Homology, Homotopy and Applications, vol.7(1), 2005, pp.87–108

MORE ABOUT THE (CO)HOMOLOGY OF GROUPS AND ASSOCIATIVE ALGEBRAS

HVEDRI INASSARIDZE

(communicated by Antonio Cegarra)

Abstract It is proved that the homology and cohomology theories of groups and associative algebras are non-abelian derived functors of the cokernel and kernel groups of higher dimensions of their defining standard chain and cochain complexes respectively. The same results are also obtained for the relative (co)homology of groups, the mod q cohomology of groups and the cohomology of groups with operators. This allowed us to give an alternative approach to higher Hopf formulas for integral homology of groups. An axiomatic characterization of the relative cohomology of groups is given and higher relative (n + 1)-th cohomology of groups is described in terms of n-fold extensions.

Introduction

The present work deals with the investigation of the (co)homology of groups and associative algebras with respect to the first non-abelian variable. It is clear that this case is significantly more difficult to study than the case of the second abelian variable and fits into the theory of non-abelian homological algebra.

One of the fundamental theorems of non-abelian homological algebra says that homology and cohomology of groups and associative algebras can be expressed as non-abelian derived functors of the group Diff of differentials and the group Der of derivations respectively. That means by using projective resolutions with respect to the non-abelian variable and then by taking the functors Diff and Der with or into an appropriate module [1-4].

The groups Diff and Der are in fact the first cokernel and kernel groups respectively in the standard chain and cochain complexes determining the corresponding homology and cohomology of groups and associative algebras. Our aim is to develop the observation that the non-abelian derived functors of the groups of cokernel and kernel of higher dimensions of the above mentioned chain and cochain complexes

The work was partially supported by INTAS grant No 00-566, FNRS grant No 7GEPJO65513.01, NATO linkage grant PST.CLG.979167 and INTAS grant No 03-51-3251. Received June 3, 2004, revised May 15, 2005; published on June 7, 2005. 2000 Mathematics Subject Classification: 18G10, 18G25,18G50, 18G55, 18G60. Key words and phrases: Cotriple derived functor, derivation, relative cohomology, crossed module. \begin{matrix}c \\ ◯ \end{matrix} 2005, Hvedri Inassaridze. Permission to copy for private use granted.

respectively could be also used to describe the (co)homology of groups and associative algebras. In this context we shall consider the classical Eilenberg-MacLane (co)homology of groups (Theorems 1 and 3), the Hochschild (co)homology of associative algebras (Theorem 16), the relative cohomology of groups (Theorem 8), the mod q cohomology of groups defined in [9,10 ] (Theorem 6) and the cohomology of groups with operators defined in [7] (Theorem 15). This approach to cotriple (co)homology theory is particularly needed to the investigation of the mod q cohomology theory of groups (see [10]). Moreover these results allow us to give an alternative purely group theoretic description of the integral homology of groups (Theorem 4). By enlarging the category of coefficients we obtain an axiomatic characterization of the relative cohomology of groups (Theorem 11) and its description by n-fold extensions in this wider class of coefficients (Theorem 14).

1. Homology and cohomology of groups

Let G be a group and A a G-module. Let us consider the following standard cochain complex defining the cohomology of groups H^{∗}(G, A):

0 → C^{0}(G, A) → C^{1}(G, A) → C^{2}(G, A) → ⋯ → C^{n}(G, A) → ⋯,  (1)

where C^{0}(G, A) = A, C^{n}(G, A) being the group of maps f : G^{n} → A, n > 0 , with coboundary operator δ^{n} : C^{n}(G, A) → C^{n+1}(G, A) given by

δ^{n}(f)(x_{1}, …, x_{n+1}) = x_{1}f(x_{2}, …, x_{n+1}) + ∑(−1)^{i}f(x_{1}, …, x_{i}x_{i+1}, …, x_{n+1})+

(−1)^{n+1}f(x_{1}, …, x_{n}) , for n > 0 and δ^{0}(a) = a − ax , x ∈ G .

Denote Z^{k}(G, A) = Ker δ^{k} and B^{k}(G, A) = Im δ^{k} for k ⩾ 0. Clearly Z^{0}(G, A) = H^{0}(G, A) and Z^{1}(G, A) = Der(G, A). Then for any short exact sequence of Gmodules

0 → A_{1} → A → A_{2} → 0

one has the following commutative diagram with exact rows and columns

0  0  0
↓  ↓  ↓
B^{k}(G, A_{1}) → B^{k}(G, A) → B^{k}(G, A_{2}) →  0
↓ ↓ ↓
0  → Z^{k}(G, A_{1}) → Z^{k}(G, A) → Z^{k}(G, A_{2})
↓ ↓ ↓ τ_{k}
→ H^{k−1}(G, A_{2}) → H^{k}(G, A_{1}) → H^{k}(G, A) → H^{k}(G, A_{2}) →^{δ^{k}} H^{k+1}(G, A_{1}) →
↓  ↓  ↓
0  0  0

This diagram induces a long exact cohomology sequence

0 → Z^{k}(G, A_{1}) → Z^{k}(G, A) → Z^{k}(G, A_{2}) → H^{k+1}(G, A_{1}) → · · · .  (2)

Let G_{A} be the category of groups acting on an abelian group A and F : G_{A} → G_{A} be the endofunctor which sends an object G of G_{A} to the free group F (G) on the underlying set of G, where the action of F (G) on A is defined by^{|g_{1}|···|g_{i}|}a = ^{g_{1}···g_{i}}a, g_{i} ∈ G, a ∈ A. Define natural morphisms τ_{G} : F (G) → G and δ_{G} : F (G) → F^{2}(G) = F (F (G)) of the category G_{A} by τ_{G}(|g|) = g and δ_{G}(|g|) = ||g||, g ∈ G. One gets a cotriple F = (F, τ, δ) on the category G_{A}. Denote F_{n}(G) = F^{n+1}(G) = F (F^{n}(G)) for n ⩾ 0. Therefore for any object G of G_{A} one has the following free simplicial group F_{∗}(G) = {F_{n}(G), ∂_{i}^{n}, s_{i}^{n}, n ⩾ 0} acting on A , where ∂_{i}^{n} = F_{i}τF_{n−i} and s_{i}^{n} = F_{i}δF_{n−i}.

Let T : G_{A} → 𝔄b be a contravariant functor to the category 𝔄b of abelian groups. Then the n-th homology group of the abelian cosimplicial group T F_{∗}(G) is called the n-th right derived functor R_{F}^{n}T of T with respect to the cotriple F. Let P be the projective class induced by F, that means an object X of G_{A} belongs to P if there is a morphism σ : X → F (X) such that τσ = 1_{X}. Then the right derived functors R_{P}^{n}T of T with respect to the projective class P are isomorphic to the cotriple derived functors R_{F}^{n}T (see [23,15]).

It is well known that the cotriple derived functors of the contravariant functor of derivations Der(−, A) : G_{A} → 𝔄b are isomorphic, up to dimension shift, to the cohomology groups with coefficients in A [1-4].

Theorem 1. There is a natural isomorphism

R_{F}^{n}Z^{k}(G, A) ≈ H^{n+k}(G, A) for all n ⩾ 1 and k ⩾ 1 .

Proof. The exact sequence (1) implies the exactness of the sequence

0 → Z^{k}(G, A_{1}) → Z^{k}(G, A) → Z^{k}(G, A_{2}) → 0 ,  (3)

if G is a free group. One deduces that any short exact sequence 0 → A_{1} → A → A_{2} → 0 of G-modules yields a long exact sequence of derived functors

0 → R_{F}^{0}Z^{k}(G, A_{1}) → R_{F}^{0}Z^{k}(G, A) → R_{F}^{0}Z^{k}(G, A_{2}) → R_{F}^{1}Z^{k}(G, A_{1}) →

R_{F}^{1}Z^{k}(G, A) → R_{F}^{1}Z^{k}(G, A_{2}) → R_{F}^{2}Z^{k}(G, A_{1}) → · · · . (4)

Now it will be shown that one has a natural isomorphism R_{F}^{0}Z^{k}(−, A) ≈ Z^{k}(−, A), k > 0.

Let F_{∗}(G) → G be the free cotriple resolution of G. Since it is left (right) contractible as augmented simplicial set (for the categorical definition of left contractibility (see [22,15]), it follows that the augmented abelian simplicial group (C^{i}(F_{∗}(G), A) → C^{i}(G, A)) is left contractible in the category of abelian groups for all i ⩾ 0 and therefore the groups H_{n}(C^{i}(F_{∗}(G), A)) are trivial for n > 0 and H_{0}(C^{i}(F_{∗}(G), A)) ≈ C^{i}(G, A) for all i ⩾ 0.

The diagram

0 → Z^{k}(F (G), A) → C^{k}(F (G), A) → Im δ_{F (G)}^{k} → C^{k+1}(F (G), A)
↓↓ ↓↓ ↓↓ ↓↓
0 → Z^{k}(F^{2}(G), A) → C^{k}(F^{2}(G), A) → Im δ_{F^{2}(G)}^{k} → C^{k+1}(F^{2}(G), A)

implies an exact sequence

R_{F}^{0}Z^{k}(G, A) → R_{F}^{0}C^{k}(G, A) → R_{F}^{0} Im δ_{G}^{k} → R_{F}^{1}Z^{k}(G, A) ,  (5)

where R_{F}^{0}C^{k}(G, A) ≈ C^{k}(G, A) and one has an injection R_{F}^{0} Im δ_{G}^{k} → C^{k+1}(G, A). It follows that R_{F}^{0}Z^{k}(G, A) ≈ Ker δ_{G}^{k}.

