Homology of \'etale groupoids, a graded approach

TL;DR

This paper develops a graded homology theory for étale groupoids, linking it to algebraic invariants of graph algebras and extending previous results to broader classes of graphs.

Contribution

It introduces a graded homology framework for étale groupoids and establishes a connection to the graded Grothendieck group of Leavitt path algebras, extending prior work.

Findings

Graded zeroth homology is isomorphic to the graded Grothendieck group of Leavitt path algebras.

The graded zeroth homology serves as a complete invariant for eventual conjugacy of shifts of finite type.

The results unify algebraic and analytic invariants in graph algebra theory.

Abstract

We introduce a graded homology theory for graded \'etale groupoids. For $\mathbb Z$-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients $\mathbb Z$ is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.