Maximal modifications and Auslander–Reiten duality for non-isolated singularities

Iyama, O. and Wemyss, M. (2014) Maximal modifications and Auslander–Reiten duality for non-isolated singularities. Inventiones Mathematicae, 197(3), pp. 521-586. (doi:10.1007/s00222-013-0491-y)

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Abstract

We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of SpecR, in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Wemyss, Professor Michael
Authors: Iyama, O., and Wemyss, M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Inventiones Mathematicae
Publisher:Springer-Verlag
ISSN:0020-9910
ISSN (Online):1432-1297
Published Online:19 November 2013
Copyright Holders:Copyright © 2013 Springer-Verlag Berlin Heidelberg
First Published:First published in Inventiones Mathematicae 197(3):521-586
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher

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