Therefore in the exact sequence (3) the zero-derived functors could be replaced by the functors Z^{k}(G, −).

Further it will be shown that R_{F}^{n}Z^{k}(G, A) = 0 for n, k > 0, if A is a coinduced G-module. In effect, to the free cotriple resolution F_{∗}(G) → G of the group G apply the standard cochain complex {C^{∗}(G, A), δ^{n}} to get the following cochain complex of abelian cosimplicial groups:

0 → C^{0}(F_{∗}(G), A) \underrightarrow{δ̃_{∗}^{0}} C^{1}(F_{∗}(G), A) \underrightarrow{δ̃_{∗}^{1}} C^{2}(F_{∗}(G), A) \underrightarrow{δ̃_{∗}^{2}}

C^{3}(F_{∗}(G), A) \underrightarrow{δ̃_{∗}^{3}} ⋯ \underrightarrow{δ̃_{∗}^{k−1}} C^{k}(F_{∗}(G), A) \underrightarrow{δ̃_{∗}^{k}} · · · . (6)

Since each F_{i}(G), i ⩾ 0, is a free group, the sequence (6) is exact except at the terms C^{0} and C^{1} for any G-module A.

The short exact sequence of abelian cosimplicial groups

0 → Der(F_{∗}(G), A) → C^{1}(F_{∗}(G), A) → Im(C^{1}F_{∗}(δ^{1})) → 0  (7)

yields the isomorphism H_{n}(Im δ̃_{∗}^{1}) ≈ H_{n+1}(Der(F_{∗}(G), A) for n > 0 and therefore, by the exactness of (6), the isomorphism H_{n}(Ker δ̃_{∗}^{2}) ≈ H_{n+1}(Der(F_{∗}(G), A) for n > 0.

Consider now the short exact sequence of abelian cosimplicial groups

0 → Ker(δ̃_{∗}^{2}) → C^{2}(F_{∗}(G), A) → Im(δ̃_{∗}^{2}) → 0  (8)

that gets the isomorphism H_{n}(Im δ̃_{∗}^{2}) ≈ H_{n+1}(Ker δ̃_{∗}^{2}) for n > 0, implying the isomorphism H_{n}(Ker(δ̃_{∗}^{3}) ≈ H_{n+2}(Der(F_{∗}(G), A)) for n > 0 and any G-module A. Continuing in this way, by using the exact sequence (6) and the triviality of the homology groups H_{n}(C^{k}(F_{∗}(G), A)), n > 0, one arrives to the isomorphism H_{n}(Ker(C^{k}(F_{∗}(G), A)) ≈ H_{n+k−1}(Der(F_{∗}(G), A)) for k > 1, n > 0, and any Gmodule A. Suppose A is a coinduced G-module. In this case it is well known that the groups H_{n}(Der(F_{∗}(G), A) are trivial for n > 0. Thus we deduce the required triviality of the derived functors R_{F}^{n}Z^{k}(G, A) for k, n > 0, when A is a coinduced G-module.

The rest of the proof is standard by using the exact sequences (2) and (4) and the fact that both functors H^{n}(−, A) and R_{F}^{n}Z^{k}(−, A) vanish for k, n > 0, if A is a coinduced G-module.

Let

C_{∗}(G) = ⋯ → C_{3}(G) → C_{2}(G) → C_{1}(G) → C_{0}(G) → ℤ → 0  (9)

be the standard bar resolution of ℤ and C_{∗}(G, A) = C_{∗}(G) ⊗_{G} A the defining standard complex for the homology of groups with coefficients in the G-module A. Denote Y_{k}(G) = Coker ∂_{k+1} and Y_{k}(G, A) = Y_{k}(G) ⊗_{G} A for k ⩾ 1, where ∂_{k} : C_{k+1}(G) → C_{k}(G), k ⩾ 0, and ε : C_{0}(G) → ℤ are homomorphisms of the bar resolution (9). Clearly Y_{k}(G, A) ≈ Coker(∂_{k+1} ⊗_{G} A), Y_{0}(G, A) = H_{0}(G, A) and Y_{1}(G, A) = I(G) ⊗_{G} A. It is well known [2] that the homology group functors H_{n}(−, A), n > 1, are isomorphic, up to dimension shift, to the non-abelian left derived functors of the functor I(−) ⊗_{G} A with respect to the free cotriple F for n > 1.

Proposition 2. There is an isomorphism

Hom_{G}(Y_{k}(G), A) ≈ Z^{k}(G, A) for k ⩾ 1 .

Proof. For k = 1 this isomorphism is well known. Take the short exact sequence

0 → Im ∂_{k+1} → C_{k}(G) → Y_{k}(G) → 0

induced by the sequence (9) for k ⩾ 1. One gets an exact sequence

0 → Hom_{G}(Y_{k}(G), A) → Hom_{G}(C_{k}(G), A) →

Hom_{G}(Im ∂_{k+1}, A) → Ext_{G}^{1}(Y_{k}(G), A) → 0 .

It follows that

Ker(Hom_{G}(C_{k}(G), A) → Hom_{G}(Im ∂_{k+1}, A)) = Z^{k}(G, A),

implying the required isomorphism for k ⩾ 1.

Theorem 3. There is a natural isomorphism

L_{n}^{F}Y_{k}(G, A) ≈ H_{n+k}(G, A) for all n ⩾ 1 and k ⩾ 1 .

Proof. Completely similar to the proof of Theorem 1 and will be omitted.

Now an application of Theorem 3 will be given in connection with the purely group theoretic interpretation of the integral homology of groups. Such description in low dimensions is well known, since one has H_{1}(G) ≈ G/[G, G] and H_{2}(G) ≈ (R ∩ [F, F])/[F, R] (Hopf formula), where R is the kernel of a free presentation F → G of the group G.

Different approaches to this problem for higher integral homology of groups have been given in [6,11,12,20,21] by constructing higher Hopf type formulas. By using Theorem 3 we provide an alternative group theoretic interpretation of the integral homology H_{∗}(G) of a group G.

Let us consider the natural surjective homomorphisms α_{n+1} : C_{n+1}(G) ⊗_{G} ℤ → Y_{n+1}(G, ℤ) induced by ∂_{n+1}, n ⩾ 0. Clearly C_{n+1}(G) ⊗_{G} ℤ is isomorphic to Σ_{α∈G^{n+1}}ℤ_{α}, ℤ_{α} ≈ ℤ, and the abelianization of F (G^{n+1}) is isomorphic to Σ_{α∈G^{n+1}}ℤ_{α}. Thus one gets functorial surjective homomorphisms

F (G^{n+1}) →^{β_{n+1}} C_{n+1}(G) ⊗_{G} ℤ →^{α_{n+1}} Y_{n+1}(G, ℤ), n ⩾ 0,

where β_{n+1} is the composite of the homomorphisms F (G^{n+1}) → F (G^{n+1})^{ab}, F (G^{n+1})^{ab} ≈ Σℤ_{α} and Σℤ_{α} ≈ C_{n+1}(G) ⊗_{G} ℤ.

Finally we obtain a functorial surjective homomorphism

γ_{n+1}(G) = α_{n+1}β_{n+1} : F (G^{n+1}) → Y_{n+1}(G, ℤ), n ⩾ 0,

and denote U_{n+1}(G) = Kerγ_{n+1}(G) for all n ⩾ 0. Therefore U_{k}(G) = [F (G^{k}), F (G^{k})] for k ⩾ 1. The functors U_{k}, k ⩾ 1, will be used to provide higher Hopf type formulas.

Let

F_{∗, G} ⟶^{τ} G = ⋯ ⟶ ; ⋮ ; ⟶ F_{2} ⟶^{∂_{0}^{2}} ; ⟶ ; ⟶_{∂_{2}^{2}} F_{1} ⟶^{∂_{0}^{1}} ; ⟶_{∂_{1}^{1}} F_{0} ⟶^{τ} G

be a P-projective resolution of G, that means this augmented pseudosimplicial group is simplicially exact and the groups F_{i}, i ⩾ 0, are free groups. The projective class P is induced by the free cotriple in the category G of groups. Thus P is consisting of free groups and a morphism of G is a P-epimorphism if and only if it is an epimorphism. It is known that the free cotriple resolution of the group G is a Pprojective resolution of G and any free presentation τ : F → G of G induces a well defined P-projective resolution P_{∗}→^{τ}G of G by using simplicial kernels [12] and constructed as follows:

⋯ ⟶^{l_{0}^{3}} ; ⋮ ; ⟶_{l_{3}^{3}} F (L_{2}G) ⟶^{τ_{L_{2}G}} L_{2}G ⟶^{l_{0}^{2}} ; ⟶ ; ⟶_{l_{2}^{2}} F (L_{1}G) ⟶^{τ_{L_{1}G}} L_{1}G ⟶^{l_{0}^{1}} ; ⟶_{l_{1}^{1}} F ⟶^{τ} G ,

where (L_{1}G, l_{0}^{1}, l_{1}^{1}) is the simplicial kernel of the morphism τ, (L_{2}G, l_{0}^{2}, l_{1}^{2}, l_{2}^{2}) is the kernel of the pair of morphisms (l_{0}^{1}τ_{L_{1}G}, l_{1}^{1}τ_{L_{1}G}) and if (L_{n}G, l_{0}^{n}, ..., l_{n}^{n}) has been constructed, then (L_{n+1}G, l_{0}^{n+1}, ..., l_{n+1}^{n+1}) is the simplicial kernel of the sequence of morphisms (l_{0}^{n}τ_{L_{n}G}, ..., l_{n}^{n}τ_{L_{n}G}). Denote P_{0} = F, P_{n} = F (L_{n}G) and ∂_{i}^{n} = l_{i}^{n}τ_{L_{n}G} for n > 0.

