Homology of Steinberg algebras

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 ${\mathcal{G}}^{\mathrm{Iso}}\setminus {\mathcal{G}}^{(0)}$ 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 $H_{\ast }(\mathcal{G})$ is a direct summand of $H{H}_{\ast }({\mathcal{A}}_k(\mathcal{G}))$; using this and the Dennis trace we obtain a map ${\overline{D}}_{\ast }:K_{\ast }({\mathcal{A}}_k(\mathcal{G}))\to H_n(\mathcal{G},k)$. 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 $H{H}_{\ast }({\mathcal{A}}_k(\mathcal{G}))$ and $H_{\ast }(\mathcal{G},k)$ in terms of the homology of $G$, and the $K$-theory of ${\mathcal{A}}_k(\mathcal{G})$ in terms of that of $k[G]$.

53 pages. References added in second version and minor corrections in the third. Fourth version fixes a mistake in Theorem 1.1, and adds some minor structural changes in the preliminary sections