Non-commutative Crepant Resolutions

By Michel van den Bergh. 2002

325 citations recorded by the source corpus.

We introduce the notion of a “non-commutative crepant” resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8f66483538e7ceec2f871497256d5959bc76345c

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Dimer Models and Calabi-Yau Algebras (2009, methodology): “Non-commutative crepant resolutions We recall the definition of an NCCR introduced by Van den Bergh in [37].”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, background): “We also note that rigid Cohen–Macaulay modules have deep relevance to the noncommutative crepant resolutions [41, 43] introduced by Van den Bergh [64, 65 ].”
Non-commutative Donaldson-Thomas theory and the conifold (2007, background): “Indeed, A is a 3-Calabi–Yau algebra, and a crepant non-commutative resolution [25] of the singular variety Z.”
CONSISTENCY CONDITIONS FOR DIMER MODELS (2011, methodology): “To model these algebras without referring to a commutative crepant resolution, Van den Bergh [24] introduced the notion of a a noncommutative crepant resolution (NCCR) of X .”
Lefschetz decompositions and categorical resolutions of singularities (2006, background): “Another question addressed in the paper is which of categorical resolutions constructed in Theorem 1 are noncommutative in the sense of Van den Bergh?”
Flops and clusters in the homological minimal model programme (2014, background): “The notion of a smooth noncommutative minimal model, called a noncommutative crepant resolution, is due to Van den Bergh [50].”
Cluster tilting for one-dimensional hypersurface singularities (2007, methodology): “3.2.11][ V2 , Th. 6.6.3] Let (R,m) be an isolated cDV singularity.”