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

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 new direction of homological algebra is called \(\Gamma \)-homological algebra. The abstract kernel of non-abelian extensions of groups, its relationship with the obstruction theory, 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 \(G\,{\rtimes }\, \Gamma \)-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 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 Kähler 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. Relations with extensions of crossed \(\Gamma \)-modules, in particular with relative extensions of group epimorphisms in the sense of Loday are established. Universal and \(\Gamma \)-equivariant universal central \(\Gamma \)-extensions of \(\Gamma \)-perfect crossed \(\Gamma \)-modules are constructed and relevant Hopf formulas are obtained. Finally, applications to the algebraic K-theory, the Galois theory of commutative rings, and the cohomological dimension of groups are given.