Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh's definition within these contexts and describe some of the current research in the area.
Leuschke, Graham J., "Non-Commutative Crepant Resolutions: Scenes from Categorical Geometry" (2011). Mathematics Faculty Scholarship. Paper 40.
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