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

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.