Theorem 4. For any P-projective resolution (F_{∗,G} →^{τ} G, ∂_{i}^{n}, s_{i}^{n}) of the group G there are isomorphisms

H_{k+1}(G) ≈ (R_{k} ∩ U_{k}(F_{0}))/U_{k}(∂_{1}^{1})(Ker U_{k}(∂_{0}^{1})), k ⩾ 1,

where R_{k} = Ker F (τ_{k}) and τ_{k} : F_{0}^{k} → G^{k} is induced by τ, k ⩾ 1.

Proof. By applying the functors U_{k}, k ⩾ 1, one gets a short exact sequence of augmented pseudosimplicial (simplicial) groups

1 → (U_{k}(F_{∗,G}) →^{U_{k}(τ)} U_{k}(G)) → (F (F_{∗,G}^{k}) →^{F (τ^{k})} F (G^{k})) →

(Y_{k}(F_{∗,G}, ℤ)) →^{Y_{k}(τ,ℤ)} Y_{k}(G, ℤ)) → 1,

for k ⩾ 1 which induces a long exact homotopy sequence. It is easily checked that the augmented pseudosimplicial group F_{∗,G} →^{τ} G is left contractible as an augmented

pseudosimplicial set. Thus the augmented pseudosimplicial group F_{∗,G}^{k} →^{τ^{k}} G^{k} is also left contractible in the category of sets implying the left contractibility of

F (F_{∗,G}^{k}) →^{F (τ^{k})} F (G^{k}) in the category of groups. It follows that π_{n}(F (F_{∗,G}^{k})) = 0 for n > 0 and π_{0}(F (F_{∗,G}^{k})) is naturally isomorphic to F (G^{k}). Therefore we have the following exact sequence

0 → π_{1}Y_{k}(F_{∗,G}, ℤ) → π_{0}U_{k}(F_{∗,G}) → π_{0}(F (F_{∗,G}^{k}))

for k ⩾ 1. By Theorem 3 one has an isomorphism π_{1}Y_{k}(F_{∗,G}, ℤ) ≈ H_{k+1}(G), k ⩾ 1.

Finally we obtain an exact sequence

0 → H_{k+1}(G) → π_{o}U_{k}(F_{∗,G}) → F (G^{k})

which gives the desired result.

Remark 5. The group (R_{k} ∩ U_{k}(F_{0}))/U_{k}(∂_{1}^{1})(Ker U_{k}(∂_{0}^{1})), k ⩾ 1, does not depend of the projective resolution of the group G.

For k = 1, 2 instead of the functors F (G) and F (G^{2}) we can take the identity functor 1_{G} and the kernel R_{G} of the canonical map γ : F (G) → G respectively. Denote U1^{′}(G) = [G, G] and U2^{′}(G) = Kerν, where ν : [F (G), F (G)] → [G, G] is induced by γ. Then there are isomorphisms

H_{k+1}(G) ≈ (R_{k} ∩ Uk^{′}(F_{0}))/Uk^{′}(∂_{1}^{1})(Ker Uk^{′}(∂_{0}^{1})), k = 1, 2,

where R_{1} is the kernel of τ and R_{2} is the kernel of F (τ). It can be proved that (R_{1} ∩ U1^{′}(F_{0}))/U1^{′}(∂_{1}^{1})(Ker U1^{′}(∂_{0}^{1})) is the Hopf formula by showing the equality

U1^{′}(∂_{1}^{1})(Ker U1^{′}(∂_{0}^{1})) = [F_{0}, R_{1}]

(see [16]). The sequence 0 → U1^{′}(G) → G → Y_{1}(G, ℤ) → 0 is exact for any group G and the sequence 0 → U2^{′}(F) → R_{G} → Y_{2}(F, ℤ) → 0 is exact for any free group F, where the homomorphism R_{G} → Y_{2}(F, ℤ) is defined as follows. For any group G one has a surjective homomorphism β1^{′} : R_{G} → Ker α_{1} induced by β_{1} and a surjective homomorphism ϑ : Y_{2}(G, ℤ) → Ker α_{1} which is an isomorphism if G is a free group. Then the required homomorphism R_{G} → Y_{2}(F, ℤ) is equal to the composite ϑ^{−1}β1^{′}. Using these short exact sequences the proof of the afore given isomorphisms for k = 1, 2 is completely similar to the proof of Theorem 4.

We shall not consider in this setting the case of Lie algebras, but it should be noted that for the (co)homology of Lie algebras results similar to Theorems 2 and 3 with respect to the standard Chevalley- Eilenberg complex can be proved and by the same way higher Hopf type formulas can be obtained for the homology H_{∗}(L, k) of a Lie algebra L over k.

2. Mod q cohomology of groups

This case has been already investigated in [10], Section 2. However, in this context it is an important example showing the necessity to use the non-abelian derived functors of higher dimensional kernels of the defining cochain complex to express the corresponding cohomology groups as cotriple cohomology. That is the reason why a short expository will be given.

The mod q cohomology H^{n}(G, A, ℤ_{q}), n ⩾ 0, of a group G with coefficients in a G-module A is defined by H^{n}(G, A, ℤ_{q}) = Ext_{G}^{n}(ℤ_{q}, A) with Z_{q} = ℤ/q as a trivial G-module [9,10].

The standard cochain complex for the mod q cohomology of groups is given by the cochain complex (Hom_{G}(q − C_{∗}, A), δ̃^{∗}), where C_{∗} → ℤ is the standard bar resolution of the trivial G-module ℤ and

q − C_{∗} → ℤ_{q} ≡ ⋯ ⟶ C_{n+1} ⊕ C_{n} ⟶^{{ \atop ∂_{n+1}}} C_{n} ⊕ C_{n−1} \underrightarrow{˜∂_{n}} C_{n−1} ⊕ C_{n−2} ⟶^{{ \atop ∂_{n−1}}} ⋯

\underrightarrow{˜∂_{2}} C_{1} ⊕ C_{0} \underrightarrow{˜∂_{1}} C_{0} \underrightarrow{ϵ̃} ℤ_{q} ⟶ 0 ,

is a free ℤ[G]-resolution of the trivial G-module ℤ_{q} with

˜∂_{n}(x_{n}, x_{n−1}) = (∂_{n}(x_{n}) + qx_{n−1}, −∂_{n−1}(x_{n−1}))

for n ⩾ 2, ˜∂_{1}(x_{1}), x_{0}) = ∂_{1}(x_{1}) + qx_{0} and ϵ̃(x_{0}) = [ϵ(x_{0})].

For mod q cohomology of groups one has the following universal coefficient formula:

0 → H^{n−1}(G, A) ⊗ ℤ_{q} → H^{n}(G, A, ℤ_{q}) → T or(H^{n}(G, A), ℤ_{q}) → 0

for all n ⩾ 0, where H^{−1}(G, A) = 0.

The cotriple derived functor R_{F}^{n}q − Z^{1}(−, A) of the functor q − Z^{1}(−, A) of qderivations is not isomorphic to the mod q group cohomology functor H^{n+1}(−, A, ℤ_{q}) for some n ⩾ 1. In effect, if F is a free group acting on A, then R_{F}^{1}q −Z^{1}(F, A) = 0, while the group H^{2}(F, A, ℤ_{q}) is isomorphic to H^{1}(F, A) ⊗ ℤ_{q}. But the following assertion holds [10].

Theorem 6. Let G be a group, A be a G-module and q a positive integer. Then there are natural isomorphisms

R_{F}^{0}(q − Z^{k})(G, A) ≈ q − Z^{k}(G, A) ,
R_{F}^{n}(q − Z^{k})(G, A) ≈ H^{n+k}(G, A, ℤ_{q}) for all k > 1 , n > 0 .

Proof. A sketch will be given. Let us denote the groups Ker δ̃^{k} and Im δ̃^{k} by q − Z^{k}(G, A) and q − B^{k+1}(G, A) respectively for k ⩾ 0. Clearly, q − Z^{0}(G, A) = q − H^{0}(G, A) and Ker δ̃^{1} is the group of q-derivations from G to A. It is shown that any short exact sequence of G-modules induces a long exact mod q cohomology sequence and a long exact sequence of cotriple derived functors R_{F}^{n}(q−Z^{k})(−, A) for k > 1. By using the q-divisibility of injective G-modules it is proved that R_{F}^{n}(q−Z^{k})(G, A) = 0 for k ⩾ 1 and n > 0, if A is an injective G-module. The rest of the proof is standard.

By Theorem 6 the mod q cohomology groups H^{3}(G, A, ℤ_{q}), H^{4}(G, A, ℤ_{q}), …, H^{2+n}(G, A, ℤ_{q}), … can be expressed in terms of the cotriple derived functors R_{F}^{n}(q − Z^{2})(−, A) of the contravariant functor q − Z^{2}(−, A).

3. Relative (co)homology of groups

Let f : G → G′ be a group homomorphism. It will be said that an abelian group A is an f-module, if the groups G and G′ act on A and ga = f(g)a for any g ∈ G and a ∈ A. If A is an f-module with f surjective, by definition the homology groups of the standard cochain complex C^{∗}(f, A) = C^{∗}(G, A)/C^{∗}(G′, A) are called the relative cohomology groups H^{n}(f, A), n ⩾ 0, of the surjective homomorphism f with coefficients in the f-module A. Similarly, the homology groups of the standard chain complex C_{∗}(f, A) = Ker(f_{∗} : C_{∗}(G, A) → C_{∗}(G′, A)) are called the relative homology groups H_{n}(f, G), n ⩾ 0, of the surjective homomorphism f with coefficients in the f-module A.

