Chain conditions on \'etale groupoid algebras with applications to Leavitt path algebras and inverse semigroup algebras

TL;DR

This paper characterizes when étale groupoid algebras are Noetherian or Artinian, extending known results to broader classes like Leavitt path algebras and inverse semigroup algebras over arbitrary rings.

Contribution

It provides new criteria for Noetherian and Artinian properties of étale groupoid algebras, generalizing previous characterizations to arbitrary coefficient rings.

Findings

Characterization of Noetherian étale groupoid algebras

Characterization of Artinian étale groupoid algebras

Extension of Leavitt path algebra classifications

Abstract

The author has previously associated to each commutative ring with unit $R$ and \'etale groupoid $\mathscr G$ with locally compact, Hausdorff and totally disconnected unit space an $R$-algebra $R\mathscr G$. In this paper we characterize when $R\mathscr G$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda~Pino and Siles~Molina of finite dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okni\'nski of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.