Gorenstein modifications and \mathds{𝑄}-Gorenstein rings

Let R R be a Cohen–Macaulay normal domain with a canonical module ω R \omega _R . It is proved that if R R admits a noncommutative crepant resolution (NCCR), then necessarily it is Q \mathds {Q} -Gorenstein. Writing S S for a Zariski local canonical cover of R R , a tight relationship between the existence of noncommutative (crepant) resolutions on R R and S 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.