ON THE COHOMOLOGY OF SPLIT EXTENSIONS OF FINITE GROUPS

STEPHEN F. SIEGEL

Abstract. Let G = H ⋊ Q be a split extension of finite groups. A theorem of Charlap and Vasquez gives an explicit description of the differentials d_{2} in the Lyndon-Hochschild-Serre spectral sequence of the extension with coefficients in a field k. We generalize this to give an explicit description of all the d_{r} (r ≥ 2) in this case. The generalization is obtained by associating to the group extension a new twisting cochain, which takes values in the kG-endomorphism algebra of the minimal kH-projective resolution induced from H to G. This twisting cochain not only determines the differentials, but also allows one to construct an explicit kG-projective resolution of k.

1. Introduction

Let G = H ⋊ Q be a split extension of finite groups, k a field, and consider the Lyndon-Hochschild-Serre (LHS) spectral sequence E_{2}^{∗∗} ≅ H^{∗}(Q, H^{∗}(H, k)) ⇒ H^{∗}(G, k). In a previous paper, we gave a simplified proof of a modified version of a theorem of Charlap and Vasquez which allows one to explicitly compute the differentials d_{2} in this case, and applied this result to a problem in the cohomology of extraspecial groups (see [4, 5] and [10, Theorem 1]). Here, we generalize the Charlap-Vasquez theorem to provide a method for explicitly computing all the differentials d_{r} (r ≥ 2).

The Charlap-Vasquez method for d_{2} proceeds as follows. Let P → k be the minimal kH-projective resolution, and for each σ ∈ Q, let ν_{1}[σ]: P → P be a chain map which commutes with the augmentation and is σ-linear, i.e. a linear map satisfying ν_{1}[σ](hx) = σhσ^{−1}ν_{1}[σ](x) (h ∈ H, x ∈ P). The existence of these maps follows from the Comparison Theorem of Homological Algebra (cf. [2, Theorem 2.4.2 and Remark]). Moreover, the uniqueness part of the Comparison Theorem says that for each σ, τ ∈ Q, there is a στ-linear map ν_{2}[σ|τ]: P → P which raises degree by one and which satisfies

∂ ∘ ν_{2}[σ|τ] + ν_{2}[σ|τ] ∘ ∂ = ν_{1}[στ] − ν_{1}[σ] ∘ ν_{1}[τ], (1.1)

The Charlap-Vasquez Theorem is an expression for d_{2} in terms of the maps ν_{2}[σ|τ].

As a corollary to our main Theorem 8.1 we obtain a direct generalization of the above. It reduces the calculation of the differentials d_{2}, …, d_{r} to the computation of certain maps ν_{n}[σ_{1}| ⋯ |σ_{n}]: P → P, one for each 1 ≤ n ≤ r and σ_{1}, …, σ_{n} ∈ Q. Each ν_{n}[σ_{1}| ⋯ |σ_{n}] is required to be σ_{1} ⋯ σ_{n}-linear and to raise degree by n − 1, and together they must satisfy

Date: January 9, 1996.

1991 Mathematics Subject Classification. Primary 20J06.

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

(1.2) ∂ ∘ ν_{n}[σ_{1}| ⋯ |σ_{n}] + (−1)^{n}ν_{n}[σ_{1}| ⋯ |σ_{n}] ∘ ∂

= ∑_{i=1}^{n−1}(−1)^{i+1}ν_{n−1}[σ_{1}| ⋯ |σ_{i}σ_{i+1}| ⋯ |σ_{n}]

+ ∑_{i=1}^{n−1}(−1)^{i}ν_{i}[σ_{1}| ⋯ |σ_{i}] ∘ ν_{n−i}[σ_{i+1}| ⋯ |σ_{n}].

Notice that when n = 2, equation (1.2) reduces to equation (1.1). The map ν_{2} can be interpreted as an obstruction to finding an action of Q on P which is consistent with the action on H, and the maps ν_{n} for n ≥ 2 can be viewed as higher-level obstructions. In particular, if Q really does act on P then all these obstructions vanish, and E_{2} = E_{∞}. (In fact a stronger result is true, namely E_{2} ≅ H^{∗}(G, k) as graded rings, cf. Evens [6, Section 2.5]). Another well-known result which follows from the main theorem is that the differentials into the horizontal edge vanish (cf. [6, Proposition 7.3.2]). Both of these results are discussed in further detail in Section 8.

The generalization of the Charlap-Vasquez theorem, as expressed above, requires the kQ-bar resolution of k to be used in constructing the LHS spectral sequence. But our main theorem actually works with a wider class of free resolutions. We call these special resolutions, and define them axiomatically in Section 4. This class includes the bar resolution, the reduced bar resolution, and the minimal resolution of an elementary abelian 2-group in characteristic 2. This last case makes our techniques particularly efficient for group extensions of the form H ⋊(ℤ/2)^{n}, which we will demonstrate in a subsequent paper.

The key to the generalization is the use of a twisting cochain (cf. Brown [3]), which is a map t from a differential graded coalgebra ℭ to a differential graded algebra 𝔄 satisfying (among other conditions) d(t) + t ∪ t = 0. This condition is designed specifically to allow one to use t to perturb the standard differential in the tensor product of a differential graded ℭ-comodule with a differential graded 𝔄module. In our case, ℭ is the special resolution for Q tensored over kQ with k, and 𝔄 is the kG-endomorphism algebra of P induced from H to G. Our twisting cochain, which is shown to be unique in an appropriate sense, captures many aspects of the cohomology of the extension. Not only does it produce the differentials in the spectral sequence, but it can be used to construct an explicit projective resolution of k over kG; in essence, this is an explicit version of a theorem of C. T. C. Wall [12].

The material is organized as follows. In Section 2 we establish our notation and review the homological algebra that we require; the propositions there are easy exercises in the definitions. In Section 3 we establish some basic facts about twisting cochains that will be fundamental for the main theorem; in particular we prove an analogue of the Comparison Theorem. Section 4 deals with the class of special resolutions mentioned above. In Section 5 we define the twisting cochain associated to an arbitrary (not necessarily split) extension, and in Section 6, which is not required for the main theorem, we show how this twisting cochain can be used to construct a kG-projective resolution of k. In Section 7 we add the assumption that the extension is split, and derive an explicit form for the twisting cochain in this case. The main theorem on the differentials is stated and proved in Section 8. We conclude in Section 9 with some small examples to demonstrate how the theorem can be applied; in a future paper we will apply the theory to more complex spectral sequences of extensions of elementary abelian 2-groups, where little had previously been known.

2. Background from homological algebra

Complexes. Let R be a commutative ring and Λ an R-algebra. (By algebra we always mean an associative algebra with unit, and similarly coalgebras are assumed to be coassociative with counit.) If C is a complex of left or right Λ-modules, let Z_{n}C = Ker(∂: C_{n} → C_{n−1}), B_{n}C = Im(∂: C_{n−1} → C_{n}), and H_{n}C = Z_{n}C/B_{n}C. If C, D are complexes of right, left Λ-modules respectively, let C ⊗_{Λ} D denote the complex with (C ⊗_{Λ} D)_{n} = ⨁_{p+q=n} C_{p} ⊗_{Λ} C_{q}, with differential given by ∂(x⊗y) = ∂(x) ⊗ y + (−1)^{p}x ⊗ ∂(y) for x ∈ C_{p}, y ∈ D_{q}.

If C and D are both complexes of left Λ-modules, then let Hom_{Λ}(C, D) denote the complex with Hom_{Λ}(C, D)_{n} = ∏_{q=p+n} Hom_{Λ}(C_{p}, D_{q}), with differential given by ∂(f) = ∂ ∘ f − (−1)^{n}f ∘ ∂, for f ∈ Hom_{Λ}(C, D)_{n}. If f − g ∈ B_{n} Hom_{Λ}(C, D) we write f ≃ g (and say f and g are chain homotopic). We also define a subcomplex hom_{Λ}(C, D) of Hom_{Λ}(C, D), which is equal to the latter in positive degrees, Z_{0} Hom_{Λ}(C, D) in degree 0, and 0 in negative degrees. In particular, f ∈ hom_{Λ}(C, D)_{0} if, and only if, f is a Λ-chain map, i.e. f is a Λ-homomorphism which preserves degree and ∂ ∘ f = f ∘ ∂. If B is a third complex of left Λmodules and f ∈ Hom_{Λ}(B, C)_{p}, then we define f^{∗}: Hom_{Λ}(C, D) → Hom_{Λ}(B, D) by f^{∗}(g) = (−1)^{pq}g ∘ f for g homogeneous of degree q. If g ∈ Hom_{Λ}(C, D) then we define g_{∗}: Hom_{Λ}(B, C) → Hom_{Λ}(B, D) by g^{∗}(f) = g ∘ f. Of course, similar definitions apply to right modules. Finally, any chain complex may be considered a cochain complex, and vice-versa, by setting C^{n} = C_{−n}.

Proposition 2.1. Let B, C, and D be complexes of left Λ-modules, and let f_{0} ∈ hom_{Λ}(B, C)_{0} and g_{0} ∈ hom_{Λ}(C, D)_{0}. Then each of the following maps is a map of complexes:

(i) C ⊗_{R} D → D ⊗_{R} C, defined by x ⊗ y ↦ (−1)^{deg(x) deg(y)}y ⊗ x (ii) Hom_{Λ}(C, D) ⊗_{R} C → D, defined by f ⊗ x ↦ f(x) (iii) C ⊗_{R} Hom_{Λ}(C, D) → D, defined by x ⊗ f↦(−1)^{deg(f) deg(x)}f(x) (iv) Hom_{Λ}(C, D) ⊗_{R} Hom_{Λ}(B, C) → Hom_{Λ}(B, D), defined by f ⊗ g ↦ f ∘ g

(v) Hom_{Λ}(B, C) ⊗_{R} Hom_{Λ}(C, D) → Hom_{Λ}(B, D), defined by f ⊗ g ↦ (−1)^{deg(f) deg(g)}g ∘ f

(vi) (f_{0})^{∗}: Hom_{Λ}(C, D) → Hom_{Λ}(B, D), defined by g ↦ g ∘ f_{0} (vii) (g_{0})_{∗}: Hom_{Λ}(B, C) → Hom_{Λ}(B, D), defined by f ↦ g_{0} ∘ f.

