1301.7059 (Charlie Beil)

Morita equivalences from Higgsing toric superpotential algebras    [PDF]

Charlie Beil

Let A and A' be superpotential algebras of dimer models, with A' cancellative and A non-cancellative, and suppose A' is obtained from A by contracting, or 'Higgsing', a set of arrows to vertices while preserving a certain associated commutative ring. A' is then a Calabi-Yau algebra and a noncommutative crepant resolution of its prime noetherian center, whereas A is not a finitely generated module over its center, often not even PI, and its center is not noetherian and often not prime. 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 not finitely generated modules over their centers, with the property that their smooth and Azumaya loci coincide.

View original: http://arxiv.org/abs/1301.7059