Non-commutative desingularization of determinantal varieties, I: Maximal minors

Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh

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We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.

This paper grew out of some observations Buchweitz and I made while working on our previous paper. Michel showed us how to verify those observations, and then much more followed. It took over five years for us to finish the paper, mostly because we kept improving the results. Finally it was declared closed, with “, I” added to the end so that we could continue with a part II.

BibTeX code:

 @article {MR2672281,
    AUTHOR = {Buchweitz, Ragnar-Olaf and Leuschke, Graham J. and Van den
              Bergh, Michel},
     TITLE = {Non-commutative desingularization of determinantal varieties
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {182},
      YEAR = {2010},
    NUMBER = {1},
     PAGES = {47–115},
      ISSN = {0020-9910},
     CODEN = {INVMBH},
   MRCLASS = {13C14 (14A22 14E15 14M12 16E10 16S38)},
  MRNUMBER = {2672281},
       DOI = {10.1007/s00222-010-0258-7},
       URL = {}