Now suppose X and Y are complexes of right Λ-modules. We define a map

(2.1)  Hom_{Λ}(X, Y) ⊗_{R} Hom_{Λ}(C, D) → Hom_{R}(X ⊗_{Λ} C, Y ⊗_{Λ} D)
f ⊗ g ↦ f × g

where (f × g)(x ⊗ y) = (−1)^{deg(x) deg(g)}f(x) ⊗ g(y) for homogeneous x, y, f, g. We then have

Proposition 2.2. Let B, C, and D be complexes of left Λ-modules, X, Y, and Z complexes of right Λ-modules, f ∈ Hom_{Λ}(B, C)_{r}, g ∈ Hom_{Λ}(X, Y)_{s}, f′ ∈ Hom_{Λ}(C, D)_{r′}, and g′ ∈ Hom_{Λ}(Y, Z)_{s′}. Then (i) ∂ ∘ (f × g) = (∂ ∘ f) × g + (−1)^{r}f × (∂ ∘ g) (ii) (f × g) ∘ ∂ = f × (g ∘ ∂) + (−1)^{s}(f ∘ ∂) × g (iii) ∂(f × g) = ∂(f) × g + (−1)^{r}f × ∂(g) (iv) (f′ × g′) ∘ (f × g) = (−1)^{rs′}(f′ ∘ f) × (g′ ∘ g).

Cup products. We now restrict our attention to the case where Λ is a Hopf algebra with antipode over a field k. All of our Λ-modules will be left Λ-modules, but these can always be considered right Λ-modules (and vice-versa), because of the antipode. All unlabeled tensor products are assumed to be tensor products over k.

Generally, cup products on the level of complexes arise as follows. Let A, B, …, F be chain complexes of Λ-modules, and let Δ: E → A ⊗ C and μ: B ⊗ D → F be Λ-chain maps. The product map (2.1)

Hom_{k}(A, B) ⊗ Hom_{k}(C, D) ⟶^{×} Hom_{k}(A ⊗ C, B ⊗ D)

restricts to a map

(2.2) Hom_{Λ}(A, B) ⊗ Hom_{Λ}(C, D) ⟶^{×} Hom_{Λ}(A ⊗ C, B ⊗ D),

which is also a map of complexes. We then define the cup product with respect to the “comultiplication” Δ and the “multiplication” μ to be the map

Hom_{Λ}(A, B) ⊗ Hom_{Λ}(C, D) ⟶^{∪} Hom_{Λ}(E, F)
f ⊗ g ⟼ f ∪ g = μ ∘ (f × g) ∘ Δ.

This is also a chain map, since it is just the composite of (2.2) followed by Δ^{∗} then μ_{∗}, each of which is a chain map, by Propositions 2.1 and 2.2.

Suppose A′, B′, C′, D′, E′, F′ form another set of complexes of Λ-modules with a multiplication and comultiplication map. Then there is a natural way to define a multiplication and comultiplication map on the pairwise tensor products. Namely, define Δ and μ as the composites

[figure: Δ: E′ ⊗ E; Δ×Δ; A′ ⊗ C′ ⊗ A ⊗ C; A′ ⊗ A ⊗ C′ ⊗ C]
[figure: μ: B′ ⊗ B ⊗ D′ ⊗ D; B′ ⊗ D′ ⊗ B ⊗ D; μ×μ; F′ ⊗ F,]

where the maps which transpose the two inner factors are defined by

w ⊗ x ⊗ y ⊗ z ↦ (−1)^{deg(x) deg(y)}w ⊗ y ⊗ x ⊗ z.

By Propositions 2.1 and 2.2, these are chain maps. Hence there is a cup product

Hom_{Λ}(A′ ⊗ A, B′ ⊗ B) ⊗ Hom_{Λ}(C′ ⊗ C, D′ ⊗ D) ⟶^{∪} Hom_{Λ}(E′ ⊗ E, F′ ⊗ F).

The cup product interacts nicely with the other operations, as in the following:

Proposition 2.3. The following hold, wherever they are defined:

(i) ∂ ∘ (f ∪ g) = (∂ ∘ f) ∪ g + (−1)^{deg(f)}f ∪ (∂ ∘ g) (ii) (f ∪ g) ∘ ∂ = f ∪ (g ∘ ∂) + (−1)^{deg(g)}(f ∘ ∂) ∪ g (iii) ∂(f ∪ g) = ∂(f) ∪ g + (−1)^{deg(f)}f ∪ ∂(g) (iv) (f′ ∪ g′) ∘ (f ∪ g) = (−1)^{deg(f) deg(g′)}(f′ ∘ f) ∪ (g′ ∘ g)

(v) (f′ ∪ g′) × (f ∪ g) = (−1)^{deg(f) deg(g′)}(f′ × f) ∪ (g′ × g).

Differential algebra. A differential graded algebra (DGA) over Λ is a complex 𝔄 of Λ-modules which is also a Λ-algebra, such that the unit map k →^{η} 𝔄 (sending the unit of k to the unit of 𝔄) and the multiplication map 𝔄 ⊗ 𝔄 →^{μ} 𝔄 are maps of complexes. (Here k is considered as a complex concentrated in degree 0.) A morphism of DGAs is a map of complexes which is also a map of Λ-algebras. An augmented DGA is a DGA which is 0 in negative degrees together with a map ϵ: 𝔄 → k which is a map of DGAs. If 𝔄 is a DGA then a differential graded left 𝔄-module (left DGM) is a complex M of left 𝔄-modules such that the multiplication map 𝔄 ⊗ M →^{μ} M is a map of complexes. Right DGMs are defined similarly.

A differential graded coalgebra (DGCA) ℭ is defined dually. Hence ℭ is a complex of Λ-modules which is also a Λ-coalgebra such that the counit map ℭ →^{ϵ} k and the comultiplication map ℭ →^{Δ} ℭ ⊗ ℭ are maps of complexes. A morphism of DGCAs, a coaugmented DGCA, and a differential graded comodule (DGCM) are also defined dually. We also use the “partition notation” for Δ: if n = a_{1} + ⋯ + a_{r} ≥ 1, then Δ_{a_{1}^{i_{1}} ,... ,a_{r}^{ir}} denotes the map

[figure: ℭ_{n}; ℭ; (1×···×1×Δ)···(1×Δ)Δ; ℭ^{⊗n}; ℭ_{a_{1}}^{⊗i_{1}} ⊗ ⋯ ⊗ ℭ_{a_{r}}^{⊗i_{r}}.]

It is immediate from the discussion above that for a DGA 𝔄 and a DGCA ℭ over Λ, there is a cup product

Hom_{Λ}(ℭ, 𝔄) ⊗ Hom_{Λ}(ℭ, 𝔄) ⟶^{∪} Hom_{Λ}(ℭ, 𝔄),

and that this gives Hom_{Λ}(ℭ, 𝔄) the structure of a DGA over k with unit ηϵ. Moreover, if M is a left DGM for 𝔄 and N is a left DGCM for ℭ, then there is a cup product

Hom_{Λ}(ℭ, 𝔄) ⊗ Hom_{Λ}(N, M) ⟶^{∪} Hom_{Λ}(N, M),

which gives Hom_{Λ}(N, M) the structure of a left DGM over Hom_{Λ}(ℭ, 𝔄). A similar statement holds for right modules.

Spectral sequences. Let Λ be an arbitrary ring and A = ⨁_{p,q∈ℤ} A^{p,q} a bigraded left Λ-module. Let π^{p,q}: A → A^{p,q} denote the projection map, set A^{n} = ⨁_{p+q=n} A^{p,q} and F^{p}A^{n} = ⨁_{p′≥p} A^{p′,n−p′}. Let δ: A → A be a Λ-map satisfying δ(F^{p}A^{n}) ⊆ F^{p}A^{n+1} and δ^{2} = 0. Hence we can write δ = δ_{0} + δ_{1} + …, where δ_{i}(A^{p,q}) ⊆ A^{p+i,q−i+1}, and the condition δ^{2} = 0 can be rewritten ∑_{i=0}^{n} δ_{i}δ_{n−i} = 0 (n ≥ 0). We call (A, δ) a bigraded complex. For each r ≥ 0, set

Z_{r}^{p,q} = {x ∈ A^{p,q} | ∃y ∈ F^{p+1}A^{p+q} s.t. δ(x + y) ∈ F^{p+r}A^{p+q+1}}
B_{r}^{p,q} = {π^{p,q}δ(y) | y ∈ F^{p−r+1}A^{p+q−1}, δ(y) ∈ F^{p}A^{p+q}}
E_{r}^{p,q} = Z_{r}^{p,q}/B_{r}^{p,q}.

Of course A = Z_{0} ⊇ Z_{1} ⊇ ⋯ ⊇ B_{1} ⊇ B_{0} = 0, so the definition of E_{r} makes sense. Define the differentials d_{r}^{p,q}: E_{r}^{p,q} → E_{r}^{p+r,q−r+1} as follows: let x ∈ Z_{r}^{p,q}, and choose y ∈ F^{p+1}A^{p+q} such that δ(x + y) ∈ F^{p+r}A^{p+q+1}. The element

π^{p+r,q−r+1} δ(x + y) + B_{r}^{p+r,q−r+1} ∈ E_{r}^{p+r,q−r+1}

is seen to be independent of the choice of y, so sending x to that element yields a well-defined map Z_{r}^{p,q} → E_{r}^{p+r,q−r+1}. One checks that B_{r}^{p,q} is in the kernel of this map, and defines the induced map to be d_{r}. Finally, it can be verified that d_{r}^{2} = 0, Ker(d_{r}) = Z_{r+1}/B_{r}, and Im(d_{r}) = B_{r+1}/B_{r}. Hence, H(E_{r}, d_{r}) ≅ E_{r+1}. We call the sequence (E_{r}, d_{r}) (r ≥ 0) the spectral sequence of the bigraded complex (A, δ). Note that if δ_{i} = 0 for i ≥ 2 then (A, δ_{0}, δ_{1}) is a double complex, and this spectral sequence is the familiar spectral sequence of a double complex.

3. Twisting cochains

Fundamentals. Let Λ be a Hopf algebra (with antipode) over a field k. As before, ⊗ means ⊗_{k}. Let 𝔄 be an augmented DGA over Λ with differential d, multiplication μ, unit map η, and augmentation ϵ. Let ℭ be a DGCA over Λ with differential d, comultiplication Δ, and counit ϵ.

Definition 3.1. A map t ∈ Hom_{Λ}(ℭ, 𝔄)_{−1} is called a twisting cochain if ϵt_{1} = 0 and d(t) + t ∪ t = 0. Two twisting cochains t̃ and t are said to be homotopic if there is a map s ∈ Hom_{Λ}(ℭ, 𝔄)_{0} such that ϵs = ϵ and d(s) = s ∪ t̃− t ∪ s, and if this is the case, one writes s: t̃≃ t.