There is another cocycle description of this defining cochain complex for the relative cohomology of groups. Let (L, l_{0}, l_{1}) be the simplicial kernel of f. Then it is easily checked that the subcomplex of C^{∗}(L, A), consisting of all cochains ψ such that there exists a cochain η ∈ C^{∗}(G, A) satisfying the equality l_{0}η − l_{1}η = ψ, is isomorphic to C^{∗}(f, A).

Denote by Z^{k}(f, A) the kernel of the differential δ̃^{k} of the cochain complex C^{∗}(f, A) for k ⩾ 0.

Definition 7. The group Der(f, A) consisting of all crossed homomorphisms f ∈ Z^{1}(L, A) such that f(x, x) = 0, x ∈ G, is called the group of derivations of the surjective homomorphism f : G → G′.

It is easily checked that Der(f, A) is naturally isomorphic to Z^{1}(f, A).

Let MG_{A} be the category whose objects are surjective homomorphisms (epimorphisms) of groups acting on a fixed abelian group A and morphisms are couples (α, α′) : f → f′ of group homomorphisms such that f′α = α′f. The free cotriple F = (F, δ, τ) defined in the category G_{A} induces in a natural way a cotriple MF given by MF (f) = F (f) and MF (α, α′) = (F (α), F (α′)) for any object f and the natural transformations of this cotriple are induced by the natural transformations of the free cotriple F.

Theorem 8. There is a natural isomorphism

R_{MF}^{n}Z^{k}(f, A) ≈ H^{n+k}(f, A)

for all n ⩾ 1 and k ⩾ 1.

Proof. The long exact relative cohomology sequence yields a short exact sequence

0 → Z^{k}(G′, A) → Z^{k}(G, A) → Z^{k}(f, A) → 0

for any k ⩾ 1 and f : G → G′ with G′ a free group, since in this case H^{n}(G′, A) = 0, n ⩾ 2. Therefore, for any object f of the category MF_{A} one has a long exact sequence of non-abelian right derived functors

0 → R_{F}^{0}Z^{k}(G′, A) → R_{F}^{0}Z^{k}(G, A) → R_{MF}^{0}Z^{k}(f, A) →

R_{F}^{1}Z^{k}(G′, A) → ⋯ → R_{MF}^{n−1}Z^{k}(f, A) → R_{F}^{n}Z^{k}(G′, A) →

R_{F}^{n}Z^{k}(G, A) → R_{MF}^{n}Z^{k}(f, A) → R_{F}^{n+1}Z^{k}(G′, A) → ⋯ (10)

where the zero-derived functors could be replaced by the corresponding functors Z^{k}(−, A).

Let C_{k}^{∗}(G, A) be the cochain complex defined by C_{k}^{n}(G, A) = C^{n+k}(G, A), n ⩾ 0, k ⩾ 1. It is clear that H_{0}(C_{k}^{∗}(G, A)) = Z^{k}(G, A) and if G is a free group, the homology groups H_{n}(C_{k}^{∗}(G, A)) are trivial for n ⩾ 1. This definition could be extended to any object f of the category MF_{A} by C_{k}^{∗}(f, A) = C_{k}^{∗}(G, A)/C_{k}^{∗}(G′, A) with f : G → G′. Take now the free cotriple resolution F_{∗}(f) of f with F_{i}(f) = F_{i}(G) → F_{i}(G′), i ⩾ 0. Then the following equalities hold:

H_{n}(C_{k}^{∗}(F_{i}(f), A) = 0, n > 0, and H_{0}(C^{∗}(F_{i}(f), A) = Z^{k}(F_{i}(f), A) for all i ⩾ 0, since the groups F_{i}(G) and F_{i}(G′) are free groups; H_{n}(C_{k}^{j}(F_{∗}(f), A) = 0, n > 0, and H_{0}(C_{k}^{j}(F_{∗}(f), A) = C_{k}^{j}(f, A) for all j ⩾ 0.

Therefore, both homology groups H_{n}(Z^{k}(F_{∗}(f), A)) and H_{n}(C_{k}^{∗}(f, A)) are isomorphic for all n, k > 0 to the n-th homology group of the total cochain complex T ot(C_{k}^{∗}(F_{∗}(f), A)) of the bicomplex C_{k}^{∗}(F_{∗}(f), A), k ⩾ 1, and one gets the required isomorphism.

In order to obtain an axiomatic description of the relative cohomology of groups the category of coefficients will be enlarged giving an extension of the classical relative cohomology theory.

Let f : G → G′ be an epimorphism of groups. Denote C_{f} the category whose objects are injections φ : A → B such that A is a G′-module, B is a G-module and φ is a morphism of G-modules induced by f. A morphism (α, β) : (φ : A → B) → (φ′ : A′ → B′) is a couple of homomorphisms α : A → A′, β : B → B′ preserving the actions and the diagram

A →^{α} A′
φ ↓  ↓ ^{φ′}
B →^{β} B′

is commutative. The homomorphism f induces a morphism f^{∗} : C^{∗}(G′, A) → C^{∗}(G, B) of cochain complexes given by f^{i}(c^{i}) = φc^{i}f, i > 0, c^{i} ∈ C^{i}(G′, A) and f^{0}(c^{0}) = φ(c^{0}). It is clear that the homomorphism f^{i} : C^{i}(G′, A) → C^{i}(G, B) is injective for all i ⩾ 0.

Define the cochain complex C^{∗}(f, φ) to be the quotient cochain complex Coker f^{∗} and its homology H_{∗}(Coker f^{∗}) gives the cohomology H^{∗}(f, φ) of the epimorphism f of groups with coefficients in φ. From this definition immediately follows a long exact cohomology sequence

0 → H^{0}(G′, A) → H^{0}(G, B) → H^{0}(f, φ) → H^{1}(G′, A) →

H^{1}(G, B) → H^{1}(f, φ) → H^{2}(G′, A) → ⋯ → H^{n−1}(f, φ) →

H^{n}(G′, A) → H^{n}(G, B) → H^{n}(f, φ) → H^{n+1}(G′, A) → …. (11)

Thus if G and G′ are free groups, then H^{n}(f, φ) = 0 for n ⩾ 2. It is also clear that there is a short exact sequence of cochain complexes

0 → C^{∗}(f, A) → C^{∗}(f, φ) → C^{∗}(G, B/A) → 0

inducing a long exact cohomology sequence

0 → Der(f, A) → Z^{1}(f, φ) → Der(G, B/A) → H^{2}(f, A) →
H^{2}(f, φ) → H^{2}(G, B/A) → H^{3}(f, A) → ⋯ → H^{n−1}(G, B/A) →
H^{n}(f, A) → H^{n}(f, φ) → H^{n}(G, B/A) → H^{n+1}(f, A) → · · · .

Denote Z^{k}(f, φ) = Ker(C^{k}(f, φ) → C^{k+1}(f, φ)), k ⩾ 1. It is easily seen that if G′ is a free group, then the sequence 0 → Z^{k}(G′, A) → Z^{k}(G, B) → Z^{k}(f, φ) → 0 is exact for k ⩾ 1.

Any short exact coefficient sequence 0 → φ_{1} → φ → φ_{2} → 0 in the category C_{f} (that means both induced short sequences of G′ and G-modules respectively are exact) induces a short exact sequence of cochain complexes 0 → C^{∗}(f, φ_{1}) → C^{∗}(f, φ) → C^{∗}(f, φ_{2}) → 0 and therefore a long exact cohomology sequence of the relative cohomology H^{∗}(f, −).

Let (φ : A → B) be an object of the category C_{f}. Let us consider the coinduced G′-module A^{∗} = Hom(ℤ[G′], A) and the coinduced G-module B^{∗} = Hom(ℤ[G], B) induced by A and B respectively. Then f induces a well defined homomorphism φ^{∗} : A^{∗} → B^{∗} and it is easily checked that φ^{∗} is an object of the category C_{f} called the coinduced f-module induced by φ. Moreover one has a canonical injection (α, β) : (φ : A → B) → (φ^{∗} : A^{∗} → B^{∗}) induced by the well known injections A → A^{∗} and B → B^{∗}. The exact cohomology sequence (11) implies that H^{n}(f, φ^{∗}) = 0 for n ⩾ 1 and any coinduced f-module φ^{∗}. By taking the cokernels one gets a homomorphism φ̄ : A^{∗}/ Im α → B^{∗}/ Im β induced by φ^{∗}.

Proposition 9. The homomorphism φ̄ is object of the category C_{f}.

Proof. It is sufficient to show that φ̄ is injective. One has a commutative diagram

\begin{matrix}A & ⟶^{α} & A^{∗} \\ || & & ↓ ^{σ} \\ A & ⟶^{α′} & Hom(ℤ[G], A) \\ ^{φ} ↓ & & ↓ ^{φ′} \\ B & ⟶^{β} & Hom(ℤ[G], B) = B^{∗} \end{matrix} ,

where σ is injective and induced by f, the group Hom(ℤ[G], A) is the coinduced Gmodule induced by A as a G-module and the composite φ′σ is equal to φ^{∗}. Therefore one gets a commutative diagram

\begin{aligned}0 ⟶ & A & ⟶^{α} & A^{∗} & ⟶ & Coker α & ⟶ 0 \\ & || & & ↓ & & ↓ σ̄ \\ 0 ⟶ & A & ⟶^{α′} & Hom(ℤ[G], A) & ⟶ & Coker α′ & ⟶ 0 \\ & ^{φ} ↓ & & ↓ ^{φ′} & & ↓ φ̄′ \\ 0 ⟶ & B & ⟶^{β} & Hom(ℤ[G], B) & ⟶ & Coker β & ⟶ 0 \end{aligned} ,

where σ̄ is injective,since σ is. It remains to show that φ̄′ is injective. But this follows from the fact that if 0 → A → B → C → 0 is an exact sequence of abelian groups, then the sequence

0 → Hom(ℤ[G], A) → Hom(ℤ[G], B) → Hom(ℤ[G], C) → 0

is also exact, since ℤ[G] is a free abelian group.

