The cyclic homology of crossed product algebras

Abstract

In their article [9] on cyclic homology, Feigin and Tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra, generalizing Burghelea’s calculation [4] of the cyclic homology of a group algebra. For an analogous spectral sequence for the Hochschild homology of a crossed product algebra, see Brylinski [2], [3]. In this article, we give a new derivation of this spectral sequence, and generalize it to negative and periodic cyclic homology HC• (A) and HP•(A). The method of proof is itself of interest, since it involves a natural generalization of the notion of a cyclic module, in which the condition that the morphism τ ∈ Λ(n,n) is cyclic of order n + 1 is relaxed to the condition that it be invertible. We call this category the paracyclic category. Given a paracyclic module P , we can define a chain complex C(P ), with differentials b and B, which respectively lower and raise degree. The condition that the module P is paracyclic translates to the condition on C(P ) that 1− (bB +Bb) is invertible. Our main result is to show that there is an analogue of the Eilenberg-Zilber theorem for bi-paracyclic modules. It is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of Feigin and Tsygan. If M is a module over a commutative ring k, we will denote by M (k) the iterated tensor product, defined by M (0) = k and M (k+1) =M (k) ⊗M . If M and N are graded modules, we will denote by M ⊗N their graded tensor product. We would like to thank J. Block, J.-L. Brylinski and C. Kassel for a number of interesting discussions on the results presented here.