(Co)homology of $\Gamma$-groups and $\Gamma$-homological algebra

We continue the study of our approach to group actions in homological algebra which we call $\Gamma$-homological algebra that was started in [27] and continued in [28]. The origin of the equivariant study of group extensions theory in homological algebra goes back to Whitehead paper [48]. Group actions on algebraic and topological objects have many important applications in K-theory and homotopy theory([6,16,31,41]). Our goal is to continue the development of extension theory in the category of $\Gamma$-groups and of the relevant equivariant (co)homology theory that has been initiated in [27,28]. A different (co)homology theory of groups with operators was provided and investigated in [7-10], motivated by the graded categorical groups classification problem [8]. The introduction of $\Gamma$-equivariant chain complexes and their homology groups substantially contribute to the realization of our aim. Moreover this approach allows us to present a version of equivariant Hochschild homology of any unital k-algebra A induced by the action of the group $\Gamma$ on the k-algebra A. The $\Gamma$-equivariant Hochschild homology is closely related to $\Gamma$- equivariant homology of groups [27]. This equivariant version differs of the equivariant Hochschild homology given in [38].

We study extensions of $\Gamma$-groups that can be viewed as a part of group actions in homological algebra, particularly of group actions on simplicial groups. Two important classes of $\Gamma$-group extensions are considered. The first class is consisting of extensions having $\Gamma$-section map. The investigation of these extensions was initiated in [27] and called $\Gamma$-equivariant extensions of $\Gamma$-groups by $\Gamma$-equivariant G-modules. For the second class we deal with extensions of $\Gamma$-groups endowed with a crossed $\Gamma$-module structure and called $\Gamma$-extensions of crossed $\Gamma$-modules (having $\Gamma$-section map). Our approach to extensions of crossed modules substantially extends the class of relative extensions of group epimorphisms introduced and investigated by Loday [33]. It should be noted that homology and cohomology of crossed modules related to extensions of crossed modules were investigated by many authors [3,7,12-13,15-17,23].

The study of $\Gamma$-group extensions having $\Gamma$-section map is closely related to the extension problem of group actions satisfying some conditions, in our case to lifting group actions that split for the given extension of groups.

Applications to algebraic K-theory, Galois theory of commutative rings and cohomological dimension of $\Gamma$-groups are given.

The paper is divided into seven sections:

2. Preliminaries

3. Extensions of $\Gamma$-groups,

4. Some computations,

5. $\Gamma$-derived functors and $\Gamma$-equivariant Hochschild homology,

6. Extensions of crossed $\Gamma$-modules,

7. Homology and central $\Gamma$-extensions of crossed $\Gamma$-modules,

8. Applications to algebraic K-theory, Galois theory of commutative rings and cohomological dimension of groups.

Some notation that will be used throughout the paper:

$[\Gamma G]$ denotes the set of elements ${}^{\gamma}gg^{-1}$ and $\Gamma G$ is the normal subgroup of G generated by the set $[\Gamma G]$. $G_{\Gamma}$ denotes the quotient group $G/\Gamma G$.

$[G,G]_{\Gamma}$ denotes the subgroup of the $\Gamma$-group G generated by the commutant subgroup [G,G] and the elements of the form ${}^{\gamma}gg^{-1}$, $g\in G$, $\gamma\in\Gamma$. It is called the $\Gamma$-commutant of the group G.

$[G,H]_{\Gamma}$ denotes the subgroup of G generated by the elements $x^{\gamma}yx^{-1}y^{-1}$, where $g\in G$, y belongs to the normal subgroup H of G, and $\gamma\in\Gamma$.

$G^{ab}_{\Gamma}$ denotes the abelianization of the group $G_{\Gamma}$.

In this section we recall some definitions and propositions given in [27] which will be used later. Moreover it is shown how equivariant versions of well known homological properties of groups are obtained by using our approach to group actions in homological algebra.

Let ${\mathbf{G}}^{\Gamma}$ be the category whose objects are groups on which a fixed group $\Gamma$ is acting, called $\Gamma$-groups, and morphisms are group homomorphisms compatible with the action of $\Gamma$.

