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

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.

arXiv comment15 pages

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Cited by in corpus (5)

• Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions
• On steady non-commutative crepant resolutions
• Grothendieck groups, convex cones and maximal Cohen-Macaulay points
• Non-commutative crepant resolutions of Hibi rings with small class group
• Semi-steady non-commutative crepant resolutions via regular dimer models