The conditions d(t) + t ∪ t = 0 and d(s) = s ∪ t̃− t ∪ s can be rewritten

d_{i−1}t_{i} + t_{i−1}d_{i} + ∑_{j=1}^{i−1} t_{j} ∪ t_{i−j} = 0, i ≥ 2, (3.1)

d_{i}s_{i} − s_{i−1}d_{i} = ∑_{j=0}^{i−1} s_{j} ∪ t̃_{i−j} − ∑_{j=0}^{i−1} t_{i−j} ∪ s_{j}, i ≥ 1, (3.2)

respectively, since t_{j} = 0 for all j ≤ 0. It can also be shown that under appropriate conditions, homotopy of twisting cochains is an equivalence relation (cf. [8, Lemma 1.4.1]), though we will not need this fact here. More important for our purposes will be the following analogy to the Comparison Theorem:

Proposition 3.2. Let 𝔄 be an augmented DGA over Λ and ℭ a DGCA over Λ. Let n ≥ 1 and assume ℭ_{n+1} is a projective Λ-module. Then

(i) Suppose the complex ⋯ → 𝔄_{1} → 𝔄_{0} →^{ϵ} k is exact at degree n − 1. Given Λ-homomorphisms t_{1}, …, t_{n} such that ϵt_{1} = 0 and (3.1) holds for 2 ≤ i ≤ n, there exists a Λ-homomorphism t_{n+1} such that (3.1) holds for 2 ≤ i ≤ n + 1.

(ii) Suppose t̃ and t are twisting cochains and 𝔄 is exact at degree n. Given Λhomomorphisms s_{0}, …, s_{n} such that ϵs_{0} = ϵ and (3.2) holds for 1 ≤ i ≤ n, there exists a Λ-homomorphism s_{n+1} such that (3.2) holds for 1 ≤ i ≤ n + 1.

Proof. To prove (i) for n ≥ 2 we need only show

d_{n−1} ∘ (t_{n}d_{n+1} + ∑_{i=1}^{n} t_{i} ∪ t_{n+1−i}) = 0. (3.3)

ON THE COHOMOLOGY OF SPLIT EXTENSIONS OF FINITE GROUPS 7

The result then follows by exactness and the projectivity of ℭ_{n+1}. However,

d ∘ ∑_{i=1}^{n} t_{i} ∪ t_{n+1−i} = ∑_{i=1}^{n}[dt_{i} ∪ t_{n+1−i} − t_{i} ∪ dt_{n+1−i}]

= ∑_{i=1}^{n}[dt_{i} ∪ t_{n+1−i} − t_{i} ∪ dt_{n+1−i}]

= ∑_{i=1}^{n}[(−t_{i−1}d − ∑_{j=1}^{i−1} t_{j} ∪ t_{i−j}) ∪ t_{n+1−i} + t_{i} ∪ (t_{n−i}d + ∑_{j=1}^{n−i} t_{j} ∪ t_{n−i−j+1})]

= ∑_{i=1}^{n−1}[t_{i} ∪ t_{n−i}d − t_{i}d ∪ t_{n−i}]

= ( ∑_{i=1}^{n−1} t_{i} ∪ t_{n−i}) ∘ d

= ( ∑_{i=1}^{n−1} t_{i} ∪ t_{n−i} + t_{n−1}d) ∘ d

= −dt_{n}d, as required. The case n = 1 is entirely similar.

For the same reason, to prove (ii) it suffices to show

d_{n} ∘ (s_{n}d_{n+1} + ∑_{i=0}^{n}(s_{i} ∪ t̃_{n+1−i} − t_{n+1−i} ∪ s_{i})) = 0. (3.4)

However, applying d on the right side of (3.2) at degree n yields

ds_{n}d = ∑_{i=0}^{n−1}(s_{i} ∪ t̃_{n−i}) ∘ d − ∑_{i=0}^{n−1}(t_{n−i} ∪ s_{i}) ∘ d, (3.5)

and

∑_{i=0}^{n−1}(s_{i} ∪ t̃_{n−i}) ∘ d = ∑_{i=0}^{n−1}(s_{i} ∪ t̃_{n−i}d − s_{i}d ∪ t̃_{n−i})

= ∑_{i=0}^{n−1}[s_{i} ∪ (−dt̃_{n+1−i} − ∑_{j=1}^{n−i} t̃_{j} ∪ t̃_{n+1−i−j})

−(ds_{i+1} − ∑_{j=0}^{i}(s_{j} ∪ t̃_{i+1−j} − t_{i+1−j} ∪ s_{j})) ∪ t̃_{n−i}] (3.6)

= − ∑_{i=0}^{n} d ∘ (s_{i} ∪ t̃_{n+1−i}) − ∑_{i=0}^{n−1} ∑_{j=0}^{i} t_{i+1−j} ∪ s_{j} ∪ t̃_{n−i}.

Similarly,

− ∑_{i=0}^{n−1}(t_{n−i} ∪ s_{i}) ∘ d = ∑_{i=0}^{n} d ∘ (t_{n+1−i} ∪ s_{i}) + ∑_{i=0}^{n−1} ∑_{j=0}^{i} t_{i+1−j} ∪ s_{j} ∪ t̃_{n−i}. (3.7)

Adding equations (3.5), (3.6), and (3.7), we obtain (3.4).

The augmentation map ϵ: 𝔄_{0} → k factors uniquely as ϵ = ϵ_{0}ϵ_{1}, where ϵ_{1} : 𝔄_{0} → 𝔄_{0}/d(𝔄_{1}) is the natural map and ϵ_{0} : 𝔄_{0}/d(𝔄_{1}) → k. As a corollary to Proposition 3.2 we obtain the following, which will apply directly to the case of a group extension.

Corollary 3.3. Let 𝔄 be an augmented DGA over Λ which is exact in positive degrees, and let ℭ be a DGCA over Λ such that ℭ_{n} is a projective Λ-module for all n ≥ 0. Then

(i) Any Λ-homomorphism t_{1} : ℭ_{1} → 𝔄_{0} satisfying ϵt_{1} = 0 and ϵ_{1}∘(t_{1}d_{2}+t_{1}∪t_{1}) = 0 can be extended to a twisting cochain t: ℭ → 𝔄.

(ii) If t̃ and t are two twisting cochains such that ϵ_{1}t̃_{1} = ϵ_{1}t_{1} then t̃ and t are homotopic, and there exists a homotopy s: t̃≃ t such that s_{0} = ηϵ.

Proof. Since Ker(ϵ_{1}) = Im(d_{1}), the hypothesis of (i) implies Im(t_{1}d_{2} + t_{1} ∪ t_{1}) ⊆ Im(d_{1}). By the projectivity of ℭ_{2}, there is a Λ-map t_{2} : ℭ_{2} → 𝔄_{1} satisfying (3.1) for i = 2; (i) now follows from Proposition 3.2(i). To prove (ii), first set s_{0} = ηϵ, and observe that ϵs_{0} = (ϵη)ϵ = ϵ. Moreover,

ϵ_{1} ∘ (s_{0}d_{1} + s_{0} ∪ t̃_{1} − t_{1} ∪ s_{0}) = ϵ_{1}t̃_{1} − ϵ_{1}t = 0,

as s_{0}d_{1} = ηϵd_{1} = 0. Hence by the projectivity of ℭ_{1}, there is a map s_{1} : ℭ_{1} → 𝔄_{1} satisfying (3.2) for i = 1, and (ii) now follows from Proposition 3.2(ii).

Perturbations. If U is a left DGCM over ℭ and V is a left DGM over 𝔄, then there is a chain map

Hom_{Λ}(ℭ, 𝔄) → Hom_{Λ}(Hom_{k}(U, V), Hom_{k}(U, V)),

given by f ↦ f∪. Let t ∈ Hom_{Λ}(ℭ, 𝔄)_{−1} be a twisting cochain, and set d_{t} = d+t∪, where d denotes the standard differential on Hom_{k}(U, V). We then have

d_{t}^{2} = d ∘ (t∪) + (t∪) ∘ d + (t∪)^{2} = d(t∪) + (t ∪ t)∪ = (d(t) + t ∪ t)∪ = 0.

Hence we may perturb d by t∪ and obtain another differential on Hom_{k}(U, V).

There is an analogous construction with tensor products in place of homomorphisms. If M is a left DGM over 𝔄 and N a right DGCM over ℭ, we define the cap product map by:

Hom_{Λ}(ℭ, 𝔄) ⟶ Hom_{Λ}(N ⊗ M, N ⊗ M)
f ⟼ f∩ = (1 × μ)(1 × f × 1)(Δ × 1) (3.8)

It follows immediately from Propositions 2.1 and 2.2 that the cap product map is a map of complexes.

Proposition 3.4. Let M be a left DGM over 𝔄, N a right DGCM over ℭ.

(i) Let f and g be homogeneous elements of Hom_{Λ}(ℭ, 𝔄). Then (f∩) ∘ (g∩) = (f ∪ g)∩.

(ii) Let t ∈ Hom_{Λ}(ℭ, 𝔄)_{−1} be a twisting cochain, and set d_{t} = d + t∩: N ⊗ M → N ⊗ M. Then d_{t}^{2} = 0.

Proof. (cf. [9, Proposition 30.5]) Using the associativity of μ and the coassociativity of Δ, we have

(f∩) ∘ (g∩) = (1 × μ)(1 × f × 1)(Δ × 1)(1 × μ)(1 × g × 1)(Δ × 1)
= (1 × μ)(1 × f × 1)(1 × 1 × μ)(Δ × 1 × 1)(1 × g × 1)(Δ × 1)

= (1 × μ)(1 × 1 × μ)(1 × f × 1 × 1)(1 × 1 × g × 1)(Δ × 1 × 1)(Δ × 1)

= (1 × μ)(1 × μ × 1)(1 × f × g × 1)(1 × Δ × 1)(Δ × 1)
= (1 × μ)(1 × μ(f × g)Δ × 1)(Δ × 1)
= (1 × μ)(1 × (f ∪ g) × 1)(Δ × 1)
= (f ∪ g)∩,

which proves (i). For (ii), since d^{2} = 0, we have

d_{t}^{2} = (t∩) ∘ (t∩) + (t∩) ∘ d + d ∘ (t∩) = (t ∪ t) ∩ +d(t∩) = −d(t) ∩ +d(t∩) = 0,

as the cap product map is a map of complexes.

We let N ⊗_{t} M denote the “twisted tensor product of N and M with respect to t,” i.e., the complex with the same graded Λ-module structure as N ⊗ M but with differential d_{t}. Alternatively, we could take M to be a right DGM and N a left DGCM. We could then define a cap product on M ⊗ N, just as in (3.8), but with

f∩ = (μ × 1)(1 × f × 1)(1 × Δ).

