Correlation between the Hochschild Cohomology and the Eilenberg–MacLane Cohomology of Group Algebras from a Geometric Point of View

Abstract

There are two approaches to the study of the cohomology of group algebras ℝ[G]: the Eilenberg-MacLane cohomology and the Hochschild cohomology. In the case of Eilenberg-MacLane cohomology, one has the classical cohomology of the classifying space BG. The Hochschild cohomology represents a more general construction, in which the so-called two-sided bimodules are considered. The Hochschild cohomology and the usual Eilenberg-MacLane cohomology are coordinated by moving from bimodules to left modules. For the Eilenberg-MacLane cohomology, in the case of a nontrivial action of the group G on the module Ml, no reasonable geometric interpretation has been known so far. The main result of this paper is devoted to an effective geometric description of the Hochschild cohomology. The key point for the new geometric description of the Hochschild cohomology is the new groupoid Gr associated with the adjoint action of the group G. The cohomology of the classifying space BGr of this groupoid with an appropriate condition for the finiteness of the support of cochains is isomorphic to the Hochschild cohomology of the algebra ℝ[G]. Hochschild homology is described in the form of homology groups of the space BGr, but without any conditions of finiteness on chains.