Cohomology of ample groupoids

TL;DR

This paper develops a new cohomology theory for ample groupoids using a cochain complex, showing its equivalence to existing theories and providing tools for computation in specific cases.

Contribution

It introduces a cochain complex for ample groupoids that aligns with continuous cocycle cohomology and proves invariance under Morita equivalence.

Findings

Cohomology coincides with continuous cocycle cohomology

Provides an exact sequence for skew product cohomology

Offers methods to compute cohomology for AF-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.