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• W2999604665 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a Cohen–Macaulay normal domain with a canonical module <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega Subscript upper R> <mml:semantics> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>omega _R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is proved that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a noncommutative crepant resolution (NCCR), then necessarily it is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathds {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Gorenstein. Writing <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a Zariski local canonical cover of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a tight relationship between the existence of noncommutative (crepant) resolutions on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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." @default.
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• W2999604665 date "2020-03-31" @default.
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• W2999604665 title "Gorenstein modifications and mathds{𝑄}-Gorenstein rings" @default.
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