# Cyclic contractions of dimer algebras always exist

## TL;DR

This paper proves that all nondegenerate dimer algebras on a torus can be cyclically contracted to cancellative dimer algebras, revealing key properties like Calabi-Yau conditions and center dimension.

## Contribution

It establishes the existence of cyclic contractions for nondegenerate dimer algebras on a torus, linking algebraic properties to geometric structures.

## Key findings

- Nondegenerate dimer algebras admit cyclic contractions to cancellative dimer algebras
- Calabi-Yau property is equivalent to being noetherian for these algebras
- Center of the algebra has Krull dimension 3

## Abstract

We show that every nondegenerate dimer algebra $A$ on a torus admits a cyclic contraction to a cancellative dimer algebra. This implies, for example, that $A$ is Calabi-Yau if and only if it is noetherian; and that the center of $A$ has Krull dimension $3$.

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Source: https://tomesphere.com/paper/1703.04450