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 BerghEmail author


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 



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.


  1. 1.
    Artin, M.: On Azumaya algebras and finite dimensional representations of rings. J. Algebra 11, 532–563 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auslander, M.: Isolated singularities and existence of almost split sequences. In: Proc. ICRA IV, Lecture Notes in Mathematics, vol. 1178, pp. 194–241, Springer (1986)Google Scholar
  3. 3.
    Auslander, M., Goldman, O.: Maximal orders. Trans. Am. Math. Soc. 97, 1–24 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ballard, M., Favero, D., Katzarkov, L.: Variation of geometric invariant theory quotients and derived categories. arXiv:1203.6643 [math.AG]
  5. 5.
    Bezrukavnikov, R.: Noncommutative counterparts of the Springer resolution. In: International Congress of Mathematicians, vol. II, pp. 1119–1144. European Mathematical Society, Zürich (2006)Google Scholar
  6. 6.
    Bezrukavnikov, R.V., Kaledin, D.B.: McKay equivalence for symplectic resolutions of quotient singularities. Tr. Mat. Inst. Steklova. Algebr. Geom. Metody, Svyazi i Prilozh 246: 20–42 (2004)Google Scholar
  7. 7.
    Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14(3), 535–554 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brion, M.: Sur les modules de covariants. Ann. Sci. École Norm. Sup. (4) 26, 1–21 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Broomhead, N.: Dimer models and Calabi-Yau algebras. Memb. Am. Math. Soc. 215(1011), viii+86 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bruns, W., Gubeladze, J.: Divisorial linear algebra of normal semigroup rings. Algebra Rep. Theory 6(2), 139–168 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Buchweitz, R.-O., Leuschke, G.J., Van den Bergh, M.: Non-commutative desingularization of determinantal varieties I. Invent. Math. 182(1), 47–115 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Buchweitz, R.-O., Leuschke, G.J., Van den Bergh, M.: Non-commutative desingularization of determinantal varieties, II: arbitrary minors. Int. Math. Res. Not. IMRN 9, 2748–2812 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chouinard II, L.G.: Krull semigroups and divisor class groups. Can. J. Math. 33(6), 1459–1468 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dao, H.: Remarks on non-commutative crepant resolutions of complete intersections. Adv. Math. 224(3), 1021–1030 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dao, H., Iyama, O., Takahashi, R., Vial, C.: Non-commutative resolutions and Grothendieck groups. J. Noncommut. Geom. 9(1), 21–34 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    de Concini, C., Procesi, C.: A characteristic free approach to invariant theory. Adv. Math. 21(3), 330–354 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Derksen, H., Weyman, J.: Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients. J. Am. Math. Soc. 13(3), 467–479 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Domokos, M., Zubkov, A.N.: Semi-invariants of quivers as determinants. Transform. Groups 6(1), 9–24 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Donkin, S.: Invariants of several matrices. Invent. Math. 110(2), 389–401 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Flenner, H.: Die Sätze von Bertini für lokale Ringe. Math. Ann. 229(2), 97–111 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Formanek, E.: Invariants and the ring of generic matrices. J. Algebra 89(1), 178–223 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fossum, R.M.: The divisor class group of a Krull domain. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, pp. viii+148. Springer, New York (1973)Google Scholar
  23. 23.
    Fulton, W., Harris, J.: Representation theory. In: Readings in Mathematics, vol. 129, pp. xvi+551. Springer, New York (1991)Google Scholar
  24. 24.
    Gordon, I., Smith, S.P.: Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities. Math. Ann. 330(1), 185–200 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Halpern-Leistner, D.: The derived category of a GIT quotient. J. Am. Math. Soc. 28(3), 871–912 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Herstein, I.N.: Notes from a ring theory conference. American Mathematical Society, Providence, R.I., 1971, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 9 (1971)Google Scholar
  27. 27.
    Iyama, O., Wemyss, M.: Maximal modifications and Auslander-Reiten duality for non-isolated singularities. Invent. Math. 197(3), 521–586 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Iyama, O., Wemyss, M.: Singular derived categories of \(\mathbb{Q}\)-factorial terminalizations and maximal modification algebras. Adv. Math. 261, 85–121 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kirwan, F.C.: Partial desingularisations of quotients of nonsingular varieties and their Betti numbers. Ann. Math. (2) 122(1), 41–85 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Knop, F.: Über die Glattheit von Quotientenabbildungen. Manuscr. Math. 56(4), 419–427 (1986)CrossRefzbMATHGoogle Scholar
  31. 31.
    Kuznetsov, A.: Semiorthogonal decompositions in algebraic geometry. arXiv:1404.3143 [math.AG]
