Morita equivalences and Azumaya loci from Higgsing dimer algebras

Let ψ:A→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.