Twists, crossed products and inverse semigroup cohomology

TL;DR

This paper establishes a correspondence between extensions of ample groupoids and Boolean inverse semigroups, classifies twists via Lausch cohomology, and introduces a new inverse semigroup crossed product concept.

Contribution

It links groupoid extensions to inverse semigroup cohomology, classifies twists with Lausch's cohomology, and defines a new crossed product framework for twisted Steinberg algebras.

Findings

Extensions of ample groupoids correspond to Boolean inverse semigroup extensions.

Discrete twists are classified by Lausch's second cohomology group.

Twisted Steinberg algebras are examples of inverse semigroup crossed products.

Abstract

Twisted \'etale groupoid algebras have been studied recently in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper, we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups and we show that they are classified by Lausch's second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists. We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.