Non-commutative Crepant Resolutions

  • Michel van den Bergh


We introduce the notion of a “non-commutative crepant” resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.


Rational Singularity Coherent Sheave Ample Line Bundle Coherent Sheaf Triangulate Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. in Math. 109 (1994), no. 2, 228–287.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1986), 195–222.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    M. Artin and W. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), 171–216.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, vol. 1, Birkhäuser, 1990, pp. 33–85.Google Scholar
  5. [5]
    M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), 335–388.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    M. Auslander and O. Goldman, Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1–24.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    V. V. Batyrev, Birational Calabi—Yau n-folds have equal Betti numbers, New trends in algebraic geometry (Warwick, 1996) (Cambridge), London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 1–11.Google Scholar
  8. [8]
    A. Beauville, Surface algébriques complexes, Astérisque, vol. 54, Soc. Math. France, 1978.Google Scholar
  9. [9]
    A. Bondal and D. Orlov, Derived categories of coherent sheaves, available as math.AG/0206295.Google Scholar
  10. [10]
    A. Bondal and D. Orlov, Semi-orthogonal decompositions for algebraic varieties, available as alggeom/950601, 1996.Google Scholar
  11. [11]
    A. Bondal and M. Van den Bergh, Generators and representability of functors in commutative and non-commutative geometry, available as math.AG/0204218.Google Scholar
  12. [12]
    N. Bourbaki, Algèbre commutative, Hermann, Paris, 1960-65.Google Scholar
  13. [13]
    T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), no. 1, 25–34.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), 613–632.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554 (electronic).MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    M. Brion, Sur les modules de covariants, Ann. Sci. École Norm. Sup. (4) 26 (1993), 1–21.MathSciNetzbMATHGoogle Scholar
  17. [17]
    K. A. Brown and C. R. Hajarnavis, Homologically homogeneous rings, Trans. Amer. Math. Soc. 281 (1984), 197–208.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    R. Elkik, Singularités rationnelles et déformations, Invent. Math. 47 (1978), no. 2, 139–147.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    G. M. Greuel, G. Pfister, and H. Schönemann, SINGULAR 2.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001,
  20. [20]
    M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), no. 3, 565–576.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Y. Kawamata, Francia’s flip and derived categories, available as math.AG/0111041.Google Scholar
  23. [23]
    B. Keller, A∞ algebras and triangulated categories, in preparation.Google Scholar
  24. [24]
    A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    M. Kontsevich, Lecture at Orsay, December 7, 1995.Google Scholar
  26. [26]
    D. Luna, Slices etales, Bull. Soc. Math. France 33 (1973), 81–105.zbMATHGoogle Scholar
  27. [27]
    P. E. Newstead, Introduction to moduli problems and orbit spaces, Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay, 1978.Google Scholar
  28. [28]
    I. Reiner, Maximal orders, Academic Press, New York, 1975.zbMATHGoogle Scholar
  29. [29]
    I. Reiten and M. Van den Bergh, Tame and maximal orders of finite representation type, Memoirs of the AMS, vol. 408, Amer. Math. Soc., 1989.Google Scholar
  30. [30]
    K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    A. Schofield, Representations of rings over skew fields, Lecture Note Series, vol. 92, London Mathematical Society, 1985.Google Scholar
  32. [32]
    R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), 175–193.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    R. P. Stanley, Combinatorics and commutative algebra, Progress in Mathematics, vol. 41, Birkhäuser Boston Inc., Boston, MA, 1983.Google Scholar
  34. [34]
    B. Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 217, Springer Verlag, Berlin, 1975.Google Scholar
  35. [35]
    D. R. Stephenson, Artin-Schelter regular algebras of global dimension three, J. Algebra 183 (1996), 55–73.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    D. R. Stephenson, Algebras associated to elliptic curves, Trans. Amer. Math. Soc. 349 (1997), 2317–2340.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    M. Van den Bergh, Three-dimensional flops and non-commutative rings, submitted. To appear in Duke Mathematical Journal.Google Scholar
  38. [38]
    M. Van den Bergh, Cohen-Macaulayness of semi-invariants for tori, Trans. Amer. Math. Soc. 336 (1993), no. 2, 557–580.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    M. Van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra (1997), 662–679.Google Scholar
  40. [40]
    K. Watanabe, Rational singularities with k*-action, Commutative algebra (Trento, 1981) (New York), Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 339-351.Google Scholar
  41. [41]
    A. Yekutieli and J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), no. 1, 1–51.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    A. Yekutieli and J. Zhang, Dualizing complexes andperverse sheaves on noncommutative ringed schemes, to appear, 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Michel van den Bergh

There are no affiliations available

Personalised recommendations