Corollary 10. For any object φ of the category C_{f} one has a short exact sequence 0 → φ → φ∗ → φ̄ → 0 in the category C_{f}, where φ∗ is a coinduced f-module.

Theorem 11. (i) There is only one (up to equivalence) connected sequence of covariant abelian group valued functors {T^{n}, δ^{n}, n ⩾ 0} defined on the category C_{f} and satisfying the following conditions:

1) T^{0} = Z^{1}(f, −).

2) Any short exact sequence 0 → φ → ψ → ω → 0 of the category C_{f} induces a long exact connected sequence of the functors T^{n}, n ⩾ 0.

3) T^{n}(φ) = 0 for n ⩾ 1 and any coinduced f-module φ. (ii) There are natural isomorphisms R^{n}_{MF}Z^{k}(f, φ) ≈ H^{n+k}(f, φ), n > 0, for any

f-module φ and k ⩾ 1.

Proof. (i) As we have seen the sequence {Z^{1}(f, −), H^{n}(f, −), δ^{n}, n > 0} satisfies conditions 1-3. The proof of the uniqueness is standard.

(ii) Similar to the proof of Theorem 8. In effect, in this case we have also the exact sequence of cotriple derived functors

0 → R^{0}_{MF}Z^{k}(G′, A) → R^{0}_{MF}Z^{k}(G, B) → R^{0}_{MF}Z^{k}(f, φ) → R^{1}_{MF}Z^{k}(G′, A) →

R^{1}_{MF}Z^{k}(G, B) → R^{1}_{MF}Z^{k}(f, φ) → R^{2}_{MF}Z^{k}(G′, A) → … →
R^{n−1}_{MF}Z^{k}(f, φ) → R^{n}_{MF}Z^{k}(G′, A) → R^{n}_{MF}Z^{k}(G, B) →
R^{n}_{MF}Z^{k}(f, φ) → R^{n+1}_{MF}Z^{k}(G′, A) → ⋯

for f : G → G′ and φ : A → B. The 0-derived functors could be replaced by Z^{k}(G′, A), Z^{k}(G, B) and Z^{k}(f, φ) respectively. If we denote C_{k}^{n}(f, φ) = C^{n+k}(f, φ), for all n ⩾ 0, k ⩾ 1, it is easily checked that H_{0}(C_{k}^{∗}(f, φ)) = Z^{k}(f, φ) and if the groups G and G′ are free groups, then H_{n}(C_{k}^{∗}(f, φ)) = 0 for n ⩾ 1. On the other hand, the short exact sequence of cochain complexes

0 → C_{k}^{j}(F_{∗}(G′), A) → C_{k}^{j}(F_{∗}(G), B) → C_{k}^{j}((F M)_{∗}(f), φ) → 0 , j ⩾ 0 ,

gives the equalities H_{n}(C_{k}^{j}((MF)_{∗}(f), φ)) = 0, n > 0, and H_{0}(C_{k}^{j}((MF)_{∗}(f), φ)) = C_{k}^{j}(f, φ) for j ⩾ 0 and k ⩾ 1.

It remains to use the bicomplex C_{k}^{∗}((MF)_{∗}(f), φ) to get the required isomorphisms via the homology of the total cochain complex T ot C_{k}^{∗}((MF)_{∗}(f), φ), k ⩾ 1, where (MF)_{∗}(f) → f is the cotriple resolution of f with respect to the cotriple MF.

It is clear if f : F → G is a surjective group homomorphism with F a free group and A is a G′-module, then for k ⩾ 1 one has an exact sequence

0 → Z^{k}(G, A) → Z^{k}(F, A) → Z^{k}(f, A) → H^{k+1}(G, A) → 0 .

Proposition 12. For any surjective group homomorphism f : G → G′ and a G′- module A there is an exact sequence

0 → Z^{1}(f, A) → Z^{1}(f, 1_{A}^{∗}) → Z^{1}(f, ¯1_{A}) → H^{2}(f, A) → 0

and H^{k}(f, ¯1_{A}) ≈ H^{k+1}(f, A), k ⩾ 2.

Proof. By Corollary 10 one can consider the short exact coefficient sequence

0 → 1_{A} → 1_{A}^{∗} → ¯1_{A} → 0

and apply Theorem 11.

Notice that the relative cohomology group H^{n}(f, A) coincides with the relative cohomology group H^{n+1}(G′, G, A), n ⩾ 2, defined in [17], where a different interpretation of the cohomology group H^{3}(G′, G, A) is given in terms of relative extensions of (G′, G).

Remark 13. It is obvious that by (ii) of Theorem 11 we recover Theorem 8 for φ = 1_{A} : A → A. To get results similar to Theorem 11 for the homology of surjective homomorphisms it is necessary to consider as coefficients the category of surjections f : B → A of G-modules with A a G′-module and the proof is similar. The case k = 1 for Theorem 8 has been also considered in [19].

Now the description of H^{2}(f, A) given in Proposition 12 will be extended to higher relative cohomology groups.

Let f : G → G′ be a surjective group homomorphism and φ : A → B be an object of the category C_{f}. Consider pairs E_{k} = (e_{k}, ψ_{k}) consisting of an exact coefficient sequence e_{k} = 0 → φ → φ_{1} → ⋯ → φ_{k+1} → 0 and an element ψ_{k} of the group Z^{1}(f, φ_{k+1}), k ⩾ 1. It will be said that a pair E_{k} = (e_{k}, ψ_{k}) is similar to a pair E′_{k} = (e′_{k}, ψ′_{k}), if there is a morphism α = (1φ, α_{1}, ⋯, α_{k+1}) : e_{k} → e′_{k} and there exists an element η′_{k−1} of the group Z^{1}(f, φ′_{k}) such that σ′_{k}(η′_{k−1}) = α_{k+1}(ψ_{k}) − ψ′_{k}, where σk^{′} : Z^{1}(f, φk^{′}) → Z^{1}(f, φk+1^{′}) is induced by the morphism φk^{′} → φk+1^{′} of the sequence ek^{′}. Denote by E^{k}(f, φ) the set of equivalence classes, k ⩾ 1, with respect to the equivalence relation generated by the relation of similar pairs. We shall introduce a ”Baer sum” on the set E^{k}(f, φ) as follows.

Let e_{1} = 0 → φ → φ_{1} → φ_{2} → 0 be a short exact sequence and (α, β) : φ → φ′ a morphism of the category C_{f}, where φ : A → B, φ_{i} : A_{i} → B_{i}, i = 1, 2. Denote by E_{1} the corresponding short exact sequence 0 → A → A_{1} → A_{2} → 0 of G′- modules and by E_{2} the short exact sequence 0 → B → B_{1} → B_{2} → 0 of G-modules respectively. One gets short exact sequences E′_{1} = E(A_{2}, α) = 0 → A′ → A′_{1} → A′_{2} → 0 and E′_{2} = E(B_{2}, β) = 0 → B′ → B′_{1} → B′_{2} → 0 of G′-modules and Gmodules respectively, where E denotes the extension functor. It is easy to see that homomorphisms φ′_{i} : A′_{i} → B′_{i}, induced by φ_{i}, i = 1, 2, respectively are objects of the category C_{f} and therefore we obtain a short exact sequence e′ = 0 → φ′ → φ′_{1} → φ′_{2} → 0 in the category C_{f} which we denote e′_{1} = E(e_{1}, (α, β)). The definition of e′_{k} = E(e_{k}, (α, β)) for e_{k} = 0 → φ → φ_{1} → ⋯ → φ_{k} → φ_{k+1} → 0 is extended in the usual way for k > 1.

For [E_{k}], [E′_{k}] ∈ E^{k}(f, φ), E_{k} = (e_{k}, ψ_{k}), E′_{k} = (e′_{k}, ψ′_{k}), define [E_{k}] + [E′_{k}] = [(E(e_{k} + e′_{k}, ∇), φ′_{k+1}(ψ_{k} + ψ′_{k}))], where the morphism ∇ : φ + φ → φ is induced by 1_{φ}. It is easy to check that this sum is correctly defined and one gets an abelian group structure on E^{k}(f, φ).

Theorem 14. For k ⩾ 1 there is a natural isomorphism ϑ_{k} : E^{k}(f, φ) → H^{k+1}(f, φ).

Proof. At first consider the case k = 1. The map ϑ_{1} : E^{1}(f, φ) → H^{2}(f, φ) is given by ϑ_{1}([(e_{1}, ψ_{1})]) = δ^{1}(ψ_{1}), where e_{1} = 0 → φ → φ_{1} → φ_{2} → 0, ψ_{1} ∈ Z^{1}(f, φ_{2}) and δ^{1} is the connecting homomorphism of the long exact cohomology sequence induced by the exact coefficient sequence e_{1}.

To define the converse map ϑ′_{1} : H^{2}(f, φ) → E^{1}(f, φ) use the short exact sequence e = 0 → φ → φ^{∗} → φ̄ → 0 implying an epimorphism δ^{1} : Z^{1}(f, φ̄) → H^{2}(f, φ) and define ϑ′_{1}(x) = [(e, ψ)], where δ^{1}(ψ) = x for x ∈ H^{2}(f, φ).

