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 \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

Nonnoetherian geometry

“…Again consider the nonnoetherian algebra R = k + xS in example (1). In Example 4.3 and Proposition 4.4, we study the endomorphism ring…”

“…From this perspective, Z(x) is a one-dimensional "smeared-out" point of R, and therefore Max R is nonlocal. More generally, let S be an integral domain and a finitely generated k-algebra, and let R be a possibly nonnoetherian subalgebra of S. In order to capture the locus where Max R "looks like" the variety Max S, we introduce the open subset In example (1), U S/R is the complement to the subvariety Z(x). We generalize this example by showing that if R is generated by a subalgebra of S and an ideal I ⊂ S, then U S/R contains the complement to the subvariety Z(I) in Max S. In addition, if I is a nonmaximal radical ideal of S and R = k + I, then U S/R = Z(I) c (Proposition 2.8).…”

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

“…Beil [Bei08] shows that square superpotential algebras, which are certain quiver algebras with relations coming from cyclic derivatives of a superpotential, are non-commutative crepant resolutions of their centers (which are three-dimensional toric Gorenstein normal domains). In fact, Broomhead [Bro09] constructs a non-commutative crepant resolution for every Gorenstein affine toric threefold, from superpotential algebras called dimer models.…”

On steady non-commutative crepant resolutions

“…An interesting family of NCCRs is given by a dimer model, which is a quiver with potential drawn on a torus and gives us an NCCR of a Gorenstein toric singularity in dimension three (see e.g. [Bei,Boc2,Bro,Dav,IU,MR]). For more results and examples of NCCRs, we refer to [BLVdB,BIKR,Dao,DFI,DH,IW1,IW3,TU,Wem].…”