Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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
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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
Email: hdao@ku.edu

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

Ryo Takahashi
Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
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
Email: michael.wemyss@glasgow.ac.uk

DOI: https://doi.org/10.1090/jag/760
Received by editor(s): January 21, 2018
Received by editor(s) in revised form: December 8, 2019, and February 18, 2020
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.
Article copyright: © Copyright 2020 University Press, Inc.