  32. 32.
    Kuznetsov, A.: Homological projective duality. Publ. Math. Inst. Hautes Études Sci. 105, 157–220 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kuznetsov, A.: Lefschetz decompositions and categorical resolutions of singularities. Sel. Math. (N.S.) 13(4), 661–696 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kuznetsov, A.: Homological projective duality, notes. (2012)
  35. 35.
    Kuznetsov, A., Lunts, V.A.: Categorical resolutions of irrational singularities. Int. Math. Res. Not. IMRN 13, 4536–4625 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Le Bruyn, L.: Trace rings of generic 2 by 2 matrices. Memb. Am. Math. Soc. 66(363), vi+100 (1987)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Le Bruyn, L.: Quiver concomitants are often reflexive Azumaya. Proc. Am. Math. Soc. 105(1), 10–16 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Le Bruyn, L., Procesi, C.: The étale local structure of matrix-invariants and concomitants. In: Algebraic Groups Utrecht 1986, Springer, Berlin (1987)Google Scholar
  39. 39.
    Le Bruyn, L., Van den Bergh, M.: Regularity of trace rings of generic matrices. J. Algebra 117, 19–29 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Leuschke, G.J.: Non-commutative crepant resolutions: scenes from categorical geometry. In: Progress in Commutative Algebra 1, pp. 293–361, de Gruyter, Berlin (2012)Google Scholar
  41. 41.
    Luna, D.: Slices étales. Bull. Soc. Math. France 33, 81–105 (1973)zbMATHGoogle Scholar
  42. 42.
    Lunts, V.A.: Categorical resolution of singularities. J. Algebra 323(10), 2977–3003 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Wiley, New York (1987)zbMATHGoogle Scholar
  44. 44.
    Mumford, D., Fogarty, J.: Geometric Invariant Theory, Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 34. Springer, Berlin (1982)zbMATHGoogle Scholar
  45. 45.
    Nastacescu, C., Van Oystaeyen, F.: Graded Ring Theory. North-Holland, Amsterdam, New York (1982)Google Scholar
  46. 46.
    Procesi, C.: Invariant theory of \(n\times n\)-matrices. Adv. Math. 19, 306–381 (1976)CrossRefzbMATHGoogle Scholar
  47. 47.
    Procesi, C.: A formal inverse to the Cayley-Hamilton theorem. J. Algebra 107(1), 63–74 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Razmyslov, Ju P.: Identities with trace in full matrix algebras over a field of characteristic zero. Izv. Akad. Nauk SSSR Ser. Mat. 38, 723–756 (1974)MathSciNetGoogle Scholar
  49. 49.
    Schofield, A., Van den Bergh, M.: Semi-invariants of quivers for arbitrary dimension vectors. Indag. Math. (N.S.) 12(1), 125–138 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Smith, K.E., Van den Bergh, M.: Simplicity of rings of differential operators in prime characteristic. Proc. London Math. Soc. (3) 75(1), 32–62 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Stafford, J.T., Van den Bergh, M.: Noncommutative resolutions and rational singularities. Mich. Math. J. 57, 659–674 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Stanley, R.P.: Combinatorics and invariant theory. Proc. Symp. Pure Math. 34, 345–355 (1979)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Špenko, Š., Van den Bergh, M.: Non-commutative crepant resolutions for some toric singularities I. arXiv:1701.05255 [math.AG]
  54. 54.
    Toda, Y., Yasuda, T.: Noncommutative resolution, \({{\rm F}}\)-blowups and \(D\)-modules. Adv. Math. 222(1), 318–330 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Van den Bergh, M.: Cohen-Macaulayness of modules of covariants. Invent. Math. 106, 389–409 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Van den Bergh, M.: Cohen-Macaulayness of semi-invariants for tori. Trans. Am. Math. Soc. 336(2), 557–580 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Van den Bergh, M.: A converse to Stanley’s conjecture for \({{\rm Sl}}_2\). Proc. Am. Math. Soc. 121(1), 47–51 (1994)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Van den Bergh, M.: Modules of covariants. In: Proceedings of the ICM 94, vol. 1, pp. 352–362, Birkäuser (1995)Google Scholar
  59. 59.
    Van den Bergh, M.: Local cohomology of modules of covariants. Adv. Math. 144(2), 161–220 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Van den Bergh, M.: Non-commutative Crepant Resolutions, The Legacy of Niels Henrik Abel. Springer, Berlin (2004)Google Scholar
  61. 61.
    Varagnolo, M., Vasserot, E.: Double affine Hecke algebras and affine flag manifolds, I. Affine flag manifolds and principal bundles, Birkhauser (2010)Google Scholar
  62. 62.
    Wemyss, M.: Aspects of the homological minimal model program. arXiv:1411.7189 [math.AG]
  63. 63.
    Wemyss, M.: Lectures on noncommutative resolutions. arXiv:1210.2564 [math.RT]
  64. 64.
    Weyl, H.: The Classical Groups. Princeton University Press, Princeton (1946)zbMATHGoogle Scholar
  65. 65.
    Weyman, J., Zhao, G.: Noncommutative desingularization of orbit closures for some representations of \(GL_n\). arXiv:1204.0488
  66. 66.
    Yasuda, T.: Noncommutative resolution of toric singularities: an application of Frobenius morphism of noncommutative blowup. arXiv:1002.0181 [math.AG]

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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