Cellular resolutions of noncommutative toric algebras from superpotentials

Craw, A. and Quintero Velez, A. (2012) Cellular resolutions of noncommutative toric algebras from superpotentials. Advances in Mathematics, 229(3), pp. 1516-1554. (doi: 10.1016/j.aim.2011.11.012)

Abstract

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998)in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric 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 Δ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution ofA. We illustrate the general construction of Δ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on Δ.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Craw, Dr Alastair and Quintero Velez, Dr Alexander
• Authors: Craw, A., and Quintero Velez, A.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Advances in Mathematics
• ISSN: 0001-8708
• Published Online: 09 August 2010
• Copyright Holders: Copyright © 2012 Elsevier
• First Published: First published in Advances in Mathematics 229(3):1516-1554
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
48001	1	Noncommutative toric geometry and multilinear series	Alastair Craw	Engineering & Physical Sciences Research Council (EPSRC)	EP/G004048/1	Mathematics