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Inventiones mathematicae

, Volume 182, Issue 1, pp 47–115 | Cite as

Non-commutative desingularization of determinantal varieties I

  • Ragnar-Olaf Buchweitz
  • Graham J. LeuschkeEmail author
  • Michel Van den Bergh
Article

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.

Mathematics Subject Classification (2000)

13C14 14A22 14E15 14C40 16S38 13D02 12G50 16G20 

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References

  1. 1.
    Beĭlinson, A.A.: Coherent sheaves on P n and problems in linear algebra. Funk. Anal. Prilozh. 12(3), 68–69 (1978). MR509388 zbMATHGoogle Scholar
  2. 2.
    Berenstein, D., Leigh, R.G.: Resolution of stringy singularities by non-commutative algebras. J. High Energy Phys. (6), Paper 30, 37 (2001). MR1849725 Google Scholar
  3. 3.
    Bezrukavnikov, R.: Noncommutative counterparts of the Springer resolution. In: International Congress of Mathematicians, vol. II, pp. 1119–1144. Eur. Math. Soc., Zürich (2006). MR2275638 Google Scholar
  4. 4.
    Bondal, A.I.: Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 25–44 (1989). MR992977 MathSciNetGoogle Scholar
  5. 5.
    Bondal, A.I., Polishchuk, A.E.: Homological properties of associative algebras: the method of helices. Izv. Ross. Akad. Nauk Ser. Mat. 57(2), 3–50 (1993). MR1230966 MathSciNetGoogle Scholar
  6. 6.
    Bourbaki, N.: Éléments de mathématique. Algèbre commutative. Springer-Verlag, Berlin (2007). Reprint of the 1998 original. Chap. 10. MR2333539 zbMATHGoogle Scholar
  7. 7.
    Buchsbaum, D.A., Rim, D.S.: A generalized Koszul complex. II. Depth and multiplicity. Trans. Am. Math. Soc. 111, 197–224 (1964). MR0159860 zbMATHMathSciNetGoogle Scholar
  8. 8.
    Buchweitz, R.-O., Leuschke, G.J.: Factoring the adjoint and maximal Cohen-Macaulay modules over the generic determinant. Am. J. Math. 129(4), 943–981 (2007). MR2343380 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eagon, J.A., Northcott, D.G.: Ideals defined by matrices and a certain complex associated with them. Proc. R. Soc. Ser. A 269, 188–204 (1962). MR0142592 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eisenbud, D., Schreyer, F.-O., Weyman, J.: Resultants and Chow forms via exterior syzygies. J. Am. Math. Soc. 16(3), 537–579 (2003) (electronic). MR1969204 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gaeta, F.: Détermination de la chaîne syzygétique des idéaux matriciels parfaits et son application à la postulation de leurs variétés algébriques associées. C. R. Acad. Sci. Paris 234, 1833–1835 (1952). MR0048093 zbMATHMathSciNetGoogle Scholar
  12. 12.
    Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II, Inst. Hautes Études Sci. Publ. Math. (17), 91 (1963). MR0163911 Google Scholar
  13. 13.
    Hille, L., Van den Bergh, M.: Fourier-Mukai transforms. In: Handbook of Tilting Theory. London Math. Soc. Lecture Note Ser., vol. 332, pp. 147–177. Cambridge University Press, Cambridge (2007). MR2384610 CrossRefGoogle Scholar
  14. 14.
    Iyama, O., Reiten, I.: Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras. Am. J. Math. 130(4), 1087–1149 (2008). MR2427009 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kaledin, D.: Derived equivalences by quantization. Geom. Funct. Anal. 17(6), 1968–2004 (2008). MR2399089 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lang, S.: Algebra, 3rd edn., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York (2002). MR1878556 zbMATHGoogle Scholar
  17. 17.
    Leuschke, G.J.: Endomorphism rings of finite global dimension. Canad. J. Math. 59(2), 332–342 (2007). MR2310620 zbMATHMathSciNetGoogle Scholar
  18. 18.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn., Graduate Texts in Mathematics, vol. 5. Springer-Verlag, New York (1998). MR1712872 zbMATHGoogle Scholar
  19. 19.
    Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. (2) 39(3), 436–456 (1989). MR1002456 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Szendrői, B.: Non-commutative Donaldson-Thomas invariants and the conifold. Geom. Topol. 12(2), 1171–1202 (2008). MR2403807 CrossRefMathSciNetGoogle Scholar
  21. 21.
    Van den Bergh, M.: Non-commutative crepant resolutions. In: The Legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004). MR2077594 Google Scholar
  22. 22.
    Van den Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3), 423–455 (2004). MR2057015 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Vetter, U.: Generic maps revised. Commun. Algebra 20(9), 2663–2684 (1992). MR1176833 zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Weyman, J.: Cohomology of Vector Bundles and Syzygies. Cambridge Tracts in Mathematics, vol. 149. Cambridge University Press, Cambridge (2003). MR1988690 zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Ragnar-Olaf Buchweitz
    • 1
  • Graham J. Leuschke
    • 2
    Email author
  • Michel Van den Bergh
    • 3
  1. 1.Dept. of Computer and Mathematical SciencesUniversity of Toronto ScarboroughTorontoCanada
  2. 2.Dept. of Math.Syracuse UniversitySyracuseUSA
  3. 3.Departement WNIUniversiteit HasseltDiepenbeekBelgium

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