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

TL;DR

This paper develops a comprehensive {\Gamma}-homological algebra framework, extending classical concepts to {\Gamma}-groups, and explores their applications in algebraic K-theory, Galois theory, and group cohomology.

Contribution

It introduces {\Gamma}-equivariant (co)homology, Hochschild homology, and extends non-abelian extension theory to {\Gamma}-groups, providing new tools and results in homological algebra.

Findings

Computed rational {\Gamma}-equivariant (co)homology of finite cyclic {\Gamma}-groups.

Proved isomorphism between {\Gamma}-equivariant extension groups and cohomology groups.

Established properties of {\Gamma}-equivariant Hochschild homology related to Kähler differentials and Morita equivalence.

Abstract

This is a further investigation of our approach to group actions in homological algebra in the settings of homology of {\Gamma}-simplicial groups, particularly of {\Gamma}-equivariant homology and cohomology of {\Gamma}-groups. This approach could be called {\Gamma}-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 {\Gamma}-extensions of {\Gamma}-groups. We compute the rational {\Gamma}-equivariant (co)homology groups of finite cyclic {\Gamma}-groups. The isomorphism of the group of n-fold {\Gamma}-equivariant extensions of a {\Gamma}-group G by a G o {\Gamma}-module A with the (n+1)th {\Gamma}-equivariant group cohomology of G with coefficients in A is proven.We define the {\Gamma}-equivariant Hochschild homology as the homology of the {\Gamma}- Hochschild complex involving the cyclic homology when the basic ring contains rational numbers and generalizing the {\Gamma}equivariant(co)homology of {\Gamma}-groups when the action of the group {\Gamma} on the Hochschild complex is induced by its action on the basic ring. Important properties of the {\Gamma}-equivariant Hochschild homology related to Kahler differentials, Morita equivalence and derived functors are established. Group (co)homology and {\Gamma}-equivariant group (co)homology of crossed {\Gamma}-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.