Morita equivalences and Azumaya loci from Higgsing dimer algebras

TL;DR

This paper explores Morita equivalences between dimer algebras related by cyclic contraction, characterizes their Azumaya loci, and provides new insights into the structure of nonnoetherian algebras with coinciding smooth and Azumaya loci.

Contribution

It introduces Morita equivalences linking the representation theories of non-cancellative and cancellative dimer algebras and characterizes their Azumaya loci.

Findings

Morita equivalences relate the representation theories of A and A'

Characterization of the Azumaya locus of A in terms of A'

Identification of conditions where smooth and Azumaya loci coincide

Abstract

Let $\psi: A \to A'$ be a cyclic contraction of dimer algebras, with $A$ non-cancellative and $A'$ cancellative. $A'$ is then prime, noetherian, and a finitely generated module over its center. In contrast, $A$ is often not prime, nonnoetherian, and an infinitely generated module over its center. We present certain Morita equivalences that relate the representation theory of $A$ with that of $A'$. We then characterize the Azumaya locus of $A$ in terms of the Azumaya locus of $A'$, and give an explicit classification of the simple $A$-modules parameterized by the Azumaya locus. Furthermore, we show that if the smooth and Azumaya loci of $A'$ coincide, then the smooth and Azumaya loci of $A$ coincide. This provides the first known class of algebras that are nonnoetherian and infinitely generated modules over their centers, with the property that their smooth and Azumaya loci coincide.