Homology of Steinberg algebras

TL;DR

This work develops a comprehensive framework for computing homological and $K$-theoretic invariants of Steinberg algebras ${\mathcal A}_k(\mathcal G)$ attached to ample groupoids by relating Hochschild and cyclic homology to groupoid homology $H_*(\mathcal G, -)$, including twisted (via 2-cocycles) and graded variants. It provides explicit chain-level descriptions of Hochschild complexes, establishes cyclic-nerve decompositions, and derives direct-sum decompositions of homology in terms of isotropy data, especially for Hausdorff groupoids. The paper then specializes to Exel–Pardo groupoids coming from self-similar group actions on graphs, giving detailed computations of $HH_*$, twisted groupoid homology, and $K$-theory via cones of explicit maps, together with Dennis trace compatibilities and Farrell–Jones-type results. A central theme is discretization: comparing universal and discretized groupoids, proving non-invariance of Hochschild homology in general, but identifying conditions under which discretization invariance holds for a broad class of functors, notably $KH$/$K$-theory in EP settings. Overall, the results deliver a robust toolkit for computing and relating algebraic invariants of Steinberg algebras to dynamical and combinatorial data of the underlying groupoids and graphs, with potential applications to index theory and noncommutative geometry.

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 $k[G]$.

Theorems & Definitions (146)

• Theorem 1.1
• Theorem 1.5
• Theorem 1.9
• Conjecture 1
• Example 2.2.1
• Example 2.2.2
• Example 2.3.3
• Remark 2.3.4
• Lemma 2.3.5
• proof