The central nilradical of nonnoetherian dimer algebras

TL;DR

This paper investigates the structure of the center of nonnoetherian dimer algebras on a torus, revealing properties of its nilradical, irreducibility of the spectrum, and relations to the ghor algebra's center.

Contribution

It characterizes the nilradical of the center, shows the irreducibility of the spectrum, and relates the reduced center to the ghor algebra's center, providing new insights into their structure.

Findings

The nilradical of the center is prime and can be nonzero.

The spectrum of the center is irreducible.

The reduced center embeds into the ghor algebra's center and their normalizations coincide.

Abstract

Let $Z$ be the center of a nonnoetherian dimer algebra $A$ on a torus. We show that the nilradical $\operatorname{nil}Z$ of $Z$ is prime, may be nonzero, and consists precisely of the central elements that vanish under a cyclic contraction of $A$. This implies that the nonnoetherian scheme $\operatorname{Spec}Z$ is irreducible. We also show that the reduced center $\hat{Z} = Z/\operatorname{nil}Z$ embeds into the center $R$ of the corresponding ghor algebra, and that their normalizations are equal. Finally, we give three characterizations of the normality of $R$, and show that if $\hat{Z}$ is normal, then it has the special form $k + J$ where $J$ is an ideal of the cycle algebra of $A$.