Noetherian criteria for dimer algebras

TL;DR

This paper establishes equivalent conditions for nondegenerate dimer algebras on a torus, linking their noetherian property, center, and module-theoretic features, using cyclic contractions.

Contribution

It provides new criteria for the noetherian property of dimer algebras, connecting algebraic, geometric, and module-theoretic aspects.

Findings

A is noetherian if and only if Z is noetherian.

A is a noncommutative crepant resolution under these conditions.

All vertex corner rings are pairwise isomorphic.

Abstract

Let $A$ be a nondegenerate dimer (or ghor) algebra on a torus, and let $Z$ be its center. Using cyclic contractions, we show the following are equivalent: $A$ is noetherian; $Z$ is noetherian; $A$ is a noncommutative crepant resolution; each arrow of $A$ is contained in a perfect matching whose complement supports a simple module; and the vertex corner rings $e_iAe_i$ are pairwise isomorphic.