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Let A be a nondegenerate dimer (or ghor) algebra on a torus, and let Z be its center. Using cyclic contractions, we show the following are equivalent: A is noetherian; Z is noetherian; A is a noncommutative crepant resolution; each arrow of A is contained in a perfect matching whose complement supports a simple module; and the vertex corner rings e i A e i are pairwise isomorphic. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: A superpotential algebra is square if its quiver admits an embedding into a two-torus such that the image of its underlying graph is a square grid, possibly with diagonal edges in the unit squares; examples are provided by dimer models in physics. Such an embedding reveals much of the algebras representation theory through a device we introduce called an impression. Let A be a square superpotential algebra, Z its center, and the maximal ideal at the origin of Spec Z. Using an impression, we [•] give a classification of all simple A-modules up to isomorphism, and give algebraic and homological characterizations of the simple A-modules of maximal k-dimension; [•] show that Z is a 3-dimensional normal toric domain and is Gorenstein, by determining transcendence bases and Z-regular sequences; and [•] show that is a noncommutative crepant resolution of , and thus a local Calabi–Yau algebra. A particular class of square superpotential algebras, the algebras, is considered in detail. We show that the Azumaya and smooth loci of the centers coincide, and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size. [Copyright &y& Elsevier]