Let k = 2 and [(e_{2}, ψ_{2})] ∈ E^{2}(f, φ), where e_{2} = 0 → φ → φ_{1} → φ_{2} → φ_{3} → 0 and ψ_{2} ∈ Z^{1}(f, φ_{3}). The exact sequence e_{2} induces two short exact sequences e_{1} : 0 → φ → φ_{1} → Imφ_{1} → 0 and e′_{1} : 0 → Imφ_{1} → φ_{2} → φ_{3} → 0. Define the map ϑ_{2} : E^{2}(f, φ) → H^{3}(f, φ) by ϑ_{2}([(e_{2}, ψ_{2})]) = δ_{e_{1}}^{2}δ_{e′_{1}}^{1}(ψ_{2}). The map ϑ′_{2} : H^{3}(f, φ) → E^{2}(f, φ) is defined as follows. Splice the short exact sequences e_{1} : 0 → φ → φ∗ → φ̄ → 0 and e′_{1} : 0 → φ̄ → φ̄∗ → ¯ φ̄ → 0 to get the exact sequence e_{2} = 0 → φ → φ∗ → φ̄∗ → ¯ φ̄ → 0. Clearly δ_{e_{1}}^{2} : H^{2}(f, φ̄) → H^{3}(f, φ) is an isomorphism and δ_{e′_{1}}^{1} : Z^{1}(f, ¯ φ̄) → H^{2}(f, φ̄) is surjective. Then ϑ′_{2} is given by ϑ′_{2}(x) = [(e, ψ_{2})], where δ_{e′_{1}}^{1}(ψ_{2}) = (δ_{e_{1}}^{2}(x))^{−1} for x ∈ H^{3}(f, φ).

For k > 2 the construction of the maps ϑ_{k} and ϑ′_{k} are similar to the case k = 2. It is routine to show that these maps for all k ⩾ 1 are correctly defined homomorphisms,both composites ϑ_{k}ϑ′_{k},ϑ′_{k}ϑ_{k} are identity maps, and left to the reader.

4. Cohomology of groups with operators

Recently in [7] a cohomology theory H_{Γ}^{∗}(G, A) of a group G with operators was defined, when a separate group Γ acts on G, with coefficients in a Γ-equivariant Gmodule. This cohomology theory which will be called shortly equivariant cohomology of groups was motivated by the graded categorical groups classification problem suggested by Frohlich and Wall [14] and solved in [8] by using the equivariant group cohomology H_{Γ}^{3}(G, A).

It was shown that H_{Γ}^{∗}(G, A) is a cotriple cohomology as the non-abelian derived functors of the group Der_{Γ}(G, A) of Γ-equivariant derivations from G to A with respect to the free cotriple defined in the category of Γ-groups acting on A.

Let G and Γ be arbitrary groups, and G a Γ-group. A Γ-module A is called a Γ-equivariant G-module, if it is a G-module such that ^{y}(^{x}a) = ^{(^{y}x)}(^{y}a) for y ∈ Γ, x ∈ G, a ∈ A. A Γ-equivariant derivation (or crossed Γ-homomorphism) from G into the G-module A is a map f : G → A satisfying the conditions f(xy) = f(x) +^{x} f(y) for x, y ∈ G and f(^{y}x) = ^{y}f(x) for y ∈ Γ, x ∈ G.

The cochain complex C_{Γ}^{∗}(G, A) defining the equivariant group cohomology H_{Γ}^{∗}(G, A) is given by

0 → C_{Γ}^{0}(G, A) → C_{Γ}^{1}(G, A) → C_{Γ}^{2}(G, A) → C_{Γ}^{3}(G, A) → ⋯ → C_{Γ}^{n}(G, A) → ⋯,

(12)

where the elements of C_{Γ}^{n}(G, A) are maps

f : ∪_{p+q=n−1} G^{p+1} × Γ^{q} → A , n ⩾ 1 , C_{Γ}^{0}(G, A) = 0 ,

and the coboundary operator δ^{n} : C^{n}Γ(G, A) → C_{Γ}^{n+1}(G, A) is defined by the formula

δ^{n}(f)(x_{1}, …, x_{p+1}, σ_{1}, …, σ_{q}) = ^{σ_{1}}f(x_{1}, …, x_{p+1}, σ_{2}, …, σ_{q})+
∑(−1)^{i}f(x_{1}, …, x_{p+1}, σ_{1}, …, σ_{i}σ_{i+1}, …, σ_{q})+
(−1)^{q}f(^{σ_{q}}x_{1}, …,^{σ_{q}} x_{p+1}, σ_{1}, …, σ_{q−1})+
(−1)^{q}[^{^{(σ_{1}···σq} x_{1}}f(x_{2}, …, x_{p+1}, σ_{1}, …, σ_{q})+
∑(−1)^{j}f(x_{1}, …, x_{j}x_{j+1}, ⋯, x_{p+1}, ⋯, σ_{1}, …, σ_{q})+
(−1)^{p+1}f(x_{1}, …, x_{p}, σ_{1}, …, σ_{q})],

with x_{1}, …, x_{p+1} ∈ G, σ_{1}, …, σ_{q} ∈ Γ and p+q = n. Then H_{Γ}^{n}(G, A) = H_{n}C_{Γ}^{∗}(G, A), n ⩾ 0. Thus H_{Γ}^{0}(G, A) = 0 and H_{Γ}^{1}(G, A) = Der_{Γ}(G, A).

Denote Z_{Γ}^{k}(G, A) = Ker δ^{k}, k ⩾ 1. One gets contravariant functors Z_{Γ}^{k}(−, A) : G_{Γ,A} → 𝔄𝔟, n ⩾ 1, from the category of Γ-groups acting on A to the category of abelian groups. Clearly Z_{Γ}^{1}(G, A) = Der_{Γ}(G, A). In [7] is defined the free cotriple F_{Γ} = (F_{Γ}, ϵ, δ) in the category G_{Γ} given by F_{Γ}(G) which is the free group generated by the set G×Γ with the action^{σ}(g, γ) = (g, σγ), the homomorphism ϵ : F_{Γ}(G) → G is given by ϵ(g, γ) = ^{γ}g and the homomorphism δ : F_{Γ}(G) → F_{Γ}^{2}(G) is defined by δ(g, γ) = ^{δ}((g, 1), γ).

Let us consider now the cotriple derived functors R_{F_{Γ}}^{n}Z_{Γ}^{k}(−, A), n ⩾ 0, k ⩾ 1, of the contravariant functors Z_{Γ}^{k}(−, A) with respect to the free cotriple F_{Γ}.

Theorem 15. There is a natural isomorphism

R_{F_{Γ}}^{n}Z_{Γ}^{k}(G, A) ≈ H_{Γ}^{n+k}(G, A)

for all n > 0, k > 0 and any Γ-equivariant G-module A.

Proof. The case k = 1 is proved in [7] (see Theorem 1.6). It follows that H_{Γ}^{n}(G, A) = 0 for n ⩾ 2, if G belongs to the projective class P_{Γ} generated by the free cotriple F_{Γ}. For k > 1 the proof is similar to the case of classical cohomology of groups (see Theorem 1) and we will shortly repeat it.

One has a functorial isomorphism R_{F_{Γ}}^{0}Z_{Γ}^{k}(−, A) ≈ Z_{Γ}^{k}(−, A), k > 0. Therefore for the equivariant group cohomology H_{Γ}^{∗}(G, A) one gets exact sequences like (2) and (4). In [6] it is shown that H_{Γ}^{n}(G, A) = 0 for n ⩾ 2, if A is an injective Γ-equivariant G-module. Since any short exact sequence of Γ-equivariant G-modules provides a long exact equivariant cohomology sequence,to complete the proof it remains to show that R_{FΓ}^{n}Z_{Γ}^{k}(G, A) = 0 for n, k > 1, if A is an injective Γ-equivariant Gmodule.

Let F_{∗}(G) → G be the free cotriple resolution of G with respect to F_{Γ}. By applying the defining cochain complex (12) to this resolution we obtain a sequence of abelian cosimplicial groups

0 → C_{Γ}^{1}(F_{∗}(G), A) → C_{Γ}^{2}(F_{∗}(G), A) → C_{Γ}^{3}(F_{∗}(G), A) → ⋯ → C_{Γ}^{n}(F_{∗}(G), A) → ⋯

(13)

which is exact for any Γ-equivariant G-module except at the first term. Assume now that A is an injective Γ-equivariant G-module. Then the short exact sequence of abelian cosimplicial groups

0 → Der_{Γ}(F_{∗}(G), A) → C_{Γ}^{1}(F_{∗}(G), A) → Im δ_{Γ}^{1} → 0

implies H_{n}(Im δ_{Γ}^{1}) = 0 for n > 0, by taking into account that H_{n}(C_{Γ}^{i}(F_{∗}(G), A)) = 0 for all n, i > 0 and any Γ-equivariant G-module, since the augmented simplicial group F_{∗}(G) → G is left contractible as augmented simplicial set. But Im δ_{Γ}^{1} = Ker δ_{Γ}^{2} providing H_{n}(Z_{Γ}^{2}(F_{∗}(G), A)) = 0, n > 0. By using the exactness of (13) and the induction on k finally one gets H_{n}(Z_{Γ}^{k}(F_{∗}(G), A)) = 0 for all n > 0, k ⩾ 1.

