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

By Osamu Iyama, M. Wemyss. 2010

158 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/116e4ee5725dc6d58381ccc234d7ea0b04e18a24

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In Algebra and Number Theory

1. M. Wemyss
2. Ryo Takahashi
3. Osamu Iyama
4. Hailong Dao
5. Oyvind Solberg
6. Charles Vial
7. Peter Jørgensen
8. I. Reiten

In Geometry and Topology

1. M. Wemyss
2. Ryo Takahashi
3. Hailong Dao
4. Oyvind Solberg
5. Osamu Iyama
6. Charles Vial
7. I. Reiten
8. B. Keller

In Mathematical Physics

1. Hailong Dao
2. Ryo Takahashi
3. Oyvind Solberg
4. M. Wemyss
5. Osamu Iyama
6. Charles Vial
7. Gustavo Jasso
8. I. Reiten

Citations in the corpus

• Auslander-Gorenstein algebras and precluster tilting (2016, background): “) See for example [1, 19, 20, 26, 28, 29, 30, 31, 34, 36] for further results in higher Auslander–Reiten theory.”
• Flops and clusters in the homological minimal model programme (2014, background): “The following, one of the main results in [23], will allow us to establish properties non-explicitly when we restrict to minimal models and mutate at single curves.”
• Non-commutative resolutions and Grothendieck groups (2012, background): “Basic examples of NCRs appeared in representation theory: Auslander algebras [1], higher Auslander algebras [19, 21] and NCCRs [33].”