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Inventiones mathematicae

, Volume 210, Issue 1, pp 3–67 | Cite as

Non-commutative resolutions of quotient singularities for reductive groups

  • Špela Špenko
  • Michel Van den Bergh
Article

Abstract

In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense. Moreover we exhibit a large class of such singularities which have (twisted) non-commutative crepant resolutions. We discuss a number of examples, both new and old, that can be treated using our methods. Notably we prove that twisted non-commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew-symmetric matrices. In contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity.

Mathematics Subject Classification

13A50 14L24 16E35 

Notes

Acknowledgements

The authors thank Roland Abuaf, Michel Brion, Hailong Dao, Johan de Jong, Craig Huneke, Jean Michel, Michael Wemyss and Gašper Zadnik for interesting discussions. The first author also thanks the University of Hasselt for its hospitality. In addition, the authors thank the referee for his careful reading of the manuscript and his helpful comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsVrije Universiteit BrusselB-1050Belgium
  2. 2.Universiteit HasseltHasseltBelgium

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