5. (Co)homology of associative algebras

We consider the category A of associative algebras over a unital commutative ring k. Let A be an associative k-algebra, M a bimodule over A and take the standard cochain complex

0 → C^{0}(A, M) →^{δ^{0}} C^{1}(A, M) →^{δ^{1}} C^{2}(A, M) →^{δ^{2}} C^{3}(A, M) →^{δ^{3}} ⋯

and the standard chain complex

⋯ →^{∂_{4}} C_{3}(A, M) →^{∂_{3}} C_{2}(A, M) →^{∂_{2}} C_{1}(A, M) →^{∂_{1}} C_{0}(A, M) → 0 ,

where C_{n}(A, M) = M ⊗ A^{⊗n}, ∂_{n} = ∑(−1)^{i}d^{i} and C^{n}(A, M) = Hom_{k}(A^{⊗n}, M), ⊗ = ⊗_{k}, with

d_{0}(m, a_{1}, …, a_{n}) = (ma_{1}, a_{2}, …, a_{n}) ,
d_{i}(m, a_{1}, …, a_{n}) = (m, a_{1}, …, a_{i}a_{i+1}, …, a_{n}) for 1 ⩽ i < n ,
d_{n}(m, a_{1}, …, a_{n}) = (a_{n}m, a_{1}, …, a_{n−1}) ,
which  give  respectively  the  Hochschild  cohomology

HH^{∗}(A, M) and homology HH_{∗}(A, M) of A with coefficients in M (see [18]).

Denote A^{e} = A ⊗_{k} A^{op} and JA = Ker(A^{e} → A) with ϵ(a ⊗ b) = ab. It is well known that for A projective over k one has the standard A^{e}-projective resolution of A:

⋯ → A^{⊗n+1} →^{b_{n−1}} A^{⊗n} →^{b_{n−2}} ⋯ →^{b_{1}} A^{⊗2} →^{b_{0}} A ,

where b_{n−2} = ∑_{i=0}^{n−1}(−1)^{i}d_{i} for n > 2 and b_{0}(a ⊗ b) = ab. Clearly Ker b_{0} = JA as A^{e}-modules. Denote Y_{k}(A) = Coker b_{k} for k ⩾ 1.

Clearly the group of derivations Der(A, M) is isomorphic to the kernel of δ^{1} and Coker ∂^{2} is isomorphic to JA ⊗_{A^{e}} M. We introduce the following notations:

Y_{k}(A, M) = Y_{k}(A) ⊗_{A^{e}} M , k > 0 , and Z^{k}(A, M) = Ker δ^{k} , k > 0 .

Thus Y_{1}(A, M) ≈ JA ⊗_{A^{e}} M and Z^{1}(A, M) ≈ Der(A, M). It is obvious that Y_{k}(−, M) and Z^{k}(−, M), k > 0, are respectively covariant and contravariant functors from the category A to the category of abelian groups.

Let F be the free cotriple defined for any associative k-algebra A by taking the free associative k-algebra F (A) over the underlying set A. Let us consider the cotriple (left) derived functors L_{n}^{F}Y_{k}(−, M) of the functors Y_{k}(−, M) for all k > 0, n ⩾ 0, and the cotriple (right) derived functors R_{F}^{n}Z^{k}(−, M) of the functors Z^{k}(−, M) for all k > 0, n ⩾ 0, with respect to the free cotriple F.

Theorem 16. There are natural isomorphisms

\begin{matrix}L_{n}^{F}Y_{k}(A, M) ≈ HH_{n+k}(A, M), \\ R_{F}^{n}Z^{k}(A, M) ≈ HH^{n+k}(A, M) \end{matrix}

for n, k > 0 and any associative k-algebra A projective over k.

These isomorphisms are well known for n > 0 and k = 1 (see [2]). For k > 1 the proof is similar to the proof of Theorem 1 and will be omitted.

6. Cohomology of crossed modules

Let ∂ : T → G be a crossed module and G the free cotriple in the category of crossed modules (see [5]). Then G(∂) is the inclusion F̄(T × G) ↪ F (T × G) ∗ F (T × G), where F̄(T × G) = Ker(F (T × G) ∗ F (T × G) → F (T × G)), F (T × G) is the free group generated by the set T × G and p_{2}u_{1} = 0, p_{2}u_{2} = 1_{F (T ×G)} with u_{1}, u_{2} : F (T, G) ↪ F (T ×G)∗F (T ×G) natural inclusions. Denote G_{i}(∂) = T_{i} ↪ G_{i} and τ_{i} : G_{i} → G_{i}/T_{i}, where G_{i} and G_{i}/T_{i} are free groups, i ⩾ 0, with G_{0}(∂) = G(∂) and G_{i} = G(G_{i−1}), i ⩾ 1.

Let us consider the category G_{1} of triples (l_{0}, l_{1}, s) consisting of group homomorphisms l_{i} : L → G, i = 0, 1, and s : G → L such that l_{i}s = 1_{G}, i = 0, 1. Morphisms of the category G_{1} are couples (α, β) : (l_{0}, l_{1}, s) → (l′_{0}, l′_{1}, s′) of group homomorphisms α : L → L′ and β : G → G′ such that βl_{i} = l′_{i}α, i = 0, 1 and αs = s′β. There are two important full subcategories G_{2} and G_{3} whose objects (l_{0}, l_{1}, s) satisfy the conditions [Ker l_{0}, Ker l_{1}] = 1 and Ker l_{0} ∩ Ker l_{1} = 1 respectively. Clearly one has the inclusions G_{1} ⊃ G_{2} ⊃ G_{3}. It is well known that these categories are equivalent in the same order to the categories PCM ⊃ CM ⊃ IJ of precrossed modules, crossed modules and injective (aspherical) crossed modules respectively. It is obvious that the latter is equivalent to the category MG of surjective group homomorphisms. The equivalence is established as follows: to any precrossed module of groups ∂ : T → G corresponds the triple (d_{0}, d_{1}, s)_{∂} consisting of d_{0}, d_{1} : T ▷◁ G → G and s : G → T ▷◁ G, where T ▷◁ G is the semidirect product of T by G, d_{0}(t, g) = g, d_{1}(t, g) = ∂(t) · g and s(g) = (1, g).

The notion of the group of derivations in an abelian group A for surjective group homomorphisms (see definition 7), or equivalently for objects of the category IJ, has been extended in [19] for crossed modules by using the above defined equivalence of the category CM with the category G_{2} and by considering the category of crossed modules acting on an abelian group A.

In order to define the cohomology of crossed modules, as for the case of the relative cohomology of groups we want to enlarge the category of coefficients. Let ∂ : T → G be a precrossed module. Denote C_{∂} the category of injections φ : A → B of abelian groups such that A is a (G/ Im ∂)-module, G acts on B and φ is a Gmodule, G acting on A via the canonical homomorphism G → G/ Im ∂. In this case it will be said that the precrossed module ∂ acts on φ. The category C_{∂} was defined above, when ∂ is injective and G acts on T by conjugation or equivalently for the canonical surjective homomorphism G → G/ Im ∂.

Let ∂ : T → G be a crossed module acting on an injection φ : A → B of abelian groups. This action induces actions of G_{i}(∂) and τ_{i} on φ for all i ⩾ 0. So the standard cochain complexes C^{∗}(τ_{i}, φ), i ⩾ 0, are well defined. Take new cochain complexes C̄^{∗}(τ_{i}, φ), i ⩾ 0, getting from C^{∗}(τ_{i}, φ) by dimension shifting, i.e. C̄^{n}(τ_{i}, φ) = C^{n+1}(τ_{i}, φ) for n, i ⩾ 0. Consider now the bicomplex C̄^{∗}(τ_{∗}, φ) and its total cochain complex T ot C̄^{∗}(τ_{∗}, φ) constructed for the cotriple resolution G_{∗}(∂) → ∂ of the crossed module ∂ with respect to the free cotriple G. Denote by Z^{1}(∂, φ) the kernel of the differential δ^{1} of the cochain complex T ot C^{∗}(τ_{∗}, φ), k ⩾ 0.

Definition 17. The cohomology groups of the crossed module (∂ : T → G) with coefficients in (φ : A → B) is given by H^{n}(∂, φ) = H_{n}(T ot C̄^{∗}(τ_{∗}, φ)) for n ⩾ 0.

It is easily checked that these cohomology groups are independent of the projective resolution of the crossed module ∂ with respect to the projective class induced by the free cotriple G.

Theorem 18. Let ∂ : T → G be a crossed module. Then

(i) Any short exact sequence of coefficients 0 → φ_{1} → φ → φ_{2} → 0 induces a long exact cohomology sequence

0 → H^{0}(∂, φ_{1}) → H^{0}(∂, φ) → H^{0}(∂, φ_{2}) → H^{1}(∂, φ_{1}) → ⋯ →
H^{n−1}(∂, φ_{2}) → H^{n}(∂, φ_{1}) → H^{n}(∂, φ) →
H^{n}(∂, φ_{2}) → H^{n+1}(∂, φ_{1}) → · · · .

(ii) If ∂ acts on (φ : A → B), there is a long exact cohomology sequence

0 → Der(∂, A) → Z^{1}(∂, φ) → Der(G, B/ Im φ) → D^{1}(∂, A) → H^{1}(∂, φ) → H^{2}(G, B/ Im φ) → D^{2}(∂, A) → H^{2}(∂, φ) → ⋯ → H^{n}(G, B/ Im φ) →

D^{n}(∂, A) → H^{n}(∂, φ) → H^{n+1}(G, B/ Im φ) → D^{n+1}(∂, A) → ⋯,

there is an isomorphism

H^{n}(∂, A → A^{∗}) ≈ H^{n+2}(B(T, G, ∂), A) , n > 0,

and an exact sequence

\begin{matrix}0 → Der(G/ Im ∂, A) → Der(G, A^{∗}) → Z^{1}(∂, A → \\ A^{∗}) → H^{2}(B(T, G, ∂), A) → 0 , \end{matrix}

where D^{∗}(∂, A) and H^{∗}(B(T, G, ∂), A) are cohomology theories of crossed modules defined in [19] and [13] respectively, and A → A^{∗} is the canonical injection of A into the coinduced G-module A^{∗} induced by the G-module A.

