# 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.

## Key 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 $A$.

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Source: https://tomesphere.com/paper/1902.11299