We then get the following analogue to Proposition 3.4:

Proposition 3.5. Let M be a right DGM over 𝔄, N a left DGCM over ℭ.

(i) Let f and g be homogeneous elements of Hom_{Λ}(ℭ, 𝔄). Then (f∩) ∘ (g∩) = (−1)^{deg(f) deg(g)}(g ∪ f)∩.

(ii) Let t ∈ Hom_{Λ}(ℭ, 𝔄)_{−1} be a twisting cochain, and set d_{t} = d − t∩: M ⊗ N → M ⊗ N. Then d_{t}^{2} = 0.

4. Special resolutions

Let k be a field, Q a group (not necessarily finite), and (Y, ∂) → k a free resolution of the trivial kQ-module k. Assume further that Y_{0} = kQ and that ϵ: Y_{0} → k is the standard augmentation, i.e., ϵ(σ) = 1 for all σ ∈ Q. Let Δ: Y → Y ⊗ Y be a diagonal approximation map (i.e., a kQ-chain map which commutes with the augmentations). Let Ȳ = k ⊗_{kQ} Y, let π: Y → Ȳ denote the natural map, and let ¯∂, Δ̄: Ȳ → Ȳ and ϵ̄: Ȳ_{0} → k denote the maps induced by ∂, Δ, and ϵ, respectively. Of course, we may identify Ȳ_{0} with k, and then ϵ̄ is just the identity map.

Definition 4.1. Let ı ∈ Hom_{k}(Ȳ , Y)_{0}. We say that (Y, ∂, Δ, ı) (or simply Y, when the maps are understood) is a special resolution if the following hold:

(i) ı_{0}(1) = 1. (ii) πı = 1Ȳ. (iii) For all n ≥ 0, ı(Ȳ_{n}) generates Y_{n} as a kQ-module. (iv) ϵ̄∪ 1Ȳ = 1Ȳ = 1Ȳ ∪ ϵ̄.

(v) (1Ȳ × Δ̄)Δ̄ = (Δ̄ × 1Ȳ)Δ̄. (vi) d(ı) = ∂_{1}ı_{1} ∪ ı.

A few remarks will clarify the definition. First, it is always possible to find a map ı satisfying (i), (ii), and (iii). This can be done, for example, by first choosing a kQ-basis {y_{n}^{j}} of Y_{n}, for each n ≥ 0 (with {y_{0}^{j}} = {1}). Then {π(y_{n}^{j})} is a k-basis for Ȳ_{n}, and one can define ı(π(y_{n}^{j})) = y_{n}^{j}. Conversely, if Y is a special resolution then these conditions imply that the image under ı of any k-basis of Ȳ is a kQ-basis of Y.

Second, conditions (iv) and (v) are equivalent to demanding that Ȳ is a DGCA over k with counit ϵ̄ and comultiplication Δ̄. We remark that (iv) is equivalent to requiring

(iv)′ ı_{0} ∪ ı = ı = ı ∪ ı_{0}.

(The cup product in (iv)′ makes sense, since Im(ı_{0}) ⊆ Y_{0} = kQ, and Y is a kQmodule.) This follows from the fact that ϵ̄∪1Ȳ_{n} = μΔ̄_{0,n} (n ≥ 0) and by considering the following commutative diagram:

[figure: Ȳ_{n}; Δ̄_{0,n}; Ȳ_{0} ⊗ Ȳ_{n}; ı_{0}×ı; μ; Y_{0} ⊗ Y_{n}; μ; Ȳ_{n}; ı; Y_{n}]

The interesting condition is (vi). Notice that the cup product here makes sense, since Im(∂_{1}) ⊆ Y_{0} = kQ. Hence ∂_{1}ı_{1} ∪ ı in degree n (n ≥ 1) is the composite

Ȳ_{n} ⟶^{Δ̄_{1,n−1}} Ȳ_{1} ⊗ Ȳ_{n−1} ⟶^{∂_{1}ı_{1}×ı} Y_{0} ⊗ Y_{n−1} = kQ ⊗ Y_{n−1} ⟶^{μ} Y_{n−1}.

Of course, d(ı) = ∂ı − ı¯∂. Hence we may interpret (vi) as an expression for ∂_{n} in terms of ¯∂_{n}, Δ̄_{1,n−1} and ı (and ∂_{1}).

Example 4.2. The bar resolution Y and the standard diagonal approximation map Δ (the “Alexander-Whitney map”) are defined by

Y_{n} = ⨁_{σ_{1},... ,σ_{n}∈Q} kQ(σ_{1}| ⋯ |σ_{n})

∂(σ_{1}| ⋯ |σ_{n}) = σ_{1}(σ_{2}| ⋯ |σ_{n}) + ∑_{i=1}^{n}(−1)^{n}(σ_{1}| ⋯ |σ_{i}σ_{i+1}| ⋯ |σ_{n})

Δ(σ_{1}| ⋯ |σ_{n}) = ∑_{i=0}^{n}(σ_{1}| ⋯ |σ_{i}) ⊗ σ_{1} ⋯ σ_{i}(σ_{i+1}| ⋯ |σ_{n}).

We identify kQ with Y_{0} = kQ() by σ ↔ σ(), and we let [σ_{1}| ⋯ |σ_{n}] denote the image of (σ_{1}| ⋯ |σ_{n}) in Ȳ. Define ı: Ȳ → Y by

ı[σ_{1}| ⋯ |σ_{n}] = (σ_{1}| ⋯ |σ_{n}), n ≥ 0.

We claim that (Y, ∂, Δ, ı) is a special resolution. It is well known that Δ is coassociative, hence Δ̄ certainly is. All of the other conditions are also easily verified, except for (vi). However,

d(ı)[σ_{1}| ⋯ |σ_{n}] = (∂ı − ı¯∂)[σ_{1}| ⋯ |σ_{n}]
= (σ_{1}| ⋯ |σ_{n}) − ı¯∂[σ_{1}| ⋯ |σ_{n}]
= (σ_{1} − 1)(σ_{2}| ⋯ |σ_{n})
= ∂_{1}(σ_{1}).(σ_{2}| ⋯ |σ_{n})
= (∂_{1}ı_{1} ∪ ı)[σ_{1}| ⋯ |σ_{n}].

Example 4.3. The reduced bar resolution is the quotient of the bar resolution by the kQ-subcomplex generated by all (σ_{1}| ⋯ |σ_{n}) such that σ_{i} = 1 for some i (1 ≤ i ≤ n). It is easily verified that the reduced bar resolution is special.

Example 4.4. Let Q′ and Q′′ be groups and set Q = Q′ × Q′′. Suppose Y′ and Y′′ are special resolutions for Q′ and Q′′, respectively. Then Y = Y′ ⊗ Y′′ is a projective resolution for the trivial kQ-module k, and we have Y_{0} = kQ′ ⊗ kQ′′ = kQ. As in Section 2, the diagonal approximations on Y′ and Y′′ induce a diagonal approximation Δ on Y. Now set ı = ı × ı: Ȳ′ ⊗ Ȳ′′ → Y′ ⊗ Y′′. We claim that Y is a special resolution. As conditions (i) through (v) are immediate, we check only (vi):

∂_{1}ı_{1} ∪ ı = (∂_{1} × 1 + 1 × ∂_{1})(ı × ı)_{1} ∪ (ı × ı)
= (∂_{1}ı_{1} × ı_{0}) ∪ (ı × ı) + (ı_{0} × ∂_{1}ı_{1}) ∪ (ı × ı)
= (∂_{1}ı_{1} ∪ ı) × (ı_{0} ∪ ı) + (ı_{0} ∪ ı) × (∂_{1}ı_{1} ∪ ı)
= (∂_{1}ı_{1} ∪ ı) × ı + ı × (∂_{1}ı_{1} ∪ ı)
= d(ı) × ı + ı × d(ı)
= d(ı × ı)
= d(ı).

Example 4.5. Let Q be an elementary abelian 2-group and k a field of characteristic 2. Then the minimal resolution of the trivial kQ-module is special. This follows from the two preceding examples, since the reduced bar resolution of a cyclic group of order 2 is minimal, and the tensor product of minimal resolutions is minimal.

Suppose, however, that p is an odd prime, k is a field of characteristic p, Q = ⟨σ | σ^{p} = 1⟩, and (Y, ∂) is the minimal resolution. Then for no choice of Δ and ı is Y special. For in this case we have Ȳ = H_{∗}(Q, k), ¯∂ = 0, and Δ̄ is the comultiplication in homology. Since p is odd, Δ̄_{1,1} = 0, so by (vi),

∂_{2}ı_{2} = ∂_{1}ı_{1} ∪ ı_{1} = μ(∂_{1} × ı_{1})Δ̄_{1,1} = 0.

Hence ı_{2}(Ȳ_{2}) ⊆ Ker(∂_{2}) = rad_{kQ}(Y_{2}), contradicting (iii).

We conclude with a technical fact concerning special resolutions which will be used in the case of a group extension.

Proposition 4.6. Let (Y, ∂, Δ, ı) be a special resolution for the trivial kQ-module k. Set ϕ = ∂_{1}ı_{1} : Ȳ_{1} → kQ. Then ϕ¯∂_{2} = ϕ ∪ ϕ.

Proof. Start with condition (vi) of the definition at degree 2, and apply ∂_{1} to obtain

∂_{2}ı_{2} − ı_{1} ¯∂_{2} = ∂_{1}ı_{1} ∪ ı_{1}
−∂_{1}ı_{1} ¯∂_{2} = −∂_{1}ı_{1} ∪ ∂_{1}ı_{1},

as claimed.

5. The twisting cochain of a group extension

Let k be a field and let 1 → H → G → Q → 1 be an extension of (not necessarily finite) groups. In this section, we show how to associate to the group extension an augmented DGA 𝔄 over k, a DGCA ℭ, and a twisting cochain t ∈ Hom_{k}(ℭ, 𝔄)_{−1}.

Recall that for a kH-module U, U^{G} = kG ⊗_{kH} U is a kG-module (the induced module). For g ∈ G, we let U^{g} denote the kH-module with the same underlying vector space structure as U, but with a new action given by h.x = ghg^{−1}x, for h ∈ H and x ∈ U. For a kG-module V, we let V_{H} denote V considered as a kH-module (the restriction of V to H; cf. Alperin [1]).

Let (Y, ∂, Δ, ı) be a special resolution for the trivial kQ-module. We let ℭ be the DGCA Ȳ = k ⊗_{kQ} Y, with differential d = ¯∂ and comultiplication Δ̄.

