Modules over etale groupoid algebras as sheaves

TL;DR

This paper establishes an equivalence between modules over etale groupoid algebras and sheaves over the groupoid, providing new insights into their structure and Morita equivalence implications.

Contribution

It proves that the category of modules over etale groupoid algebras is equivalent to sheaves over the groupoid, offering a new proof of Morita equivalence results.

Findings

Category of modules is equivalent to sheaves over the groupoid

Provides a new proof for Morita equivalence of groupoid algebras

Includes various algebra classes like Leavitt path algebras

Abstract

The author has previously associated to each commutative ring with unit $\Bbbk$ and \'etale groupoid $\mathscr G$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk$-algebra $\Bbbk\mathscr G$. The algebra $\Bbbk\mathscr G$ need not be unital, but it always has local units. The class of groupoid algebras includes group algebras, inverse semigroup algebras and Leavitt path algebras. In this paper we show that the category of unitary $\Bbbk\mathscr G$-modules is equivalent to the category of sheaves of $\Bbbk$-modules over $\mathscr G$. As a consequence we obtain a new proof of a recent result that Morita equivalent groupoids have Morita equivalent algebras.