(Co)homology of Γ-groups and Γ-homological algebra

Abstract

This is a further investigation of our approach to group actions in homological algebra in the settings of homology of Γ-simplicial groups, particularly of Γ-equivariant homology and cohomology of Γ-groups. This approach could be called Γ-homological algebra. The abstract kernel of non-abelian extensions of groups, its relation with the obstruction to the existence of non-abelian extensions and with the second group cohomology are extended to the case of non-abelian Γ-extensions of Γ-groups. We compute the rational Γ-equivariant (co)homology groups of finite cyclic Γ-groups. The isomorphism of the group of n-fold Γ-equivariant extensions of a Γ-group G by a G o Γ-module A with the (n+1)th Γ-equivariant group cohomology of G with coefficients in A is proven.We define the Γ-equivariant Hochschild homology as the homology of the Γ- Hochschild complex involving the cyclic homology when the basic ring contains rational numbers and generalizing the Γequivariant(co)homology of Γ-groups when the action of the group Γ on the Hochschild complex is induced by its action on the basic ring. Important properties of the Γ-equivariant Hochschild homology related to Kahler differentials, Morita equivalence and derived functors are established. Group (co)homology and Γ-equivariant group (co)homology of crossed Γ-modules are introduced and investigated by using relevant derived functors Finally, applications to algebraic K-theory, Galois theory of commutative rings and cohomological dimension of groups are given.