Gorenstein modifications and \mathds{𝑄}-Gorenstein rings

Dao, Hailong; Iyama, Osamu; Takahashi, Ryo; Wemyss, Michael

doi:10.1090/jag/760

Gorenstein modifications and $\mathds {Q}$-Gorenstein rings

Authors: Hailong Dao, Osamu Iyama, Ryo Takahashi and Michael Wemyss
Journal: J. Algebraic Geom. 29 (2020), 729-751
DOI: https://doi.org/10.1090/jag/760
Published electronically: March 31, 2020
MathSciNet review: 4158464
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Abstract | References | Additional Information

Abstract: Let $R$ be a Cohen–Macaulay normal domain with a canonical module $\omega _R$. It is proved that if $R$ admits a noncommutative crepant resolution (NCCR), then necessarily it is $\mathds {Q}$-Gorenstein. Writing $S$ for a Zariski local canonical cover of $R$, a tight relationship between the existence of noncommutative (crepant) resolutions on $R$ and $S$ is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of any NCCR is log-terminal, and the Auslander–Esnault classification of two-dimensional CM-finite algebras can be deduced from Buchweitz–Greuel–Schreyer.

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Additional Information

Hailong Dao
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-7523
MR Author ID: 828268
Email: hdao@ku.edu

Osamu Iyama
Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
MR Author ID: 634748
Email: iyama@math.nagoya-u.ac.jp

Ryo Takahashi
Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
MR Author ID: 674867
Email: takahashi@math.nagoya-u.ac.jp

Michael Wemyss
Affiliation: School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow, G12 8QW, United Kingdom
MR Author ID: 893224
Email: michael.wemyss@glasgow.ac.uk

Received by editor(s): January 21, 2018
Received by editor(s) in revised form: December 8, 2019
Published electronically: March 31, 2020
Additional Notes: Part of this work was completed during the AIM SQuaRE: Cohen-Macaulay representations and categorical characterizations of singularities. The authors thank AIM for funding and for their kind hospitality. The first author was further supported by NSA H98230-16-1-0012. The second author was supported by JSPS Grant-in-Aid for Scientific Research 16H03923. The third author was supported by JSPS Grant-in-Aid for Scientific Research 16K05098. The fourth author was supported by EPSRC grant EP/K021400/1.
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