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.