Any exact sequence E of $\Gamma$-groups

is called $\Gamma$-extension of the $\Gamma$-group G by the $\Gamma$-group A. The extension E is an extension with $\Gamma$-section map if there is a map $\beta:G\rightarrow X$ such that $\tau\beta=1_{G}$ and $\beta$ is compatible with the action of $\Gamma$, $\beta(^{\gamma}g)=^{\gamma}{\beta(g),g\in G,\sigma\in\Gamma}$. In addition if $\beta$ is a homomorphism then the extension E is called split extension.

The class $\mathcal{P}$ of relatively projective $\Gamma$-equivariant G-modules is a projective class with respect to proper sequences of $\Gamma$-equivariant G-modules.

For the cohomological description of the set $E^{1}_{\Gamma}(G,A)$ of equivalence classes of $\Gamma$-equivariant extensions of G by A the $\Gamma$-equivariant homology and cohomology of $\Gamma$-groups have been introduced as relative $Tor_{n}^{\mathcal{P}}$ and $Ext^{n}_{\mathcal{P}},n\geq 0$ in the category of $\Gamma$-equivariant G-modules [27], namely

Let

be the bar resolution of $\mathbb{Z}$, where $B_{0}=\mathbb{Z}(G)$ and $B_{n},n>0$,is the free $\mathbb{Z}(G)$-module generated by $[g_{1},g_{2},...,g_{n}]$, $g_{i}\in G$. The action of $\Gamma$ on G induces an action of $\Gamma$ on the sequence (2.2) defined by ${}^{\gamma}(mg)=m^{\gamma}g$ and ${}^{\gamma}(g[g_{1},g_{2},...,g_{n}])=^{\gamma}{g}[^{\gamma}{g_{1}},^{\gamma}{g_{2}},...,^{\gamma}{g_{n}}]$ for $B_{0}$ and $B_{n}$ respectively, $n\geq 1$. Then the sequence 2.2 is the $\Gamma$-equivariant bar resolution of $\mathbb{Z}$, the groups $B_{n}$ being relatively free $\Gamma$-equivariant G-modules and there are isomorphisms $H_{n}^{\Gamma}(G,A)\cong H_{n}(B_{*}{\bigotimes}_{G\ltimes\Gamma}A)$, $H^{n}_{\Gamma}(G,A)\cong H_{n}(Hom_{G\ltimes\Gamma}(B_{*},A))$.

For the $\Gamma$-equivariant cohomology of $\Gamma$-groups an alternative description by cocycles is provided. To this end the group $C^{n}_{\Gamma}(G,A)$ of $\Gamma$-maps, $f:G^{n}\rightarrow A$ for $n>0$, called $\Gamma$-cochains is considered. By using the classical cobord operators $\delta^{n}:C^{n}_{\Gamma}(G,A)\rightarrow C^{n+1}_{\Gamma}(G,A)$, $n>0$, we obtain a cochain complex

where $C^{0}_{\Gamma}(G,A)=A^{\Gamma}$, $Ker\delta^{1}=Der_{\Gamma}(G,A)$ is the group of $\Gamma$-derivations and the homology groups of the complex $C^{*}_{\Gamma}(G,A)$ are isomorphic to the $\Gamma$-equivariant cohomology groups of the $\Gamma$-group G with coefficients in the $\Gamma$-equivariant G-module A.

Two $\Gamma$-equivariant extensions E and E’ of G by A are called equivalent if there is a morphism $E\rightarrow E^{\prime}$ which is the identity on G and A. We denote by $E^{1}_{\Gamma}(G,A)$ the set of equivalence classes of $\Gamma$-equivariant extensions of G by A.

Any free group F(G) generated by a $\Gamma$-group G becomes a $\Gamma$-free group by the following action of $\Gamma$: ${}^{\gamma}{\mid g\mid}=\mid^{\gamma}{g}\mid$, $g\in G,\gamma\in\Gamma$. The defining property of the $\Gamma$-free group F with basis E is that every $\Gamma$-map $E\overset{f}{\rightarrow}G$ to a $\Gamma$-group G is uniquely extended to a $\Gamma$-homorphism $F\overset{f^{\prime}}{\rightarrow}G$.