Let (P, ∂) →^{ϵ} k be a projective resolution for the trivial kH-module, and set

𝔄 = hom_{kG}(P^{G}, P^{G})^{op}. (5.1)

Thus 𝔄 is a DGA over k with multiplication 𝔄 ⊗ 𝔄 →^{μ} 𝔄 given by right composition (which involves a sign). To define the augmentation ϵ: 𝔄_{0} → k we first define a map ϵ_{1} : 𝔄_{0} → kQ, as follows. Since any f ∈ 𝔄_{0} is a map of complexes, there exists a unique α ∈ kQ such that right multiplication by α (·α) makes the following diagram commute:

[figure: ⋯; P_{1}^{G}; f_{1}; P_{0}^{G}; ϵ^{G}; f_{0}; k^{G} = kQ; ·α; ⋯; P_{1}^{G}; P_{0}^{G}; ϵ^{G}; k^{G} = kQ.]

We define ϵ_{1}(f) = α, and we set ϵ = ϵ_{0}ϵ_{1}, where ϵ_{0} : kQ → k is the standard augmentation. It follows that

ϵ_{1}μ(f ⊗ g) = ϵ_{1}(gf) = ϵ_{1}(f)ϵ_{1}(g),

hence ϵ_{1} is a map of algebras (which was the reason for choosing the opposite endomorphism algebra in (5.1)). Moreover, for f ∈ 𝔄_{1}, d_{1}(f) = ∂f + f∂, which in degree 0 is just ∂_{1}f_{0}, so ϵ_{1}d_{1}(f) = 0. We conclude that ϵ_{1} is a map of DGAs, and since ϵ_{0} is clearly a map of DGAs, so is ϵ = ϵ_{0}ϵ_{1}.

Since P^{G} → k^{G} = kQ is just a projective resolution of the kG-module kQ, the Comparison Theorem implies that the sequence

⋯ → 𝔄_{2} → 𝔄_{1} → 𝔄_{0} →^{ϵ_{1}} kQ → 0,

is exact. Hence 𝔄 is exact in positive degrees, we may identify 𝔄_{0}/ Im(d_{1}) with kQ, and the factorization ϵ = ϵ_{0}ϵ_{1} agrees with the notation of Section 3.

We now turn to the definition of t. Let ϕ = ∂_{1}ı_{1} : ℭ_{1} → Y_{0} = kQ. Since ℭ_{1} is k-projective, there exists a k-linear map t_{1} : ℭ_{1} → 𝔄_{0} such that ϵ_{1}t_{1} + ϕ = 0. By Proposition 4.6,

ϵ_{1} ∘ (t_{1} ∪ t_{1} + t_{1}d_{2}) = ϵ_{1}t_{1} ∪ ϵ_{1}t_{1} + ϵ_{1}t_{1}d_{2}
= ϕ ∪ ϕ − ϕd_{2}
= 0.

Hence we can apply Corollary 3.3 to immediately prove Theorem 5.1. Let 1 → H → G → Q → 1 be a group extension, k a field, Y → k a special kQ-resolution, and P → k an arbitrary kH-projective resolution. Let ℭ = k ⊗_{kQ} Y and 𝔄 = hom_{kG}(P^{G}, P^{G})^{op}, and let ϵ_{1} and ϕ be defined as above. Then there exists a twisting cochain t: ℭ → 𝔄 with ϵ_{1}t_{1} + ϕ = 0. Moreover, any two such twisting cochains are homotopic, and the homotopy can be chosen to be ηϵ in degree 0.

We will call a map t satisfying the conditions of Theorem 5.1 a twisting cochain associated to the group extension 1 → H → G → Q → 1.

6. Projective resolutions

We continue with the notation of Section 5: k is a field, 1 → H → G → Q → 1 is a group extension, P is a kH-projective resolution, Y is a special kQ-resolution, and 𝔄, ℭ and t are defined as before. We now show how the twisting cochain t relates to a construction of C. T. C. Wall in [12]. In that paper, Wall showed that for any free resolution Y, a differential could be imposed on P^{G} ⊗ Ȳ to form a projective resolution of the trivial kG-module, and he provided an inductive method for finding such a differential. Here, we show—with our additional assumption that Y is special—that this differential can be constructed as the “twisted differential” d_{t} arising from our twisting cochain t.

By Proposition 2.1(iii), right evaluation (with a sign) P^{G} ⊗ 𝔄 →^{μ} P^{G} is a chain map, and therefore gives P^{G} the structure of a right DGM over 𝔄. Since Ȳ is a left DGCM over itself, we can apply Proposition 3.5(ii) to form the twisted tensor product X = P^{G} ⊗_{t} Ȳ. We define ϵ_{X}: X → k as the composite

P^{G} ⊗_{t} Ȳ ⟶^{ϵ^{G}×1} kQ ⊗ Ȳ ⟶^{ϵ_{0}×ϵ̄} k ⊗ k ⟶^{μ} k.

Now P^{G}, ℭ = Ȳ, and 𝔄 are all kG-modules, with G acting trivially on ℭ and 𝔄. In particular, each X_{n} is a kG-module—in fact, X_{n} is a projective kG-module, as it is isomorphic to the direct sum of modules of the form P_{i}^{G}. Moreover, ϵ_{X}, as a composite of kG-homomorphisms, is a kG-homomorphism. Finally, the differential d_{t} = d − t∩ is a kG-homomorphism. This follows from the observation that P^{G} ⊗ 𝔄 →^{μ} P^{G} preserves the action of G (and t trivially preserves the G-action), so one can apply the perturbation construction of Section 3 with Λ = kG.

Theorem 6.1. Let k be a field, 1 → H → G → Q → 1 a group extension, P →^{ϵ} k a kH-projective resolution, Y →^{ϵ} k a special kQ-resolution, and t an associated twisting cochain. Then P^{G} ⊗_{t} Ȳ ⟶^{ϵ_{X}} k is a projective resolution of the trivial kGmodule.

Proof. We first check that ϵ_{X} ∘(d_{t})_{1} = 0. For this it suffices to show ϵ_{X} ∘ (t_{1}∩) = 0. But

\begin{aligned}(ϵ^{G} × 1) ∘ (t_{1}∩) & = (ϵ^{G} × 1)(μ × 1)(1 × t_{1} × 1)(1 × Δ̄) \\ & = (μ × 1)(ϵ^{G} × ϵ_{1} × 1)(1 × t_{1} × 1)(1 × Δ̄) \\ & = −(μ × 1)(ϵ^{G} × ϕ × 1)(1 × Δ̄) \end{aligned} (6.1)

Hence

ϵ_{X} ∘ (t_{1}∩) = μ(ϵ_{0} × ϵ̄)(ϵ^{G} × 1)(t_{1}∩)
= −μ(ϵ_{0} × ϵ̄)(μ × 1)(ϵ^{G} × ϕ × 1)(1 × Δ̄)
= μ(1 × ϵ̄)(μ × 1)(ϵ_{0}ϵ^{G} × ϵϕ × 1)(1 × Δ̄)
= 0,

as ϵϕ = (ϵ∂_{1})ı_{1} = 0.

We next show H_{∗}(X, d_{t}) = k, completing the proof. To see this, consider the spectral sequence arising from the bigraded complex defined by

A^{p,q} = P_{−q}^{G} ⊗ Ȳ_{−p}, δ = d_{t} = d − t ∩ .

Writing δ = δ_{0} + δ_{1} + ⋯, as in Section 2, we see that

δ_{r}(x ⊗ ȳ) = \begin{cases}∂(x) ⊗ ȳ & if r = 0 \\ (−1)^{q}x ⊗ ¯∂(ȳ) − t_{1} ∩ (x ⊗ ȳ) & if r = 1 \\ −t_{r} ∩ (x ⊗ ȳ) & if r ≥ 2, \end{cases}

for x ∈ P_{−q}^{G}, ȳ ∈ Ȳ_{−p}. Each column is a direct sum of copies of P^{G}, and d_{0} = δ_{0} induces the standard differential on each copy, hence

E_{1}^{p,q} ≅ \begin{cases}kQ ⊗ Ȳ_{−p} & if q = 0 \\ 0 & if q ≠ 0. \end{cases}

The isomorphism is given as follows: α ⊗ ȳ ∈ kQ ⊗ Ȳ_{−p} is represented in A^{p,q} by x ⊗ ȳ, where x is any element of P_{0}^{G} such that ϵ^{G}(x) = α.

Consider the kQ-homomorphism μ ∘ (1 × i): Ȳ^{Q} = kQ ⊗ Ȳ → Y. By Definition 4.1(iii), this map is surjective, and by comparing dimensions, we see that it is an isomorphism. Using this isomorphism to identify E_{1} with Y, we claim that d_{1} induces the standard differential ∂ on Y. This will complete the proof, since we will then have H_{∗}(X, d_{t}) = E_{∞} = E_{2} = k (concentrated in degree 0). But since d_{1} is a kQ-homomorphism, it suffices to check this for y = ı(ȳ), for any ȳ ∈ Ȳ. Then y corresponds to ζ ∈ E_{1}^{p,0}, where ζ is represented by x ⊗ ȳ, where x ∈ P_{0}^{G} and ϵ^{G}(x) = 1. But by Definition 4.1(vi) and (6.1),

−μ(1 × ı)(ϵ^{G} × 1)(t_{1} ∩ (x × ȳ)) = μ(1 × ı)(μ × 1)(ϵ^{G} × ϕ × 1)(1 × Δ̄)(x × ȳ)
= μ(1 × ı)(ϕ × 1)Δ̄(ȳ)
= (ϕ ∪ ı)(ȳ)
= d(ı)(ȳ)
= ∂ı(ȳ) − ı¯∂(ȳ).

Hence d_{1}(ζ) corresponds to

μ(1 × ı)(ϵ^{G} × 1)δ_{1}(x ⊗ ȳ) = μ(1 × ı)(ϵ^{G} × 1)(x ⊗ ¯∂(ȳ) − t_{1} ∩ (x ⊗ ȳ))
= ı¯∂(ȳ) + ∂ı(ȳ) − ı¯∂(ȳ)
= ∂(y),

as claimed.

7. The explicit twisting cochain of a split extension of finite groups

We now concentrate on the case where G = H ⋊ Q is a split extension of finite groups. In this case another description of 𝔄 is useful. Consider the augmented DGA

𝔄̃ = kQ ⊗ hom_{k}(P, P)^{op}.

