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

Hailong Dao; Osamu Iyama; Ryo Takahashi; Michael Wemyss

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arXiv · 1611.04137

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

Hailong Dao·Osamu Iyama·Ryo Takahashi·Michael Wemyss

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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 $\mathbb{Q}$-Gorenstein. Writing $S$ for a Zariski local canonical cover of $R$, then 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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BibTeX RIS

Hailong Dao, Osamu Iyama, Ryo Takahashi, Michael Wemyss. 2016-11-13. Gorenstein modifications and $\mathbb{Q}$-Gorenstein rings. https://arxiv.org/abs/1611.04137

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