Let $\mathbb{F}$ be the projective class of $\Gamma$-free groups in the category ${\mathbf{G}}^{\Gamma}$ of $\Gamma$-groups.

We also recall some results on $\Gamma$-equivariant integral homology [27]. These homology groups are simply denoted $H^{\Gamma}_{n}(G)$ for $H^{\Gamma}_{n}(G,\mathbb{Z})$, the groups G and $\Gamma$ acting trivially on $\mathbb{Z}$.

The Brown - Ellis formula is also obtained extending Hopf formula to higher $\Gamma$-equivariant homology of groups (see [28]).

The consideration of retracts of $\Gamma$-free groups has motivated the following

$Question$: Let G be a $\Gamma$-subgroup of a $\Gamma$-free group. What are reasonable conditions, automatically satisfied in the case of trivial G, under which G is a $\Gamma$-free group? In other words, we are asking whether the well-known Nielsen–Schreier theorem on free groups can be nicely extended to $\Gamma$-free groups.

As G. Janelidze informed me, at least some partial answers, including the counter-example and Theorem 2.12 below, should be known (although we could not find a proper reference):

We take $\Gamma$ = ( 1, $\gamma$) a two-element cyclic group; F to be the free group on a two-element set, say $\{x,y\}\}$ and $f:F\rightarrow\Gamma$ to be the group homomorphism defined by f(x) = $\gamma$ = f(y); define G to be Ker f. We make F a $\Gamma$-group, defining the $\Gamma$-action on it by ${}^{\gamma}x=y$ and ${}^{\gamma}y=x.$ The group $\Gamma$ is a $\Gamma$-group, assuming that $\Gamma$ acts on itself trivially; this makes f a $\Gamma$-group homomorphism, and so G is a $\Gamma$-subgroup of F. After that we observe: (a) G is a free group on a three element set. Indeed, according to Nielsen—Schreier formula, we have rank(G) = (F : G)(rank(F) — 1) + 1 = 2(2 — 1) + 1 = 3. (b) For $\gamma\in\Gamma$ and $t\in F$ , we have ${}^{\gamma}t=t\Rightarrow t=1$. Indeed, if t is a non-empty word, the t and ${}^{\gamma}t$ must begin with different letters. (c) Let B be any basis of G; since it is a basis, 1 doesn’t belongs to B. If B is a $\Gamma$-subset of G, then, presenting it as the disjoint union of $\Gamma$-orbits, we obtain a contradiction between (a) and (b). Indeed, (a) implies that at least one orbit must have exactly one element, which contradicts to (b) since 1 doesn’t belongs to B. That means G is not $\Gamma$-free group.

Since Nielsen - Schreier theorem on free groups doesn’t hold for $\Gamma$-free groups, there is a meaning to study the structure of $\Gamma$-subgroups of $\Gamma$-free groups, in particular to establish conditions for $\Gamma$-subgroups of $\Gamma$-free groups to be $\Gamma$-free groups.

For instance, let F be a $\Gamma$-group and G a $\Gamma$-subgroup of F. The action of $\Gamma$ on F induces an action of $\Gamma$ on the set $G/F=\{Gt|t\in F\}$ of right cosets such that the canonical map $p:F\rightarrow G/F$ is a surjection of $\Gamma$-sets.

As an application of this theorem consider the following simple example: We take $\Gamma=\{1,\gamma,\gamma^{2}\}$ a three-element cyclic group, F to be the $\Gamma$-free group on the three-element set {x,y,z}and defining the $\Gamma$-action on it by ${}^{\gamma}x=y,^{\gamma}y=z,^{\gamma}z=x$. We also take $\mathds{Z}/2\mathds{Z}\times\mathds{Z}/2\mathds{Z}$ to be a $\Gamma$-group assuming that $\Gamma$ acts on it by ${}^{\gamma}(c,0)=(0,c)$ and ${}^{\gamma}(0,c)=(c,c)$ where c denotes the generator of $\mathds{Z}/2\mathds{Z}$, and $f:F\rightarrow\mathds{Z}/2\mathds{Z}\times\mathds{Z}/2\mathds{Z}$ to be a $\Gamma$-group homomorphism defined by f(x) = (c,0), f(y) = (0,c) and f(z) = (c,c). Then $G=Ker(f)$ is a $\Gamma$-free group.

