Partial Actions of Free Groups and Groupoid Homology

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 ) $(X,T)$ left parenthesis upper X comma upper T right parenthesis , where X X upper X is compact Hausdorff and totally disconnected, and T T upper T is a local homeomorphism with domain a clopen subset of X X upper X . We also show that algebra of such a partial action groupoid over a field has global dimension at most 2 $2$ 2 when the space is second countable.