On the homology of graded algebras

Abstract

Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies of S have canonical decompositions with components labeled by the set T(G) of conjugacy classes of G : for Hochschild homology and similarly for cyclic homology. In this article, we describe the components of Hochschild homology in the case where S is strongly G graded.The description is given in terms of a spectral sequence where Hq (R,Sg ) is the Hochschild homology of the identity component R = Se of S with coefficients in the bimodule Sg and Hp (CG (g),.) is the group homology of the centralizer CG (g) of g in G. If R is a separable algebra then the spectral sequence degenerates and yields an isomorphism