Some results related to this problem are obtained in [49,50].

Finally we provide an assertion establishing relation of the $\Gamma$-equivariant cohomology of groups with the well known equivariant cohomology of topological spaces.

Let G be a $\Gamma$-group and X a topological space on which the groups G and $\Gamma$ are acting such that G is acting properly and ${}^{\gamma}(^{g}x)=^{\gamma}g(^{\gamma}x)$. $\gamma\in\Gamma,g\in G,x\in X.$

We introduce an internal property of $\Gamma$-group extensions possessing a $\Gamma$-section map that will be used through out the paper.

$Proof.$ Let $B_{\Gamma(\tau)}$ denote the normal subgroup of B generated by the elements ${}^{\gamma}gg^{-1}$ such that $\tau(^{\gamma}gg^{-1})=1,\gamma\in\Gamma,g\in G.$ It is evident that $B_{\Gamma(\tau)}$ is a $\Gamma$-subgroup of B and the canonical map $\delta:B\rightarrow B/B_{\Gamma(\tau)}$ is a $\Gamma$-homomorphism.

Let $\alpha:G\rightarrow B$ be a section map for the $\Gamma$-homomorphism $\tau$, then the sequence

is a $\Gamma$-extension of $\Gamma$-groups with $\Gamma$-section map. where $\tau^{\prime}$ is induced by $\tau$. In effect, for that it suffices to show that if ${}^{\sigma}g=^{\gamma}g$ then ${}^{\sigma}(\delta\alpha(g))=^{\gamma}(\delta\alpha(g))$, $\sigma,\gamma\in\Gamma,g\in G$. One has $\tau(^{\sigma}\alpha(G))=^{\sigma}{(\tau\alpha(g))}=^{\sigma}g$ and $\tau(^{\gamma}\alpha(G))=^{\gamma}{(\tau\alpha(g))}=^{\gamma}g$. Thus $\tau(^{\sigma}\alpha(G))=\tau(^{\gamma}\alpha(G))$ and therefore $\tau(^{\gamma^{-1}\sigma}\alpha(g))=^{\gamma^{-1}}\tau(^{\sigma}\alpha(g))=^{\gamma^{-1}}\tau(^{\tau}\alpha(g))=g.$ It follows that ${}^{\gamma^{-1}\sigma}\alpha(g).\alpha(g)^{-1}\in B_{\Gamma(\tau)}$ implying the equality $\delta(^{\gamma^{-1}\sigma}\alpha(g))=\delta(\alpha(g))$ and finally the required equality.

Now assume the sequence E possesses the $\Gamma$-property. Then the equali $B_{\Gamma(\tau)}=1$ implies the isomorphism of the sequences $E\cong E_{\Gamma}$. Conversely let the sequence E satisfies the conditions of the theorem. That means it admits a $\Gamma$-section map $\alpha:G\rightarrow B$ and the group $\Gamma$ acts trivially on $Ker\tau.$ Let $\tau(^{\gamma}b\cdot b^{-1})=1$ for some $b\in B,\gamma\in\Gamma.$ This yields the equality ${}^{\gamma}(\alpha\tau(b))=\alpha\tau(b).$ By using the equality $b=\alpha\tau(b)\cdot c,c\in A$, one obtains ${}^{\gamma}b=^{\gamma}(\alpha\tau(b))\cdot^{\gamma}c=\alpha\tau(b)\cdot c=b.$ This completes the proof.

