Non-commutative crepant resolutions: scenes from categorical geometry

Download Paper (final arxiv version of 2011-08-22, updated with bugfixes and adjustments)

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.


I was invited to write this expository paper for the proceedings that was put together for three AMS special sessions held in Boca Raton in 2009. I now think I was just too ambitious — it would have been better if I’d made some hard decisions at the beginning, about what to assume and what to say. As it happened, the thing just kept growing and growing. Also, I don’t remember why I included “scenes” in the title; that should have been changed, since it’s basically a non sequitur.

Bibtex code:

@incollection {MR2932589,
    AUTHOR = {Leuschke, Graham J.},
     TITLE = {Non-commutative crepant resolutions: scenes from categorical
 BOOKTITLE = {Progress in commutative algebra 1},
     PAGES = {293--361},
 PUBLISHER = {de Gruyter},
   ADDRESS = {Berlin},
      YEAR = {2012},
   MRCLASS = {14A22 (14E15 14F05 16E35 16S38)},
  MRNUMBER = {2932589},