Homology of Steinberg algebras

TL;DR

This paper investigates the homological invariants of Steinberg algebras associated with ample groupoids, providing explicit computations of Hochschild and cyclic homology in various cases and relating them to groupoid and group homology.

Contribution

It offers new formulas for Hochschild and cyclic homology of Steinberg algebras, linking them to groupoid homology and extending to self-similar group actions.

Findings

Hochschild and cyclic homology computed for principal or Hausdorff groupoids.

Groupoid homology appears as a direct summand of Hochschild homology.

Established a connection between K-theory of the algebra and groupoid homology.

Abstract

We study homological invariants of the Steinberg algebra $\mathcal{A}_k(\mathcal{G})$ of an ample groupoid $\mathcal{G}$ over a commutative ring $k$. For $\mathcal{G}$ principal or Hausdorff with ${\mathcal{G}}^{\rm{Iso}}\setminus{\mathcal{G}}^{(0)}$ discrete, we compute Hochschild and cyclic homology of $\mathcal{A}_k(\mathcal{G})$ in terms of groupoid homology. For any ample Hausdorff groupoid $\mathcal{G}$, we find that $H_*(\mathcal{G})$ is a direct summand of $HH_*(\mathcal{A}_k(\mathcal{G}))$; using this and the Dennis trace we obtain a map $\overline{D}_*:K_*(\mathcal{A}_k(\mathcal{G}))\to H_n(\mathcal{G},k)$. We study this map when $\mathcal{G}$ is the (twisted) Exel-Pardo groupoid associated to a self-similar action of a group $G$ on a graph, and compute $HH_*(\mathcal{A}_k(\mathcal{G}))$ and $H_*(\mathcal{G},k)$ in terms of the homology of $G$, and the $K$-theory of $\mathcal{A}_k(\mathcal{G})$ in terms of that of $k[G]$.