Cohomology of ample groupoids

Abstract

We introduce a cochain complex for ample groupoids $\mathcal G$ using a flat resolution defining their homology with coefficients in $\mathbb Z$. We prove that the cohomology of this cochain complex with values in a $\mathcal G$-module $M$ coincides with the previously introduced continuous cocycle cohomology of $\mathcal G$. In particular, this groupoid cohomology is invariant under Morita equivalence. We derive an exact sequence for the cohomology of skew products by a $\mathbb Z$-valued cocycle. We indicate how to compute the cohomology with coefficients in a $\mathcal G$-module $M$ for $AF$-groupoids and for certain action groupoids.