Gorenstein modifications and ℚ-Gorenstein rings

2016/11/13 by Hailong Dao, Dao, Hailong, Osamu Iyama +5 · 3 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1611.04137

openalex publication_date 2016/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be a Cohen--Macaulay normal domain with a canonical module ω_R. It is proved that if R admits a noncommutative crepant resolution (NCCR), then necessarily it is 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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