is needed to prove (i). It suffices to prove (ii) when u and v are basis elements, since ∘ is bilinear and s is linear. So let u = X_{S}x^{α}, v = X_{T} x^{β}, with m(β) < m(T). The definitions show

s(u) ∘ v = X_{S∪T −{m(S)}}x_{m(S)}x^{α+β}(P)
s(u ∘ v) = X_{S∪T −{m(S∪T)}}x_{m(S∪T)}x^{α+β}(Q),

where

P = S ≠ ∅ and m(S) ≤ m(α) and m(S) ≤ m(β) and (S − {m(S)}) ∩ T = ∅
and M(α) ≤ m(β),
Q = S ∪ T ≠ ∅ and m(S ∪ T) ≤ m(α + β) and S ∩ T = ∅ and M(α) ≤ m(β).

But P and Q are easily seen to be equivalent, and if they hold then m(S) = m(S ∪ T), establishing (ii). Finally, if m(S) > m(α) and m(T) > m(β) then m(S ∪ T) > m(α + β), and (iii) follows.

Proposition 5.2. Let {f_{α}} be the standard twisting system for the extension G = H ⋊ Q. Then for |α| = 1 we have f_{α}(x_{1}^{i_{1}} … x_{r}^{i_{r}}) = f_{α}(x_{1})^{∘i_{1}} ∗ ⋯ ∗ f_{α}(x_{r})^{∘i_{r}}.

Proof. Let f = f_{α}, and let σ be the automorphism induced on kH by g^{α}. Suppose 1 ≤ i ≤ r. We show by induction on n that f(x_{i}^{n}) = f(x_{i})^{∘n} for n ≥ 1. This is empty for n = 1, so suppose it holds for n. By construction, f(x_{i}^{n}) ∈ Im s. So by Lemma 5.1(ii),

f(x_{i}^{n+1}) = sf∂(x_{i}^{n+1}) = sf(X_{i}x_{i}^{n}) = s(σ(X_{i})f(x_{i}^{n})) = s(σ(X_{i})) ∘ f(x_{i}^{n})
= f(x_{i})^{∘(n+1)}.

We next show that f(u ∗ v) = f(u) ∗ f(v) for all u, v ∈ P, completing the proof. We do this by induction on deg(u)+deg(v) for homogeneous u, v. The statement being trivial in total degree 0, we need only show the inductive step. By kH-linearity, we may assume u = x^{β} and v = x^{γ}. Then

f(u ∗ v) = sf∂(u ∗ v) = sf(∂(u) ∗ v + u ∗ ∂(v))
= s(f∂(u) ∗ f(v) + f(u) ∗ f∂(v))
= s(∂f(u) ∗ f(v) + f(u) ∗ ∂f(v))
= s∂(f(u) ∗ f(v))
= f(u) ∗ f(v).

The last step follows because s∂(w) = w − ∂s(w) for w of positive degree, but since f(u), f(v) ∈ Im s, Lemma 5.1(iii) implies w = f(u) ∗ f(v) ∈ Im s, hence s(w) = 0 by Lemma 5.1(i).

References

[1] D. J. Benson, Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, Cambridge, 1991.

[2] D. J. Benson and J. F. Carlson, Periodic modules with large period, Quart. J. Math. Oxford (2) 43, 283–296 (1992).

[3] D. J. Benson and M. Feshbach, On the cohomology of split extensions, Proc. Amer. Math. Soc. 121, 687–690 (1994).

[4] K. S. Brown, Cohomology of Groups, GTM 87, Springer-Verlag, New York, 1982.

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

[6] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons, New York, 1962.

[7] L. Evens, The Cohomology of Groups, Oxford Univ. Press, Oxford, 1991.

[8] S. F. Siegel, The spectral sequence of a split extension and the cohomology of an extraspecial group of order p^{3} and exponent p, J. Pure App. Alg. 106 (2), 185–198 (1996).

[9] S. F. Siegel, On the cohomology of split extensions of finite groups, Trans. Amer. Math. Soc., to appear (1996).

[10] S. F. Siegel, Alg, MIT Scheme 7.3.0 code available on the World Wide Web, URL

   http://math.umass.edu/~siegel/math (1996).

[11] B. Totaro, Cohomology of semidirect product groups, J. Algebra 182, 469–475 (1996).

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515

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