Maximal Modifications and Auslander-Reiten Duality for Non-isolated Singularities

Osamu Iyama, Michael Wemyss

Research output: Contribution to journalArticlepeer-review

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, but in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.
Original languageEnglish
Pages (from-to)521-586
JournalInventiones mathematicae
Volume197
Early online date19 Nov 2013
DOIs
Publication statusPublished - 30 Sep 2014

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