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 \mathfrak{m} 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 Z_{\mathfrak{m}} is Gorenstein, by determining transcendence bases and Z-regular sequences; and show that A_{\mathfrak{m}} is a noncommutative crepant resolution of Z_{\mathfrak{m}}, and thus a local Calabi-Yau algebra. A particular class of square superpotential algebras, the Y^{p,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.Comment: 53 pages. Final version. To appear in J. Algebr

Communicated by P. SmithWe introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of A are all isomorphic if and only if A is noetherian, if and only if the center Z of A is noetherian, if and only if A is a finitely generated Z-module. Furthermore, we show that Z is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.

We show that every nondegenerate dimer algebra on a torus admits a cyclic contraction to a cancellative dimer algebra. This implies, for example, that a nondegenerate dimer algebra is Calabi-Yau if and only if it is noetherian, if and only if its center is noetherian; and the Krull dimension of the center of every nondegenerate dimer algebra (on a torus) is 3.2010 Mathematics Subject Classification. 13C15, 14A20.