Example

Let F(G) be the $\Gamma$-free group generated by the $\Gamma$-group G. The short exact sequence of $\Gamma$-groups

where $\tau(\mid g\mid)=g$, has a $\Gamma$-section map $\alpha:G\rightarrow F(G)$, sending any element g to $\mid g\mid$. Then the short sequence of $\Gamma$-groups

is a central $\Gamma$-equivariant extension of G having a $\Gamma$-section map $\eta\alpha$ and $\Gamma$ is trivially acting on $R/[F(G),R]_{\Gamma}$, where $\tau^{\prime}$ is induced by $\tau$ and $\eta:F(G)\rightarrow F(G)/[F(G),R]_{\Gamma}$ is the canonical $\Gamma$-homorphism. Therefore by Theorem 3.2 the sequence (3.1) has the $\Gamma$-property.

If the $\Gamma$-group G is $\Gamma$-perfect the sequence (3.1) yields the following $\Gamma$-extension of G

Since the group $[F(G),F(G)]_{\Gamma}/[F(G),R]_{\Gamma}$ is a $\Gamma$-subgroup of $F(G)/[F(G),R]_{\Gamma}$, the sequence (3.2) also has the $\Gamma$-property and therefore it is a $\Gamma$-extension of G (with $\Gamma$-section map). The sequence (3.2) is the universal central $\Gamma$-equivariant extension of the $\Gamma$-perfect group G and the group $R\cap[F(G),F(G)]_{\Gamma}/[F(G),R]_{\Gamma}$ is isomorphic to $H^{\Gamma}_{2}(G)$ [27]. As we see in this example the $\Gamma$-property has been used substantially.

Now we continue our investigation of $\Gamma$-extensions of $\Gamma$-groups by considering the non-abelian case. Let

be an extension of $\Gamma$-groups having the $\Gamma$-property. By Theorem 3.2 the group $\Gamma$ acts trivially on J and any section map for the sequence (3.3) is a $\Gamma$-map. By conjugation one gets a $\Gamma$-homomorphism $\theta:X\rightarrow AutJ$ implying the $\Gamma$-homorphism $\psi:G\rightarrow AutJ/InJ$, where $\Gamma$ is assuming trivially acting on AutJ and InJ denotes the group of inner automorphisms of J.

$Proof.$ We will follow the classical proof (when the action of $\Gamma$ on G is trivial).

First of all it should be noted that for a given extension (3.3) of the abstract kernel $(G,J,\psi)$ a section map $\alpha:G\rightarrow X$ with $\alpha(1)=1$ induces by conjugation an automorphism $\varphi(x)\in\psi(x),x\in G$ of the group J and maps $f,\mu:G\times G\rightarrow J$ satisfying the well known equalities

$\alpha(x)+l=\varphi(x)(l)+\alpha(x)$,$l\in J,x\in G,$

$\alpha(x)+\alpha(y)=f(x,y)+\alpha(xy),x,y\in G,$

(1) $\varphi(x)(f(x,y))+f(x,yz)=f(x,y)+f(xy,z),z\in G,$

(2) $\varphi(x)\varphi(y)=\mu(f(x,y))\varphi(xy).$

Taking into account the action of $\Gamma$ one has ${}^{\gamma}(\alpha(x)+^{\gamma}l=^{\gamma}(\varphi(x)(l))+^{\gamma}(\alpha(x))$. Therefore $\alpha(^{\gamma}(x))+l=^{\gamma}\varphi(x)(l)+\alpha(^{\gamma}x)$. On the other hand $\alpha(^{\gamma}x)+l=\varphi(^{\gamma}x)(l)+\alpha(^{\gamma}x).$ Thus $\varphi(^{\gamma}x)(l)=^{\gamma}\varphi(x)(l)$ showing that $\varphi$ is a $\Gamma$-map. Similarly it can be proved that the maps f and $\mu$ are $\Gamma$-maps.

Conversely, for given $\Gamma$-maps $\varphi:G\rightarrow Aut(J),f,\mu:G\times G\rightarrow J$ satisfying the equalities (1) and (2), and $\varphi(1)=1,f(1,g)=f(x,1)=0,x\in J,g\in G,$ we can construct an extension of the $\Gamma$-group G with $\Gamma$-property by considering the set $B(J,\varphi,f,G)$ of couples $(x,g),x\in J,g\in G,$ and defining the group structure on it as follows:

$(x,g)+(x_{1},g_{1})=(x+\varphi(g)x_{1}+f(g,g_{1}),gg_{1})$, and $\Gamma$ is acting on $B(J,\varphi,f,G)$ componentwise ${}^{\gamma}(x,g)=(x,^{\gamma}g),\gamma\in\Gamma$.This $\Gamma$-extension is called semi-direct product $\Gamma$-extension of the $\Gamma$-group G by J. It is evident that it has the $\Gamma$-property.

