Skip to main content
Log in

Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings

Algebras and Representation Theory Aims and scope Submit manuscript

Cite this article

Abstract

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but generalizes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aleksandrov, A.G.: Nonisolated Saito singularities. Math. USSR Sbornik 65(2), 561–574 (1990)

    Article  MATH  Google Scholar 

  2. Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1969)

  3. Auslander, M.: On the dimension of modules and algebras. III. Global dimension. Nagoya Math. J. 9, 67–77 (1955)

    MATH  MathSciNet  Google Scholar 

  4. Auslander, M.: Rational singularities and almost split sequences. Trans. AMS 293(2), 511–531 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Auslander, M., Platzeck, M.I., Todorov, G.: Homological theory of idempotent ideals. Trans. Amer. Math. Soc. 332(2), 667–692 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Auslander, M., Reiten, I.: Grothendieck groups of algebras with nilpotent annihilators. Proc. AMS 103(4), 1022–1024 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bass, H.: Algebraic K-theory. W. A. Benjamin, Inc., New York-Amsterdam (1968)

    MATH  Google Scholar 

  8. Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Amer. Math. Soc 9(2), 473–527 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Böhm, J., Decker, W., Schulze, M.: Local analysis of Grauert–Remmert-type normalization algorithms. Internat. J. Algebra Comput. 24(1), 69–94 (2014)

    Article  MathSciNet  Google Scholar 

  10. Bourbaki, N.: Algebra II. Chapters 4–7. Elements of Mathematics (Berlin). In: Cohn, P.M., Howie, J. (eds.) Translated from the 1981 French edition. [Reprint of the 1990 English edition]. Springer, Berlin (2003)

    Google Scholar 

  11. Bridgeland, T.: Flops and derived categories. Invent. Math. 147(3), 613–632 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bridgeland, T., King, A., Reid, M.: The MacKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc. 14(3), 535–554 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Brown, K.A., Hajarnavis, C.R.: Homologically homogeneous rings. Trans. Amer. Math. Soc. 281(1), 197–208 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bruns, W., Herzog, J.: Cohen-Macaulay rings Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  15. Buchweitz, R.-O.: Desingularizing free divisors Talk at Free divisors workshop. University of Warwick (2011)

  16. Buchweitz, R.-O., Conca, A.: New free divisors from old. J. Commut. Algebra 5(1), 17–47 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Buchweitz, R.-O., Ebeling, W., von Bothmer, H.G.: Low-dimensional singularities with free divisors as discriminants. J. Algebraic Geom. 18(2), 371–406 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Buchweitz, R.-O., Leuschke, G.J., van den Bergh, M.: Non-commutative desingularization of determinantal varieties I. Invent. Math. 182(1), 47–115 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Buchweitz, R.-O., Mond, D.: Linear free divisors and quiver representations. In: Singularities and computer algebra, vol. 324 London Math. Soc. Lecture Note Ser. pp. 41–77. Cambridge University Press, Cambridge (2006)

  20. Buchweitz, R.-O., Pham, T. The Koszul Complex Blows up a Point (2013). In preparation

  21. Burban, I., Iyama, O., Keller, B., Reiten, I.: Cluster tilting for one-dimensional hypersurface singularities. Adv. Math. 217(6), 2443–2484 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Curtis, C.W., Reiner, I.: Methods of representation theory. Vol. I. John Wiley & Sons, Inc., New York (1981) With applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication

  23. Dao, H.: Remarks on non-commutative crepant resolutions of complete intersections. Adv. Math. 224(3), 1021–1030 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Dao, H.: What is Serre’s condition (S n ) for sheaves? (2010). URL http://mathoverflow.net/questions/22228/what-is-serres-condition-s-n-for-sheaves

  25. Dao, H., Huneke, C.: Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings. Amer. J. Math. 135(2), 561–578 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Dao, H., Iyama, O., Takahashi, R., Vial, C.: Non-commutative resolutions and Grothendieck groups. To appear in J. Noncommut. Geom., arXiv:1205.4486 (2012)

  27. de Jong, T., van Straten, D.: Deformations of the normalization of hypersurfaces. Math. Ann. 288(3), 527–547 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  28. Doherty, B., Faber, E., Ingalls, C.: Computing global dimension of endomorphism rings via ladders . In preparation (2014)

