The periodic cyclic homology of crossed products of finite type algebras

TL;DR

This paper investigates the periodic cyclic homology of crossed product algebras, establishing a complete description for commutative cases with finite groups and relating it to orbifold cohomology, with insights into noncommutative cases.

Contribution

It provides a comprehensive analysis of cyclic homology for crossed products of finite type algebras, linking it to orbifold cohomology in the commutative case and exploring noncommutative examples.

Findings

Complete description of cyclic homology for commutative algebras with finite groups

Identification of cyclic homology with orbifold cohomology in certain cases

Examples showing limitations of the orbifold cohomology identification

Abstract

We study the periodic cyclic homology groups of the cross-product of a finite type algebra $A$ by a discrete group $\Gamma$. In case $A$ is commutative and $\Gamma$ is finite, our results are complete and given in terms of the singular cohomology of the strata of fixed points. These groups identify our cyclic homology groups with the \dlp orbifold cohomology\drp\ of the underlying (algebraic) orbifold. The proof is based on a careful study of localization at fixed points and of the resulting Koszul complexes. We provide examples of Azumaya algebras for which this identification is, however, no longer valid. As an example, we discuss some affine Weyl groups.