On the noncommutative geometry of square superpotential algebras

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 m the maximal ideal at the origin of SpecZ. 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 Zm is Gorenstein, by determining transcendence bases and Z-regular sequences; and•show that Am is a noncommutative crepant resolution of Zm, and thus a local Calabi–Yau algebra.A particular class of square superpotential algebras, the Yp,q 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.