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NONNOETHERIAN HOMOTOPY DIMER ALGEBRAS AND NONCOMMUTATIVE CREPANT RESOLUTIONS

Part of: Theory of modules and ideals Algebraic geometry: Foundations

Published online by Cambridge University Press:  30 October 2017

CHARLIE BEIL
CHARLIE BEIL*
Affiliation:
Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, NAWI Graz, Heinrichstrasse 36, A-8010 Graz, Austria e-mail: charles.beil@uni-graz.at
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Abstract

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Noetherian dimer algebras form a prominent class of examples of noncommutative crepant resolutions (NCCRs). However, dimer algebras that are noetherian are quite rare, and we consider the question: how close are nonnoetherian homotopy dimer algebras to being NCCRs? To address this question, we introduce a generalization of NCCRs to nonnoetherian tiled matrix rings. We show that if a noetherian dimer algebra is obtained from a nonnoetherian homotopy dimer algebra A by contracting each arrow whose head has indegree 1, then A is a noncommutative desingularization of its nonnoetherian centre. Furthermore, if any two arrows whose tails have indegree 1 are coprime, then A is a nonnoetherian NCCR.

MSC classification

Primary: 13C15: Dimension theory, depth, related rings (catenary, etc.)
Secondary: 14A20: Generalizations (algebraic spaces, stacks)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 447 - 479
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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