For any $\Gamma$-extension of G by J with $\Gamma$-property and abstract kernel $(G,J,\psi)$ the arising $\Gamma$-maps $\varphi$, f and $\psi$ should satisfy the equalities (1) and (2).That is not the case in general and the obstruction is defined by the equality

(3) $\varphi(x)(f(x,y))+f(x,yz)=k(x,y,z)+f(x,y)+f(xy,z),$

where k(x,y,z) is an element of the center C of J which is the kernel of $\mu.$ It is evident that $k:G\times G\times G\rightarrow C$ is a $\Gamma$-map and it can be proved similarly to the classical case that it is 3-th $\Gamma$-cocycle of the chain complex $C^{*}_{\Gamma}(G,C).$ The $\Gamma$-cocycle k is called obstruction $Ob(G,J,\psi)$ for the abstract kernel $(G,J,\psi).$

Finally, any $\Gamma$-extension of the abstract kernel $(G,J,\psi)$ is equivalent to a semi-direct product $\Gamma$-extension and the set of semi-direct product $\Gamma$-extensions of G by J is bijective to $H^{2}_{\Gamma}(G,C)$. The proof of theses two assertions completely follows the well known case when $\Gamma$ is acting trivially and it is left to the reader. This completes the proof.

As noted in Preliminaries the bijection of the set of $\Gamma$-equivariant extensions of G by A with the second $\Gamma$-equivariant homology group of the $\Gamma$-group G could be extended to higher dimensions. To this aim we need the notion of n-fold $\Gamma$-equivariant extension of G by A which is defined as a long exact sequence of $\Gamma$-groups:

where $0\rightarrow A\rightarrow B_{1}\rightarrow Im\alpha_{1}\rightarrow 0$, $Im\alpha_{i-1}\rightarrow B_{i}\rightarrow Im\alpha_{i}\rightarrow\rightarrow 0$, $2\leq i\leq n-1$, are proper sequences of $\Gamma$-equivariant G-modules and $0\rightarrow Im\alpha_{n-1}\rightarrow B_{n}\rightarrow G\rightarrow e$ is a $\Gamma$-equivariant extension of G by $Im\alpha_{n-1}$.

Let $E^{n}_{\Gamma}(G,A)$, $n\geq 1$, denote the class of equivalence classes of n-fold $\Gamma$-equivariant extensions of G by A. It becomes an abelian group by using the Baer sum operation for all $n\geq 1$.

$Proof.$ We need the following property of the relative derived functors $Ext^{*}_{\Gamma}(A,B)$ of $Hom_{G\propto\Gamma}(A,B)$ defined in the category of $\Gamma$-equivarianr G-modules by using relatively projective $\Gamma$-resolutions with respect to the class $\mathcal{P}$ of proper sequences.

Every proper sequence of $\Gamma$-equivariant G-modules $0\rightarrow A\rightarrow D\rightarrow B\rightarrow 0$ and any $\Gamma$-equivariant G-module L give rise long exact sequences

$0\rightarrow Ext^{0}_{\Gamma}(B,L)\rightarrow Ext^{0}_{\Gamma}(D,L)\rightarrow Ext^{0}_{\Gamma}(A,L))\overset{\theta^{0}}{\rightarrow}Ext^{1}_{\Gamma}(B,L)\rightarrow...\rightarrow Ext^{n}_{\Gamma}(A,L))\overset{\theta^{0}}{\rightarrow}Ext^{n+1}_{\Gamma}(B,L)\rightarrow...,$

