Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Leuschke, Graham

doi:10.1515/9783110250404.293

Cited by 29 publications

(28 citation statements)

References 146 publications

(119 reference statements)

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“…A proof of this claim is provided at [14, Proposition 2.14]. The interested reader should consult these papers, and also [27], for further background of the connections with noncommutative resolution of singularities.…”

Section: Remarks 52 (I)mentioning

confidence: 97%

The Cohen Macaulay Property for Noncommutative Rings

Brown

Macleod

2017

Algebr Represent Theor

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Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres.

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Section: Remarks 52 (I)mentioning

confidence: 97%

The Cohen Macaulay Property for Noncommutative Rings

Brown

Macleod

2017

Algebr Represent Theor

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“…Sequence (1) is a projective resolution of A. Hence Λ has global dimension e. On the other hand, from [Ra] (see also [Le2]), it has global dimension d. Hence if d 2, then the equality in the theorem holds.…”

Section: Preliminariesmentioning

confidence: 98%

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

Yasuda

2012

Journal of Algebra

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The aim of this paper is threefold: first, to prove that the endomorphism ring associated to a pure subring of a regular local ring is a noncommutative crepant resolution if it is maximal CohenMacaulay; second, to see that in that situation, a different, but Morita equivalent, noncommutative crepant resolution can be constructed by using Frobenius morphisms; finally, to study the relation between Frobenius morphisms of noncommutative rings and the finiteness of global dimension. As a byproduct, we will obtain a result on wild quotient singularities: If the smooth cover of a wild quotient singularity is unramified in codimension one, then the singularity is not strongly F-regular.

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“…Moreover if n is odd then Λ is a CohenMacaulay R := T Sp(V ) -module. In other words, in the terminology of [14,18], when n is odd Λ is a non-commutative crepant resolution (NCCR) of R.…”

Section: 1) D(λ) ֒→ D(z)mentioning

confidence: 99%

“…Similarly for a noetherian scheme/stack X we write D(X) := D b coh (X). If Y is the determinantal variety of n × n-matrices of rank ≤ r then in [2] (and independently in [5]) a "non-commutative crepant resolution" [14,18] Λ for k[Y ] was constructed. Such an NCCR is a k[Y ]-algebra which has in particular the property that D(Λ) is a "strongly crepant categorical resolution" of Perf(Y ) (the derived category of perfect complexes on Y ) in the sense of [12,Def.…”

Section: Introductionmentioning

confidence: 99%