As usual, kQ is considered to be an augmented DGA concentrated in degree 0 with the usual augmentation map ϵ_{0}. We define 𝔄̂ to be the k-submodule of 𝔄̃ generated by all σ ⊗ ϕ, where σ ∈ Q and ϕ ∈ hom_{kH}(P, P^{σ^{−1}}). Notice that 𝔄̂ contains the unit element, is closed under multiplication and the differential, and hence is an augmented DGA. As complexes,

𝔄̂ = ⨁_{σ∈Q} σ ⊗ hom_{kH}(P, P^{σ^{−1}}) ≅ hom_{kH} (⨁_{σ∈Q} σ ⊗ P, P )
= hom_{kH}((P^{G})_{H}, P) ≅ hom_{kG}(P^{G}, P^{G}) = 𝔄.

The isomorphism 𝔄̂ → 𝔄 is given as follows: σ ⊗ f ∈ 𝔄̂ corresponds to the map γ ∈ 𝔄 defined by γ(1 ⊗ x) = σ ⊗ f(x). One may check that this isomorphism preserves the multiplication and commutes with the augmentations, hence 𝔄̂ and 𝔄 are isomorphic as augmented DGAs.

We next show how to define an explicit twisting cochain for the extension. Because H is finite, we can take P to be the minimal resolution. This resolution is characterized by the following property: if X → k is any projective resolution of the trivial kH-module, then there exist kH-chain maps ι: P → X and π: X → P commuting with the augmentations such that πι is the identity on P. In particular, we can take X to be any complex of kG-modules which becomes a kH-projective resolution upon restriction to H (for example, a kG-projective resolution of the trivial module). Fix such a choice of X, ι, and π for the remainder of this section. We then have

Proposition 7.1. There exists a map s ∈ Hom_{kH}(X, X)_{1} such that sι = 0, πs = 0, s^{2} = 0, and ∂s + s∂ = 1 − ιπ.

Proof. Let W = Ker(π), so X = ι(P) ⊕ W as kH-complexes. W is a projective resolution of 0, and is therefore contractible. Hence if we let V_{n} = Ker(∂_{n}) (n ∈ ℤ), then there is a commutative diagram

⋯ ⟶ W_{2} ⟶ W_{1} ⟶ W_{0} ⟶ 0
↓≅  ↓≅  ↓≅
⋯ ⟶ V_{2} ⊕ V_{1} ⟶ V_{1} ⊕ V_{0} ⟶ V_{0} ⊕ 0 ⟶ 0

where the maps along the bottom row are given by (v_{n}, v_{n−1}) ↦ (v_{n−1}, 0) (v_{i} ∈ V_{i}). Define s′ ∈ Hom_{kH}(W, W)_{1} by s′(v_{n}, v_{n−1}) = (0, v_{n}). It is clear that (s′)^{2} = 0 and ∂s′ + s′∂ is the identity on W. Extend s′ to a map s ∈ Hom_{kH}(X, X)_{1} by s(x + y) = s′(y) (x ∈ ι(P), y ∈ W).

Fix a map s as in Proposition 7.1, let ϕ = ∂_{1}ı_{1}, and define t ∈ Hom_{k}(ℭ, 𝔄)_{−1} by letting t_{n} (n ≥ 1) be (−1)^{n} times the composite

ℭ_{n} = Ȳ_{n} ⟶^{Δ̄_{1^{n}}} Ȳ_{1}^{⊗n} ⟶^{ϕ×···×ϕ} kQ^{⊗n} ⟶^{w_{n}} 𝔄̂_{n−1} ≅ 𝔄_{n−1},

where w_{n}(σ_{1} ⊗ σ_{2} ⊗ σ_{3} ⊗ ⋯ ⊗ σ_{n}) = σ_{1}σ_{2}σ_{3} ⋯ σ_{n} ⊗ πσ_{n}^{−1}s ⋯ σ_{3}^{−1}sσ_{2}^{−1}sσ_{1}^{−1}ι.

Theorem 7.2. Let G = H ⋊ Q be a split extension of finite groups, k a field, P → k the minimal kH-projective resolution, Y → k a special kQ-resolution, and define a map t as above. Then t is a twisting cochain associated to the extension.

For proving the theorem, it is useful to have another description of t. Define

ι̂ ∈ Hom_{k}(Ȳ_{0}, hom_{kG}(P^{G}, (X_{H})^{G})_{0})  by  ι̂(1) = ι^{G},
π̂ ∈ Hom_{k}(Ȳ_{0}, hom_{kG}((X_{H})^{G}, P^{G})_{0})  by  π̂(1) = π^{G},
ŝ ∈ Hom_{k}(Ȳ_{0}, hom_{kG}((X_{H})^{G}, (X_{H})^{G})_{1}^{op})  by  ŝ(1) = s^{G},
ϕ̂ ∈ Hom_{k}(Ȳ_{1}, hom_{kG}((X_{H})^{G}, (X_{H})^{G})_{0}^{op})  by  ϕ̂ = β ∘ ϕ,

where β: kQ → hom_{kG}((X_{H})^{G}, (X_{H})^{G})_{0}^{op} is the morphism of DGAs defined by β(σ)(1 ⊗ x) = σ ⊗ σ^{−1}x (σ ∈ Q, x ∈ X). Then we may form cup products using right compositions for multiplications, and we see that

t_{n} = −ι̂∪ (ϕ̂ ∪ ŝ)^{∪(n−1)} ∪ ϕ̂ ∪ π̂, n ≥ 1. (7.1)

Notice that for (7.1) to hold requires not only the coassociativity of Δ̄, but part (iv) of Definition 4.1 as well.

Proof of Theorem 7.2. For notational convenience, we set a = ι̂, b = ϕ̂, c = ŝ, z = π̂, and suppress the cup product symbol. Let x = bc, so

t = − ∑_{n≥0} ax^{n}bz.

We have d(a) = 0 = d(z), as ι and π are chain maps, while Proposition 4.6 implies d(b) = b^{2}, and Proposition 7.1 implies d(c) = 1 − za. Now,

d(x) = d(b)c − bd(c) = bx − b + bza,

whence

∑_{n≥0} d(x^{n}) = ∑_{0≤i≤n−1} x^{i}d(x)x^{n−i−1}

= ∑ x^{i}bx^{n−i} − ∑ x^{i}bx^{n−i−1} + ∑ x^{i}bzax^{n−i−1}

= − ∑_{n≥0} x^{n}b + ∑_{0≤i≤n−1} x^{i}bzax^{n−i−1}.

It follows that

d(t) = − ∑_{n≥0} ad(x^{n})bz − ∑_{n≥0} ax^{n}d(b)z

= ∑_{n≥0} ax^{n}b^{2}z + ∑_{0≤i≤n−1} ax^{i}bzax^{n−i−1}bz − ∑_{n≥0} ax^{n}b^{2}z

= −t^{2}.

Finally, ϵ_{1}t_{1} = −ϵ_{1}w_{1}ϕ = −ϕ, so t is a twisting cochain associated to the group extension.

8. The LHS spectral sequence of a split extension of finite groups

We continue to examine the split extension of finite groups G = H⋊Q. As before, let Y be a special kQ-resolution, P the minimal kH-resolution, X a kG-resolution, and t an arbitrary associated twisting cochain. Let M be a kG-module such that M_{H} is semisimple, i.e. rad_{kH}(M) = 0. Construct the LHS spectral sequence of the extension with coefficients in M from the double complex

E_{0} = Hom_{kQ}(Y, Hom_{kH}(X, M)). (8.1)

In accordance with the standard sign conventions, the differentials d′: E_{0}^{p,q} → E_{0}^{p+1,q} and d′′: E_{0}^{p,q} → E_{0}^{p,q+1} are given by

d′(f)(y) = (−1)^{p+q+1}f(∂(y)) = −∂^{∗}(f)(y)
d′′(f)(y) = (−1)^{q+1}f(y) ∘ ∂ = −∂^{∗}(f(y)),

for f ∈ E_{0}^{p,q} and y ∈ Y_{p}. We have

E_{1} ≅ Hom_{kQ}(Y, H^{∗}(H, M)), (8.2)

and since P is the minimal resolution and M_{H} is semisimple,

H^{∗}(H, M) = Hom_{kH}(P, M) ≅ Hom_{kG}(P^{G}, M).

At the same time we consider the spectral sequence (Ẽ_{r}, d̃_{r}) arising from the bigraded complex A = Hom_{k}(Ȳ , H^{∗}(H, M)) endowed with the differential δ, which is defined as follows: Composition

hom_{kG}(P^{G}, P^{G})^{op} ⊗ Hom_{kG}(P^{G}, M) → Hom_{kG}(P^{G}, M).

gives H^{∗}(H, M) the structure of a left DGM over 𝔄, hence there is a cup product Hom_{k}(ℭ, 𝔄) ⊗ A →^{∪} A (and this makes A a left DGM over Hom_{k}(ℭ, 𝔄)). We let δ = d + t∪, where d is the usual differential on A. As we saw in Section 3, δ^{2} = 0.

Since the differential in H^{∗}(H, M) is 0, we have

δ_{r}(f) = \begin{cases}0 & if r = 0 \\ (−1)^{p+q+1}fd + t_{1} ∪ f & if r = 1 \\ t_{r} ∪ f & if r ≥ 2, \end{cases}

for f ∈ A^{p,q}. For the same reason, we have

(dg) ∪ f = d ∘ (g ∪ f) = 0,

for all g ∈ Hom_{k}(ℭ, 𝔄) and f ∈ A (keep in mind dg = d ∘ g in our notation). Since δ_{0} = 0, Ẽ_{1} = A. Moreover, the map Φ: A → E_{1}, defined by

Φ(f)(ı(ȳ)) = f(ȳ),  ȳ ∈ Ȳ , f ∈ A, (8.3)

is an isomorphism. We can now state our main theorem:

Theorem 8.1. Let G = H ⋊ Q be a split extension of finite groups, k a field, and M a kG-module which is semisimple as a kH-module. Let Y → k be a special kQ-resolution, P → k the minimal kH-resolution, and construct the LHS spectral sequence (E_{r}, d_{r}) as above. Let t be a twisting cochain associated to the extension, and construct the spectral sequence (Ẽ_{r}, d̃_{r}) using t as above. Then the map Φ: Ẽ_{1} → E_{1} induces an isomorphism Φ_{r}: Ẽ_{r} → E_{r} for all r ≥ 1.

To prove the theorem, we first show that the spectral sequence Ẽ is independent of the choice of twisting cochain t. This leaves us free to use the explicit twisting cochain defined in Section 7.

Lemma 8.2. Suppose t̂ is another twisting cochain associated to the group extension, and let (Ê_{r}, d̂_{r}) denote the spectral sequence arising from the bigraded complex A endowed with the differential δ̂ = d + t̂∪. Then d̂_{r} = d̃_{r} for all r ≥ 1.

