Geometric description of the Hochschild cohomology of group algebras

Abstract

There are two approaches to the study of the cohomology of group algebras R [ G ] R[G] , the Eilenberg–MacLane cohomology and the Hochschild cohomology. The Eilenberg–MacLane cohomology gives the classical cohomology of the classifying space B G BG (or the Eilenberg–MacLane complex K ( G , 1 ) K(G,1) ). Note that the space B G BG can be interpreted as a classifying space of the groupoid of the trivial action of the group G G . The Hochschild cohomology is a more general construction, which considers the so-called bimodules of the algebra R [ G ] R[G] and their derivative functors Ext ⁡ ( R [ G ] , M ) \operatorname {Ext}(R[G],M) , for which no geometric interpretation has been known so far. The key point for calculating the Hochschild cohomology H H ∗ ( R [ G ] ) HH^*(R[G]) is the new groupoid G r Gr associated with the adjoint action of the group G G . For this groupoid, the classical cohomology of the corresponding classification space B G r BGr with the finiteness condition for the supports of cochains is isomorphic to the Hochschild cohomology of the algebra R [ G ] R[G] : H H ∗ ( R [ G ] ) ≈ H f ∗ ( B G r ) . \begin{equation*} HH^*(R[G])\approx H^*_f(BGr). \end{equation*} This result represents a fundamental contribution to understanding the geometry of the cohomological properties of group algebras, in particular, understanding the differences between the homology and cohomology of group algebras. The paper is devoted to the motivation of the Hochschild (co)homology group of the group algebra R [ G ] R[G] and its description in terms of the classical (co)homology of the classifying space of the groupoid of the adjoint action of the original group G G under a suitable finiteness assumption on the supports of the cohomology group.