Chain Conditions for Etale Groupoid Algebras

TL;DR

This paper characterizes chain conditions such as noetherian, artinian, and semisimple properties for etale groupoid algebras, extending known results for Leavitt path algebras using categorical methods.

Contribution

It provides a categorical characterization of chain conditions for etale groupoid algebras, generalizing prior results for specific algebra classes.

Findings

Characterization of noetherian and artinian etale groupoid algebras

Identification of conditions for semisimplicity

Extension of results from Leavitt path algebras

Abstract

Let $R$ be a unital commutative ring with unit and $\mathscr{G}$ an ample groupoid. Using the topology of the groupoid $\mathscr{G}$, Steinberg defined an etale groupoid algebra $R\mathscr{G}$. These etale groupoid algebras generalize various algebras including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for etale groupoid algebras. We characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple etale groupoid algebras, generalizing existing results for Leavitt path algebras.