Proof. By Corollary 3.3, there is a homotopy s: t̂ ≃ t with s_{0} = ηϵ. For r = 1, equation (3.2) yields

d_{1}s_{1} = s_{0} ∪ t̂_{1} − t_{1} ∪ s_{0} = t̂_{1} − t_{1},

for f ∈ A^{p,q}, so that indeed

δ̂_{1}(f) − δ_{1}(f) = (t̂_{1} − t_{1}) ∪ f = d_{1}s_{1} ∪ f = 0.

Now assume r ≥ 2 and d̂_{n} = d̃_{n} for 1 ≤ n < r. Let ζ ∈ Ẽ_{r}^{p,q} and choose f̂ ∈ F^{p}A^{p+q} such that ζ = π^{p,q}(f̂) + B̃_{r}^{p,q} and δ̂(f̂) ∈ F^{p+r}A^{p+q+1}. By definition,

d̂_{r}(ζ) = π^{p+r,q−r+1}δ̂(f̂) + B̃_{r}^{p+r,q−r+1}.

Let f = s ∪ f̂. Since s_{0} = ηϵ, π^{p,q}(f̂) = π^{p,q}(f). Moreover,

δ(f) = d(s ∪ f̂) + t ∪ s ∪ f̂
= d(s) ∪ f̂+ s ∪ d(f̂) + (s ∪ t̂− d(s)) ∪ f̂
= s ∪ (d(f̂) + t̂∪ f̂)
= s ∪ δ̂(f̂).

Hence π^{p+r,q−r+1}δ̂(f̂) = π^{p+r,q−r+1}δ(f), and therefore d̂_{r}(ζ) = d̃_{r}(ζ).

Lemma 8.3. There exist k-linear maps η: E_{1} → E_{0} of bidegree (0, 0), ρ: E_{0} → E_{1} of bidegree (0, 0), and T: E_{0} → E_{0} of bidegree (1, −1) such that

(i) η(E_{1}) ⊆ Z_{1}, ρ|_{Z_{1}} is the natural map, and ρη = 1_{E_{1}}. (ii) d′(f) + d′′T (f) = 0, for all f ∈ E_{0} such that d′(f) ∈ Im(d′′). (iii) δ_{r} = Φ^{−1}ρd′T^{r−1}ηΦ, for all r ≥ 1.

Proof. Let a = ι̂∪, b = ϕ̂∪, c = ŝ∪, and z = π̂∪. For any kQ-module U, let R = ı^{∗}: Hom_{kQ}(Y, U) → Hom_{k}(Ȳ , U) denote precomposition with ı. It follows from Definition 4.1 that R is an isomorphism. Hence we may set η = R^{−1}zR and ρ = R^{−1}aR, and then (i) is easily verified. To define T, begin with the observation that if γ ∈ B Hom_{kH}(X, M), then γ represents the zero element of H^{∗}(H, M), so γι = 0. Therefore

γ = γ(1 − ιπ) = γ(∂s + s∂) = (γs)∂.

(So s^{∗} is a section of the map ∂: Hom_{kH}(X, M) → B Hom_{kH}(X, M).) Hence we let T = −R^{−1}cRd′. Then if d′(f) ∈ Im(d′′) = Hom_{kQ}(Y_{p+1}, B_{−q} Hom_{kH}(X, M)), we have

(d′′T (f))(y) = (−1)^{q}T (f)(y) ∘ ∂ = −d′(f)(y) ∘ s∂ = −d′(f)(y)

for all y ∈ ι(Ȳ), proving (ii).

We now turn to the proof of (iii). By Definition 4.1(vi), we have

−Rd′ = ı^{∗}∂^{∗} = (∂ı)^{∗} = (ı¯∂ + ϕ ∪ ı)^{∗} = ¯∂^{∗}R + bR,

Furthermore, ¯∂^{∗} commutes with a, c, and z, and az = 1. Hence

Φ^{−1}ρd′ηΦ = RR^{−1}a(Rd′)R^{−1}zRR^{−1} = −a(¯∂^{∗} + b)z = −¯∂^{∗} − abz = d + t_{1}∪,

and we have verified (iii) for r = 1. Now, Proposition 7.1 implies ac = c^{2} = cz = 0, so for r ≥ 2,

Φ^{−1}ρd′T^{r−1}ηΦ = aRd′(−R^{−1}cRd′)^{r−1}R^{−1}z
= −a(¯∂^{∗} + b)c(¯∂^{∗} + b) ⋯ c(¯∂^{∗} + b)z
= −a(bc)^{r−1}bz
= t_{r}∪,

which proves (iii).

Proof of Theorem 8.1. The proof is by induction on r. By Lemma 8.3(iii), Φδ_{1} = ρd′ηΦ = d_{1}Φ, so the theorem holds for r = 1.

Let π_{i,j}: Z_{i}/B_{j} → E_{i} denote the natural map (j ≤ i). Fix r ≥ 2, and assume Φ_{s}d̃_{s} = d_{s}Φ_{s} for all 1 ≤ s < r, where Φ_{i}: Ẽ_{i} → E_{i} satisfies Φ_{i}(x̃ + B̃_{i}) = π_{i,1}Φ(x̃) (x̃ ∈ Z̃_{i}, 1 ≤ i ≤ r). We must show Φ_{r}d̃_{r} = d_{r}Φ_{r}.

Let ζ̃ ∈ Ẽ_{r}^{p,q}, ζ = Φ_{r}(ζ̃). By definition, there are x̃_{i} ∈ Ẽ_{0}^{p+i,q−i} (0 ≤ i ≤ r − 2) with x̃_{0} ∈ Z̃_{r}^{p,q}, ζ̃ = x̃_{0} + B̃_{r}^{p,q},

∑_{j=0}^{i} δ_{i−j+1}(x̃_{j}) = 0, 0 ≤ i ≤ r − 2 (8.4)

and

d̃_{r}(ζ̃) = ∑_{i=0}^{r−2} δ_{r−i}(x̃_{i}) + B̃_{r}^{p+r,q−r+1}.

We next produce z_{j} ∈ E_{0}^{p+j,q−j} (0 ≤ j ≤ r − 1) satisfying ζ_{r} = π_{r,0}(z_{0}) and

d′(z_{i}) + d′′(z_{i+1}) = 0, 0 ≤ i ≤ r − 2. (8.5)

It will then follow that d_{r}(ζ) = π_{r,0}(d′(z_{r−1})). To do this, let x_{i} = Φ(x̃_{i}),

z_{i} = ∑_{j=0}^{i} T^{i−j}η(x_{j}), 0 ≤ i ≤ r − 2,

and z_{r−1} = T (z_{r−2}). We prove (8.5) by induction on i. For i = 0, we are to show

d′η(x_{0}) + d′′T η(x_{0}) + d′′η(x_{1}) = 0. (8.6)

But by Lemma 8.3(i), d′′η = 0, and since η(x_{0}) ∈ Z_{r} ⊆ Z_{1}, equation (8.6) follows from Lemma 8.3(ii).

Now suppose 1 ≤ i ≤ r − 3 and d′(z_{i−1}) + d′′(z_{i}) = 0. Then d′′d′(z_{i}) = −d′d′′(z_{i}) = 0, so d′(z_{i}) ∈ Ker(d′′) = Z_{1}. But

ρd′(z_{i}) = ∑_{j=0}^{i} ρd′T^{i−j}η(x_{j}) = ∑_{j=0}^{i} Φδ_{i−j+1}(x̃_{j}) = 0,

by Lemma 8.3(iii) and (8.4). Since ρ|_{Z_{1}} is the natural map, d′(z_{i}) ∈ Im(d′′). Now since

d′′T (z_{i}) = d′′ (∑_{j=0}^{i} T^{i−j+1}η(x_{j})) = d′′ (∑_{j=0}^{i} T^{i−j+1}η(x_{j}) + η(x_{i+1})) = d′′(z_{i+1}),

Lemma 8.3(ii) implies

0 = d′(z_{i}) + d′′T (z_{i}) = d′(z_{i}) + d′′(z_{i+1}),

which completes the inductive step. Finally, d′(z_{r−3}) + d′′(z_{r−2}) = 0, so as above one obtains d′(z_{r−2}) ∈ Im(d′′), and in the same way completes the proof of (8.5).

Finally,

d_{r}(ζ) = π_{r,0}d′(z_{r−1}) = π_{r,0} ∑_{j=0}^{r−2} d′T^{r−j−1}η(x_{j}) = π_{r,1} ∑_{j=0}^{r−2} ρd′T^{r−j−1}η(x_{j})

= π_{r,1}Φ ∑_{j=0}^{r−2} δ_{r−j}(x̃_{j}) = Φ_{r}d̃_{r}(ζ̃),

completing the proof of the theorem.

Corollary 8.4. Let Y be the bar resolution and (E_{r}, d_{r}) the LHS spectral sequence constructed using Y.

(i) There exist maps ν_{n}[σ_{1}| ⋯ |σ_{n}] ∈ hom_{kH}(P, P^{σ_{1}···σ_{n}})_{n−1} (n ≥ 1, σ_{1}, …, σ_{n} ∈ Q) which satisfy ϵν_{1}[σ_{1}] = ϵ and equation (1.2).

(ii) Let (Ẽ_{r}, d̃_{r}) be the spectral sequence of the bigraded complex (A, δ), where A = Hom_{k}(Ȳ , Hom_{kH}(P, M)) and δ is defined by

δ_{r}(f)[σ_{1}| ⋯ |σ_{p+r}] = (−1)^{γ}σ_{1} ⋯ σ_{r} ∘ f[σ_{r+1}| ⋯ |σ_{p+r}] ∘ ν_{r}[σ_{r}^{−1}| ⋯ |σ_{1}^{−1}],

where γ = r(r+1)/2+pr+q, f ∈ A^{p,q} and r ≥ 2. Then the map Φ: Ẽ_{1} → E_{1} induces an isomorphism Φ_{r}: Ẽ_{r} → E_{r} for all r ≥ 1.

Proof. By setting

t[σ_{1}| ⋯ |σ_{n}] = (−1)^{n(n+1)/2}σ_{1} ⋯ σ_{n} ⊗ ν_{n}[σ_{n}^{−1}| ⋯ |σ_{1}^{−1}] ∈ 𝔄̂_{n−1},

it is easily seen that the existence of ν_{n} is equivalent to the existence of a twisting cochain associated to the extension, so (i) follows from Theorem 5.1, and (ii) is just a restatement of Theorem 8.1.

