Cohomologie et homologie non abéliennes des groupes. (Nonabelian cohomology and homology of groups) (Q1107637)

Let G be a group and A a crossed G-module i.e. A is a group on which G operates and there is given a homomorphism \(\delta\) : \(G\to A\) satisfying \(\delta\) ( \(ga)=g\delta (a)g^{-1}\) and \(^{\delta (a)}(a')=aa'a^{- 1}\) for all a,a'\(\in A\), \(g\in G\). The usual notion of derivation from A to B where A, B are non-Abelian groups on which the group G operates is suitably modified to define derivations from crossed G-module (A,\(\rho)\) to crossed G-module (B,\(\mu)\) so that a product is defined in \(Der_ G(A,B)\)- the set of derivations from (A,\(\delta)\) to (B,\(\mu)\) to make it a group. Two derivations (\(\alpha\),g), \((\beta,h)\in Der_ G(A,B)\) are said to be equivalent if there exists \(a\in A\) such that for every \(x\in G\), \(\beta (x)=a^{-1}\alpha (x)\) xa. This becomes an equivalence relation `\(\sim '\) and the group \(Der_ G(G,A)/\sim\) is called the first cohomology group of G with coefficients in the crossed G-module (A,\(\delta)\). This group becomes isomorphic to the usual first cohomology group H 1(G,A) when A is Abelian. Corresponding to a short exact sequence \(1\to (A,1)\to (B,\mu)\to (C,\lambda)\to 1\) of crossed G-modules, a seven term exact sequence of groups \[ 1\quad \to \quad H\quad 0(G,A)\quad \to \quad H\quad 0(G,B)\quad \to \quad H\quad 0(G,C)\quad \to \quad H\quad 1(G,A)\quad \to \quad H\quad 1(G,B)\quad \to \quad H\quad 1(G,C)\quad \to^{\Delta}\quad H\quad 2(G,A) \] (where H \(0(G,X)=X\) G the group of fixed elements of X) is obtained. The notion of tensor product of non- Abelian groups defined by \textit{R. Brown} and \textit{J.-L. Loday} [C. R. Acad. Sci., Paris, Sér. I 298, 353-356 (1984; Zbl 0573.55011) and Topology 26, 311-335 (1987; Zbl 0622.55009)] is generalized to give a tensor product of pre-crossed G-modules. Using the tensor product \(G\otimes A\), when G is a group and A a crossed G-module, homology groups \(H_ 0(G,A)\) and \(H_ 1(G,A)\) are defined. A six term exact sequence in these homology groups is obtained corresponding to a short exact sequence of crossed G-modules which is then used to obtain a certain relation in the algebraic K-theory of non-commutative local rings.