On the central geometry of nonnoetherian dimer algebras

TL;DR

This paper investigates the geometric properties of the center of nonnoetherian dimer algebras on a torus, revealing its Krull dimension, local noetherianity, and singularity structure.

Contribution

It demonstrates that the center has Krull dimension 3, is locally noetherian on a dense set, and its reduced center exhibits a Gorenstein singularity with a unique positive-dimensional closed point.

Findings

Center has Krull dimension 3.

Center is locally noetherian on an open dense set.

Reduced center has a Gorenstein singularity with one positive-dimensional closed point.

Abstract

Let $Z$ be the center of a nonnoetherian dimer algebra on a torus. Although $Z$ itself is also nonnoetherian, we show that it has Krull dimension $3$, and is locally noetherian on an open dense set of $\operatorname{Max}Z$. Furthermore, we show that the reduced center $Z/\operatorname{nil}Z$ is depicted by a Gorenstein singularity, and contains precisely one closed point of positive geometric dimension.