Proof. (i) Let φ_{1} : A_{1} → B_{1}, φ : A → B and φ_{2} : A_{2} → B_{2}. Then the commutative diagram

\begin{aligned}& 0 & & 0 \\ & ↓ & & ↓ \\ 0 ⟶ & C̄^{∗}(G_{i}/T_{i}, A_{1}) & ⟶ & C̄^{∗}(G_{i}, B_{1}) & ⟶ & C̄^{∗}(τ_{i}, φ_{1}) & ⟶ 0 \\ & ↓ & & ↓ & & ↓ \\ 0 ⟶ & C̄^{∗}(G_{i}/T_{i}, A) & ⟶ & C̄^{∗}(G_{i}, B) & ⟶ & C̄^{∗}(τ_{i}, φ) & ⟶ 0 \\ & ↓ & & ↓ & & ↓ \\ 0 ⟶ & C̄^{∗}(G_{i}/T_{i}, A_{2}) & ⟶ & C̄^{∗}(G_{i}, B_{2}) & ⟶ & C̄^{∗}(τ_{i}, φ_{2}) & ⟶ 0 \\ & ↓ & & ↓ \\ & 0 & & 0 \end{aligned} ,

consisting of cochain complexes with exact rows and exact left two columns for i ⩾ 0, since the groups G_{i} and G_{i}/T_{i} are free groups, implies the short exact sequence

0 → C̄^{∗}(τ_{i}, φ_{1}) → C̄^{∗}(τ_{i}, φ) → C̄^{∗}(τ_{i}, φ_{2}) → 0

for all i ⩾ 0.

Hence one gets a short exact sequence of bicomplexes

0 → C̄^{∗}(τ_{∗}, φ_{1}) → C̄^{∗}(τ_{∗}, φ) → C̄^{∗}(τ_{∗}, φ_{2}) → 0

implying a short exact sequence of their total cochain complexes

0 → T ot C̄^{∗}(τ_{∗}, φ_{1}) → T ot C̄^{∗}(τ^{∗}, φ) → T ot C̄^{∗}(τ_{∗}, φ_{2}) → 0

and therefore giving the required long exact cohomology sequence.

(ii) For any τ_{i} : G_{i} → G_{i}/T_{i}, i ⩾ 0, there is a short exact sequence

0 → Der(G_{i}/T_{i}, A) → Der(G_{i}, A) → Der(τ_{i}, A) → 0,

where the groups G_{i} and G_{i}/T_{i}, i ⩾ 0, are free groups. Hence we have a short exact sequence of cochain complexes

0 → Der(τ_{∗}, A) → Der(τ_{∗}, φ) → Der(G_{∗}, B/ Im φ) → 0

which induces a long exact cohomology sequence

0 → Der(∂, A) → H_{0}(Der(τ_{∗}, φ)) → Der(G, B/ Im φ) → D^{1}(∂, A) →
H_{1}(Der(τ_{∗}, φ)) → H^{2}(G, B/ Im φ) → D^{2}(∂, φ) →
H_{2}(Der(τ_{∗}, φ)) → ⋯ → H^{n}(G, B/ Im φ) →
D^{n}(∂, φ) → H_{n}(Der(τ_{∗}, φ)) → H^{n+1}(G, B/ Im φ) → D^{n+1}(∂, φ) → · · · .

In the bicomplex C̄(τ_{∗}, φ) the groups H_{n}(C̄^{∗}(τ_{i}, φ)) are trivial and H_{0}(C̄^{∗}(τ_{i}, φ)) ≈ Der(τ_{i}, φ) for all i ⩾ 0. It follows that there are isomorphisms H^{n}(∂, φ) ≈ H_{n}(Der(τ_{∗}, φ)), n ⩾ 0, and we get the required long exact cohomology sequence.

Taking as coefficients the injection φ : A → A^{∗} of a G/ Im ∂-module A into its coinduced G-module A^{∗} = Hom(ℤ[G], A) gives rise to a short exact sequence of cochain complexes

0 → Der(G_{∗}/T_{∗}, A) → Der(G_{∗}, A^{∗}) → Der(τ_{∗}, A → A^{∗}) → 0

and therefore to a long exact cohomology sequence

0 → Der(G/ Im ∂, A) → Der(G, A^{∗}) → Z^{1}(∂, A → A^{∗}) →
H_{1}(Der(G_{∗}/T_{∗}, A)) → H_{1}(Der(G_{∗}, A^{∗})) → ⋯ H_{n}(Der(G_{∗}, A^{∗})) →

H_{n}(Der(τ_{∗}, A → A^{∗})) → H_{n+1}(Der(G_{∗}/T_{∗})) → H_{n+1}(Der(G_{∗}, A^{∗})) → · · · .

As we have seen H^{n}(∂, A → A^{∗}) is isomorphic to H_{n}(Der(τ_{∗}, A → A^{∗})), n ⩾ 0. On the other hand, the groups H_{n}(Der(G_{∗}, A^{∗})) are isomorphic to H^{n+1}(G, A^{∗}) for n ⩾ 1, since G_{∗} → G is a projective resolution of the group G, and therefore are trivial for n ⩾ 1. Finally, in [19] it is proved that the groups H_{n}(Der(G_{∗}/T_{∗}, A)) are isomorphic to the cohomology groups H^{n+1}(B(T, G, ∂), A), n ⩾ 1, of crossed modules defined in [13]. Now from the above long exact cohomology sequence we can deduce the last two statements of (ii).

Note that in [19] it is proved that the cohomology theory D^{∗}(∂, A) of crossed modules is closely related with the cohomology theories of crossed modules defined in [5] and [13].

Define the group Z^{k}(∂, φ) as H_{0}(Z^{k}(τ_{∗}, φ)) for any crossed module ∂ acting on φ and all k ⩾ 1. Clearly for k = 1 we recover the above definition of Z^{1}(∂, φ). It is also easily seen that if τ : G → G′ is a surjective group homomorphism acting on φ, the group Z^{k}(Ker τ ↪ G, φ) is isomorphic to Z^{k}(τ, φ), k ⩾ 1.

Theorem 19. There is a natural isomorphism

R_{G}^{n}Z^{1}(∂, φ) ≈ H^{n}(∂, φ) for all n > 0

and a long exact sequence

0 → Z^{k}(∂, φ) → H_{0}(C^{k}(τ_{∗}, φ)) → Z^{k+1}(∂, φ) → R_{G}^{1}Z^{k}(∂, φ) →

H_{2}(C^{k}(G_{∗}/T_{∗}, A)) → R_{G}^{1}Z^{k+1}(∂, φ) → ⋯ → R_{G}^{n}Z^{k+1}(∂, φ) → R_{G}^{n+1}Z^{k}(∂, φ) →

H_{n+2}(C^{k}(G_{∗}/T_{∗}, A)) → R_{G}^{n+1}Z^{k+1}(∂, φ) → R_{G}^{n+1}Z^{k}(∂, φ) → ⋯

for all k ⩾ 1.

Proof. Since the groups G_{i} and G_{i}/T_{i} are free groups, i ⩾ 0, from the short exact sequence of cochain complexes

0 → C̄^{∗}(G_{i}/T_{i}, A) → C̄^{∗}(G_{i}, B) → C̄^{∗}(τ_{i}, φ) → 0 , i ⩾ 0 ,

it follows that H_{n}(C̄^{∗}(τ_{i}, φ)) = 0 for n > 0, and H_{0}(C̄^{∗}(τ_{i}, φ)) = Z^{1}(τ_{i}, φ), i ⩾ 0. Therefore the homology group H_{n}(T ot C̄^{∗}(τ_{∗}, φ)) is isomorphic to H_{n}(Z^{1}(τ_{∗}, φ)) for n ⩾ 0.

The short exact sequence of cochain complexes

0 → C^{k}(G_{∗}/T_{∗}, A) → C^{k}(G_{∗}, B) → C^{k}(τ_{∗}, φ) → 0

induces an isomorphism H_{n}(C^{k}(τ_{∗}, φ)) ≈ H_{n+1}(C^{k}(G_{∗}/T_{∗}, A)) for n, k ⩾ 1, because G_{∗} → G as a projective resolution of G is simplicially exact and therefore left (right) contractible as augmented simplicial set, implying the triviality of the groups H_{n}(C^{k}(G_{∗}, B)) for n > 0. On the other hand, for k ⩾ 1 one has the following short exact sequence of cochain complexes:

0 → Z^{k}(τ_{∗}, φ) → C^{k}(τ_{∗}, φ) → Z^{k+1}(τ_{∗}, φ) → 0 ,

inducing a long exact sequence

0 → Z^{k}(∂, φ) → H_{0}(C^{k}(τ_{∗}, φ)) → Z^{k+1}(∂, φ) → R_{G}^{1}Z^{k}(∂, φ) → H_{1}(C^{k}(τ_{∗}, φ)) →

R_{G}^{1}Z^{k+1}(∂, φ) → ⋯ R_{G}^{n}Z^{k+1}(∂, φ) → R_{G}^{n+1}Z^{k}(∂, φ) →

H_{n+1}(C^{k}(τ_{∗}, φ)) → R_{G}^{n+1}Z^{k+1}(∂, φ) → R_{G}^{n+1}Z^{k}(∂, φ) · · · .

This completes the proof.

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

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Hvedri Inassaridze hvedri@rmi.acnet.ge

A.Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Alexidze Str. 1, Tbilisi 0193, Georgia