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

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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 \(\operatorname{Spec}R\), 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.

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Acknowledgements

We would like to thank Michel Van den Bergh, Vanya Cheltsov, Constantin Shramov, Ryo Takahashi and Yuji Yoshino for stimulating discussions and valuable suggestions. We also thank the anonymous referee for carefully reading the paper, and offering many useful insights and suggested improvements.

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Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, 464-8602, Chikusa-ku, Nagoya, Japan

    Osamu Iyama

  2. The Maxwell Institute, School of Mathematics, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, EH9 3JZ, Edinburgh, UK

    Michael Wemyss

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  1. Osamu Iyama
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  2. Michael Wemyss
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Correspondence to Michael Wemyss.

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In memory of Kentaro Nagao

The first author was partially supported by JSPS Grant-in-Aid for Scientific Research 24340004, 23540045, 20244001 and 22224001. The second author was partially supported by a JSPS Postdoctoral Fellowship and by the EPSRC.

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

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