$0\rightarrow Ext^{0}_{\Gamma}(L,A)\rightarrow Ext^{0}_{\Gamma}(L,D)\rightarrow Ext^{0}_{\Gamma}(L,B))\overset{\delta^{0}}{\rightarrow}Ext^{1}_{\Gamma}(L,A)\rightarrow...\rightarrow Ext^{n}_{\Gamma}(L,B))\overset{\delta^{0}}{\rightarrow}Ext^{n+1}_{\Gamma}(L,A)\rightarrow....$

The connected sequence of contravariant functors $(Ext^{n}_{\Gamma}(-,A),\theta^{n},n=1,2,...)$ is the right universal sequence of contravariant functors (or right satellite of of the functor $Ext^{1}_{\Gamma}(-,L)$) relative to the class $\mathcal{P}$ of proper sequences of $\Gamma$-equivariant G-modules [26]. In effect, let $(U_{n},\mu^{n},n=1,2,...)$ be a connected sequence of contravariant functors related to the class $\mathcal{P}$ and $f^{1}:Ext^{1}_{\Gamma}(-,A)\rightarrow U^{1}(-)$ be a morphism of functors. It will be shown that there exists a unique extension of the morphism $f^{1}$ to $f^{n}:Ext^{n}_{\Gamma}(-,A)\rightarrow U^{n}(-),n=2,3,...$ compatible with the connecting homomorphisms. To define the extension $f^{2}:Ext^{2}(J,A)\rightarrow U^{2}(J)$ consider the short exact sequence $0\rightarrow L\rightarrow P\rightarrow J\rightarrow 0$, where P is a relatively projective $\Gamma$-equivariant G-module, implying the isomorphism $\theta^{1}:Ext^{1}_{\Gamma}(L,A)\rightarrow Ext^{2}_{\Gamma}(J,A)$. Then the homomorphism $f^{2}$ is given by $\mu^{1}f^{1}(\theta^{1})^{-1}$. It is easily checked that $f^{2}$ is correctly defined, compatible with the connecting homomorphisms $\theta^{1}$ and $\delta^{1}$, and it is the unique extension of $f^{1}$. The extension of $f^{n}$ to $f^{n+1}$ for $n>1$ is constructed similarly.

Now it will be shown that the connected sequence of contravariant functors $(E^{n}_{\Gamma}(-,A),\sigma^{n},n=1,2,...)$ in the category of $\Gamma$-equivariant G-modules is also the right universal sequence of contravariant functors relative to the class $\mathcal{P}$ of proper sequences of $\Gamma$-equivariant G-modules. For the proper sequence $E=0\rightarrow J\rightarrow B\rightarrow L\rightarrow 0$ the connecting homomorphism $\delta^{n}:E^{n}_{\Gamma}(J,A)\rightarrow E^{n+1}_{\Gamma}(L,A)$ is given by $\sigma^{n}([E_{n}])=[E_{n}\otimes E]$, where $E_{n}=0\rightarrow A\rightarrow X_{1}\rightarrow X_{2}\rightarrow...\rightarrow X_{n}\rightarrow J\rightarrow 0$ and $E_{n}\otimes E$ is an n+1- fold extension of L by A obtained by splicing the n-fold extension $E_{n}$ with the proper sequence E.

Let $\varphi^{1}:E^{1}_{\Gamma}(-,A)\rightarrow U^{1}(-)$ be a morphism of functors. Its uniquely defined extension $\varphi^{2}:E^{1}_{\Gamma}(-,A)\rightarrow U^{1}(-)$ is realized as follows: for $[E_{2}]\in Ext^{2}_{\Gamma}(J,A)$, $E_{2}=0\rightarrow A\rightarrow X_{1}\overset{\alpha_{1}}{\rightarrow}X_{2}\rightarrow J\rightarrow 0$, by using connecting homomorphisms with respect to the proper sequence $0\rightarrow Im\alpha_{1}\rightarrow X_{2}\rightarrow J\rightarrow 0$ we define $\varphi^{2}([E_{2}])=\mu^{1}\varphi^{1}([E_{1}])$, where $E^{1}=0\rightarrow A\rightarrow X_{1}\rightarrow Im\alpha_{1}\rightarrow 0$. The extension of $\varphi^{n}$ to $\varphi^{n+1}$ for $n>1$ is constructed in a completely similar way and it is omitted.

