# \'Etale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings

## TL;DR

This paper establishes a correspondence between skew inverse semigroup rings and convolution algebras of ample groupoids with sheaf coefficients, generalizing known results for Steinberg algebras over fields.

## Contribution

It introduces a new framework linking skew inverse semigroup rings with groupoid convolution algebras with sheaf coefficients, extending existing algebraic structures.

## Key findings

- Skew inverse semigroup rings can be realized as groupoid convolution algebras with sheaf coefficients.
- Convolution algebras of ample groupoids are isomorphic to certain skew inverse semigroup rings.
- Known results for Steinberg algebras over fields are recovered as special cases.

## Abstract

Given an action $\varphi$ of of inverse semigroup $S$ on a ring $A$ (with domain of $\varphi(s)$ denoted by $D_{s^*}$) we show that if the ideals $D_e$, with $e$ an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.