Equivariant periodic cyclic homology for ample groupoids

TL;DR

This paper develops a new equivariant periodic cyclic homology theory for ample groupoids, demonstrating key properties like homotopy invariance, stability, excision, and establishing an analogue of the Green-Julg theorem.

Contribution

It introduces a bivariant equivariant cyclic homology framework for ample groupoids, extending classical results to this broader setting.

Findings

The theory satisfies homotopy invariance, stability, and excision.

An analogue of the Green-Julg theorem is proved for proper groupoid actions.

The framework generalizes known results from group actions to ample groupoids.

Abstract

We define and study bivariant equivariant periodic cyclic homology for actions of ample groupoids. In analogy to the group case, we show that the theory satisfies homotopy invariance, stability, and excision in both variables. We also prove an analogue of the Green-Julg theorem for actions of proper groupoids.