Let us see how Corollary 8.4 implies the result, mentioned in Section 1, that if there is an action of Q on P which commutes with the differentials and the augmentation, and satisfies σ(hx) = (σhσ^{−1})σ(x) (h ∈ H, x ∈ P), then E_{2} = E_{∞}. Indeed, if this is the case, we may set ν_{1}[σ](x) = σ(x). We then have ν_{1}[σ] ∘ ν_{1}[τ] = ν_{1}[στ], so we may take ν_{2} = 0. It follows by induction on n that we may take ν_{n} = 0 for n ≥ 2: suppose ν_{i} = 0 for 2 ≤ i ≤ n − 1; then the right side of equation (1.2) vanishes, and we may therefore set ν_{n} = 0. Hence δ_{n} = 0 for n ≥ 2, and therefore Ẽ_{2} = Ẽ_{∞}, so the same is true in the LHS spectral sequence.

We can also see that if H acts trivially on M then all the differentials into the horizontal edge vanish. To do this we must first make the following observation: throughout we have assumed that P → k is the minimal resolution and rad_{kH}(M) = 0. This hypothesis was needed to ensure that the differentials in Hom_{kH}(P, M) vanish, so that the latter is H^{∗}(H, M), and this was all that was required in the proofs. If H acts trivially on M, we can increase the size of P slightly so that in degree 0 we have kH. To do this, let W be a complement of the projective cover of k in kH. Then the projective cover of k is not a summand of W. Form the exact complex which in degrees 0 and 1 is W (and zero elsewhere) and which has the identity map for the differential. Replace P with the direct sum of P and this complex, and we have a projective resolution of the required form which is kH is degree 0. But now there is an obvious action of Q on P_{0} induced by the action of Q on H, and we use this to define each ν_{1}[σ] on P_{0}. It follows that we can choose all the ν_{2}[σ|τ] to vanish on P_{0}, and then an inductive proof like the one above shows we can take all ν_{n}[σ_{1}| ⋯ |σ_{n}] (n ≥ 2) to vanish on P_{0}. Thus all the δ_{n} (n ≥ 2) which land in the horizontal edge vanish, so the same is true for the d̃_{n}, and therefore the d_{n}.

9. Examples

Now suppose that k is a field of characteristic 2, Q = ⟨b | b^{2} = 1⟩, and G = H⋊Q. Let Y → k be the minimal kQ-resolution, say Y_{n} = kQy_{n}, ∂(y_{n+1}) = (b + 1)y_{n}, and Δ(y_{n}) = ∑_{i=0}^{n} y_{i} ⊗ b^{i}y_{n−i} (n ≥ 0). As we saw in Example 4.5, (Y, ∂, Δ, ı) is a special resolution, where ı: Ȳ → Y is defined by ı(ȳ_{n}) = y_{n}. In this case, the twisting cochain formula becomes

d_{n−1}t_{n} = ∑_{i=1}^{n−1} t_{i} ∪ t_{n−i}, n ≥ 2, (9.1)

as ¯∂ = 0. Thus the calculation of t reduces to finding maps α_{n} ∈ hom_{kH}(P, P^{b^{n}})_{n−1} (n ≥ 1) satisfying

(i) ϵα_{1} = ϵ
(ii) ∂α_{2} + α_{2}∂ = α_{1}α_{1} + 1_{P}
(iii) ∂α_{n} + α_{n}∂ = ∑_{i=1}^{n−1} α_{i}α_{n−i}, for n ≥ 3.

(9.2)

For given such maps, we may define t_{n}(ȳ_{n}) = b^{n} ⊗ α_{n} ∈ 𝔄̂_{n−1}, and then t satisfies (9.1). We now have

A^{p,q} = Hom_{k}(Ȳ_{p}, Hom_{kH}(P_{q}, M)) ≅ Hom_{kH}(P_{q}, M),

and if ζ ∈ A^{p,q} corresponds to f ∈ Hom_{kH}(P_{q}, M) then δ_{r}(ζ) corresponds to

b^{r} ∘ f ∘ α_{r} ∈ Hom_{kH}(P_{q−r+1}, M).

Now suppose that in addition we have M = k, H = ⟨a | a^{2^{m−1}}⟩ for some m > 3, and that bab = a^{d} (0 ≤ d < 2^{m−1}). There are three interesting cases to consider:

(i) d = 2^{m−1} −1, so G is the dihedral group of order 2^{m}, (ii) d = 2^{m−2} −1, in which case G is the semidihedral group of order 2^{m}, and (iii) d = 2^{m−2} + 1, in which case G = M_{m}(2) (in the notation of Gorenstein [7]). Together with the direct product, these represent the four isomorphism classes of groups which are split extensions of a cyclic group of order 2 by a cyclic group of order 2^{m−1} (cf. [7, Corollary 5.4.2]).

In any case, we have (say) P_{n} = kHe_{n} (n ≥ 0), and ∂(e_{n}) = (a + 1)e_{n−1} if n is positive and even, and ∂(e_{n}) = Ne_{n−1}, where N = ∑_{i=1}^{2^{m−1}} a^{i}, if n is odd. Since H^{n}(H, k) = k (n ≥ 0), Q acts trivially on H^{∗}(H, k), and E_{1} = E_{2} = H^{∗}(Q, k) ⊗ H^{∗}(H, k), which is depicted in the left hand side of Figure 1 (ignoring for the

[figure: •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; x_{2}; •; 1; •; 1; •; 1; •; 1; •; 1; x_{1}; •; 1; •; 1; •; 1; •; 1; •; 1; •; 1; y; •; 1; •; 1; •; 1]  •1  •1  •1  •1  •1 •0  •0  •0  •0  •0 •1  •1  •0  •0  •0 •1 v = x_{2}^{2}  •1  •1  •1  •1 •1 u = x_{1}x_{2}  •1  •1  •1  •1 •0  •0  •0  •0  •0 •1 x_{1}  •1  •0  •0  •0 •1  •1 y  •1  •1  •1

Figure 1. Spectral sequence for ℤ/2^{m−1} ⋊ ℤ/2.

moment the arrows). The integers in the figure represent the k-dimension at each point, and the generators x_{1}, y, and x_{2} are chosen in the natural way; for example, x_{2} corresponds to the map f ∈ Hom_{kH}(P_{2}, k), where f(e_{2}) = 1.

Now if we let

α_{1}(e_{n}) = \begin{cases}e_{n} & if n is even \\ ∑_{i=0}^{d−1} a^{i}e_{n} & if n is odd, \end{cases}

an easy calculation shows that α_{1} commutes with the differentials and that ϵα_{1} = ϵ. To find a suitable α_{2}, we first calculate 1_{P} + α_{1}^{2}. Clearly this maps e_{n} to 0 for even n, while for n odd,

α_{1}^{2}(e_{n}) = α_{1} (∑_{i=0}^{d−1} a^{i}e_{n}) = (∑_{i=0}^{d−1} a^{di}) (∑_{j=0}^{d−1} a^{j}) e_{n} = κe_{n},

where κ = 1 in the dihedral case, and κ = N + 1 in the other two cases.

Hence in the dihedral case, we have α_{1}^{2} + 1_{P} = 0, i.e. we have an actual action of Q on P, and as we have seen this implies E_{2} = E_{∞}.

In the other two cases, we may define α_{2} by

e_{4i} ↦ 0, e_{4i+1} ↦ e_{4i+2}, e_{4i+2} ↦ N_{2}e_{4i+3}, e_{4i+3} ↦ 0,

for i ≥ 0, where

N_{2} = N/(a + 1) = ∑_{j=0}^{2^{m−2}−1} a^{2j},

and a quick calculation verifies that (9.2) is satisfied for n = 2. Hence by Theorem 8.1, d_{2}(x) corresponds to f ∘ α_{2} : e_{1} ↦ e_{2} ↦ 1, i.e. d_{2}(x_{2}) = x_{1}y^{2}. Similarly, d_{2}(x_{1}) = 0, since α_{2}(e_{0}) = 0. (This also follows from the fact that all differentials into the horizontal edge vanish in the case of a split extension.) This determines the effect of d_{2} on all of E_{2}, as d_{2} is a derivation. Finally, the positions of the generators of E_{3} reveal that E_{3} = E_{∞} (see Figure 1).

We remark that even though d_{r} = 0 for all r ≥ 3 the same is not true of t_{r}. For α_{1}α_{2} + α_{2}α_{1} is defined by

e_{4i} → 0, e_{4i+1} ↦ ∑_{i=1}^{d−1} a^{i}e_{4i+2}, e_{4i+2} ↦ λe_{4i+3}, e_{4i+3} ↦ 0,

for i ≥ 0, where λ = 0 if G = M_{m}(2), and λ = N if G is semidihedral. Hence α_{3}, and therefore t_{3}, is nonzero.

Acknowledgments

The work here is based on part of my Ph.D. dissertation [11]. I am grateful to my advisor, Jon Alperin, for the advice and encouragement he provided, as well as several others for numerous conversations and suggestions, especially Len Evens, Jon Carlson, Dave Benson, and Burt Totaro. Len Evens brought the Charlap-Vasquez theory to my attention in the first place, and Dave Benson, when presented with an earlier version of this work, suspected equation (1.2) came from a twisting cochain (and launched into a spontaneous twisting cochain lecture), thus sending me in exactly the right direction.

References

[1] J. L. Alperin, Local representation theory, Cambridge studies in advanced mathematics 11, Cambridge University Press, 1986.

[2] 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, 1991.

[3] E. H. Brown, Jr., Twisted tensor products, I, Ann. Math. 69 (1959), 223–246.

[4] L. S. Charlap and A. T. Vasquez, The cohomology of group extensions, Trans. Amer. Math. Soc. 124 (1966), 24–40.

[5] L. S. Charlap and A. T. Vasquez, Characteristic classes for modules over groups. I, Trans. Amer. Math. Soc. 127 (1969), 533–549.

[6] L. Evens, Cohomology of groups, Oxford University Press, 1991.

[7] D. Gorenstein, Finite groups, Chelsea Publishing Company, 1980.

[8] J. Huebschmann, Perturbation theory and free resolutions for nilpotent groups of class 2, J. Algebra 126

[9] J. P. May, Simplicial objects in algebraic topology, University of Chicago Press, 1982.

[10] 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 and App. Alg., to appear.

[11] S. F. Siegel, Cohomology and group extensions, Ph.D. thesis, University of Chicago (1993).

[12] C. T. C. Wall, Resolutions for extensions of groups, Math. Proc. Camb. Phil. Soc. 57 (1961), 251–255.

Department of Mathematics, Northwestern University, Evanston IL 60208-2730

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