We conclude that the isomorphism $f^{1}:Ext^{1}_{\Gamma}(-,A)\rightarrow E^{1}_{\Gamma}(-,A)$ implies the isomorphism of these two right universal sequences of contravariant functors and this yields the isomorphism $Ext^{n}_{\Gamma}(B,A)\cong E^{n}_{\Gamma}(B,A)$ for $n\geq 1$ and $\Gamma$-equivariant G-modules A and B. It follows that there is an isomorphism $h^{1}:E^{1}_{\Gamma}(G,-)\rightarrow E^{2}_{\Gamma}(\mathbb{Z},-)$.

If we consider the functors $E^{n}_{\Gamma}(B,A)$ with respect to the second variable then the sequence $E^{n}_{\Gamma}(\mathbb{Z},-),\eta^{n},n\geq 2$, and the sequence $E^{n}_{\Gamma}(G,-),\lambda^{n},n\geq 1$, are right universal sequences of covariant functors on the category of $\Gamma$-equivariant G-modules with respect to the class $\mathfrak{P}$ of proper sequences of $\Gamma$-equivariant G-modules. The connecting homomorphisms $\eta^{n}$ and $\lambda^{n}$ are defined and the universality is shown similarly to the case of the previous sequence $(E^{n}_{\Gamma}(-,A),\sigma^{n},n=1,2,...)$. Therefore the isomorphism $h^{1}:E^{1}_{\Gamma}(G,-)\rightarrow E^{2}_{\Gamma}(\mathbb{Z},-)$ induces isomorphisms $h^{n}:E^{n}_{\Gamma}(G,-)\rightarrow E^{n+1}_{\Gamma}(\mathbb{Z},-)$ for all $n\geq 1$. It remains to apply the isomorphism $E^{n+1}_{\Gamma}(\mathbb{Z},-)\rightarrow Ext^{n+1}_{\Gamma}(\mathbb{Z},-)$. This completes the proof.

It is reasonable to ask for the computation of the $\Gamma$-equivariant (co)homology of groups introduced in [27]. As noted above $\Gamma$-equivariant homology and cohomology groups of retracts of $\Gamma$-free groups are trivial for $n>1$. Here we provide an attempt to the investigation of this problem for finite cyclic $\Gamma$-groups.

It is well known that for the computation of the (co)homology of finite cyclic groups the following free resolution of $\mathds{Z}$ is used:

where $\mathrm{\mathds{Z}_{m}}$ is a finite cyclic group of order m and generator t, D = t - 1 and $N=1+t+t^{2}+...+t^{m-1}$. Assume now that a group $\Gamma$ is acting on $\mathds{Z}_{m}$ with trivial action on $\mathds{Z}$. Similarly to the action of $\Gamma$ on the bar resolution of $\mathds{Z}$ it induces an action of $\Gamma$ on the resolution (4.1) of $\mathds{Z}$. It is easily checked that the homomorphism D is compatible with trivial action of $\Gamma$ only. This case is not interesting, since it is reduced to the usual (co)homology of groups. Therefore the resolution (4.1) is unsuitable for the computation of $\Gamma$-equivariant (co)homology of finite cyclic $\Gamma$-groups.

By slightly changing the value of D we are able to compute the rational $\Gamma$-equivariant (co)homology of the finite cyclic $\Gamma$-group $\mathds{Z}_{m}$.

Let $B_{*}\otimes\mathds{Q}\rightarrow\mathds{Q}$ be the bar resolution of the field $\mathds{Q}$ of rational numbers obtained by tensoring the bar resolution $B_{*}\rightarrow\mathds{Z}$ of $\mathds{Z}$ by $\mathds{Q}$, where $B_{0}\otimes\mathds{Q}\cong\mathds{Q}(G)$, and $B_{n},n\geq 1,$ is the free $\mathds{Q}(G)$- module generated by the elements $[g_{1},g_{2},...,g_{n}],g_{i}\in G$.