Partial actions of free groups and groupoid homology

Abstract

We give a length one projective resolution of the trivial module for the groupoid of a semi-saturated partial action (in the sense of Exel) of a free group on a compact Hausdorff and totally disconnected space. As a consequence we obtain an elementary computation of the homology of these groupoids, which include transformation groupoids of free group actions and Deaconu-Renault groupoids of systems $(X,T)$ where $X$ is compact Hausdorff and totally disconnected and $T$ is a local homeomorphism with domain a clopen subset of $X$. We also show that algebra of such a partial action groupoid over a field has global dimension at most $2$ when the space is second countable.