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)

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Publisher's URL: http://dx.doi.org/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

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
480011Noncommutative toric geometry and multilinear seriesAlastair CrawEngineering & Physical Sciences Research Council (EPSRC)EP/G004048/1Mathematics