Chain Conditions for Etale Groupoid 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.