Cellular resolutions of noncommutative toric algebras from superpotentials

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998) [2] in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional tone algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex Delta in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of A. We illustrate the general construction of Delta for an example in dimension four arising from a tilting bundle on a smooth tone Fano threefold to highlight the importance of the incidence function on Delta. (C) 2011 Elsevier Inc. All rights reserved.