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Michel Van den Bergh

  1. Notes on formal deformations of abelian categories.

    Tue, 02/02/2010 - 16:01 — SomNambul
    Authors: Michel Van den Bergh
    Subjects: Rings and Algebras
    Abstract

    In these notes we provide the foundation for the deformation theoretic parts
    of arXiv:0807.375 and arXiv:math/0102005.

  2. Non-commutative desingularization of determinantal varieties, I.

    Mon, 11/16/2009 - 16:01 — SomNambul
    Authors: Michel Van den Bergh, Ragnar-Olaf Buchweitz, Graham J. Leuschke
    Subjects: Commutative Algebra
    Abstract

    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.

  3. Deformed Calabi-Yau Completions.

    Wed, 08/26/2009 - 14:58 — SomNambul
    Authors: Michel Van den Bergh, Bernhard Keller
    Subjects: Representation Theory
    Abstract

    We define and investigate deformed n-Calabi-Yau completions of homologically
    smooth differential graded (=dg) categories. Important examples are: deformed
    preprojective algebras of connected non Dynkin quivers, Ginzburg dg algebras
    associated to quivers with potentials and dg categories associated to the
    category of coherent sheaves on the canonical bundle of a smooth variety. We
    show that deformed Calabi-Yau completions do have the Calabi-Yau property and
    that their construction is compatible with derived equivalences and with
    localizations.

  4. Hochschild cohomology for Lie algebroids.

    Thu, 08/20/2009 - 23:06 — SomNambul
    Authors: Damien Calaque, Carlo A. Rossi, Michel Van den Bergh
    Subjects: Algebraic Geometry
    Abstract

    We define the Hochschild (co)homology of a ringed space relative to a locally
    free Lie algebroid. Our definitions mimic those of Swan and Caldararu for an
    algebraic variety. We show that our (co)homology groups can be computed using
    suitable standard complexes.

    Our formulae depend on certain natural structures on jetbundles over Lie
    algebroids. In an appendix we explain this by showing that such jetbundles are
    formal groupoids which serve as the formal exponentiation of the Lie algebroid.

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