Cyclic homology of crossed products

TL;DR

This paper introduces a simplified mixed complex for computing various cyclic homologies of crossed product algebras involving Hopf algebras and cocycles, extending to relative cyclic homology with spectral sequence applications.

Contribution

It develops a more straightforward mixed complex for Hochschild and cyclic homology of crossed products, generalizes previous spectral sequence results, and handles relative cyclic homology with a subalgebra K.

Findings

A new simplified mixed complex for cyclic homology calculations.

Two spectral sequences converging to the cyclic homology of crossed products.

Generalization of existing spectral sequences when cocycles take values in K.

Abstract

We obtain a mixed complex, simpler that the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E=A#fH, where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. Actually, we work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homology of E relative to K. The first one in the general setting and the second one (which generalizes those previously found by several authors) when f takes its values in K.