  29. Evans, E.G., Griffith, P.: Syzygies London Mathematical Society Lecture Note Series, vol. 106. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  30. Faber, E.: Characterizing normal crossing hypersurfaces. To appear in Math. Ann., arXiv:1201.6276 (2014)

  31. Flenner, H.: Rationale quasihomogene Singularitäten. Arch. Math. 36, 35–44 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  32. Goodearl, K.R., Small, L.W.: Krull versus global dimension in Noetherian P.I. rings. Proc. Amer. Math. Soc. 92(2), 175–178 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  33. Granger, M., Mond, D., Nieto Reyes, A.M.S.: Linear free divisors and the global logarithmic comparison theorem. Ann. Inst. Fourier 59(2), 811–850 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  34. Granger, M., Mond, D., Schulze, M.: Free divisors in prehomogeneous vector spaces. Proc. Lond. Math. Soc. (3) 102(5), 923–950 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  35. Green, E.L., Martínez-Villa, R.: Koszul and Yoneda algebras. II. In: Algebras and modules, II (Geiranger, 1996), vol. 24 CMS Conf. Proc. pp. 227–244. Amer. Math. Soc., Providence, RI (1998)

  36. Hartshorne, R.: Algebraic Geometry Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

    Google Scholar 

  37. Hazewinkel, M., Gubareni, N., Kirichenko, V.V.: Algebras, rings and modules. Vol. 1, Mathematics and its Applications vol. 575. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  38. Huneke, C., Wiegand, R.: Tensor products of modules, rigidity and local cohomology. Math. Scand. 81(2), 161–183 (1997)

    MATH  MathSciNet  Google Scholar 

  39. Ingalls, C., Paquette, C.: Homological dimensions for co-rank one idempotent subalgebras. arXiv:1405.5429v1 (2014)

  40. Ingalls, C., Yasuda, T.: Log Centres of Noncommutative Crepant Resolutions are Kawamata Log Terminal: Remarks on a paper of Stafford and van den Bergh (2013). Preprint

  41. Iyama, O.: Representation dimension and Solomon zeta function. In: Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun., vol. 40, pp. 45–64. Amer. Math. Soc., Providence, RI (2004)

  42. Iyama, O.: Auslander correspondence. Adv. Math. 210(1), 51–82 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  43. Iyama, O., Wemyss, M.: The classification of special Cohen-Macaulay modules. Math. Z. 265(1), 41–83 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. Iyama, O., Wemyss, M.: Maximal modifications and Auslander-Reiten duality for non-isolated singularities. Invent. Math. 197(3), 521–586 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  45. Leuschke, G.J.: Endomorphism rings of finite global dimension. Canad. J. Math. 59(2), 332–342 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  46. Leuschke, G.J.: Non-commutative crepant resolutions: scenes from categorical geometry. In: Progress in commutative algebra 1, pp. 293–361. de Gruyter, Berlin (2012)

  47. Leuschke, G.J., Wiegand, R.: Cohen-Macaulay representations, Mathematical Surveys and Monographs, vol. 181. American Mathematical Society, Providence, RI (2012)

  48. Looijenga, E.J.N.: Isolated singular points on complete intersections London Mathematical Society Lecture Note Series, vol. 77. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  49. McConnell, J.C., Robson, J.C.: Noncommutative Noetherian rings, Graduate Studies in Mathematics, vol. 30, revised edn. American Mathematical Society, Providence, RI (2001). With the cooperation of L. W. Small

  50. Pierce, R.S.: Associative algebras Graduate Texts in Mathematics. Studies in the History of Modern Science, 9, vol. 88. Springer, New York (1982)

    Google Scholar 

  51. Reiner, I.: Maximal orders. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York (1975). London Mathematical Society Monographs, No. 5

  52. Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo 27(2), 265–291 (1980)

    MATH  MathSciNet  Google Scholar 

  53. Saito, K.: Primitive forms for a universal unfolding of a function with an isolated critical point. J. Fac. Sci. Univ. Tokyo Sect. IA Math 28(3), 775–792 (1981)

    MATH  MathSciNet  Google Scholar 

  54. Sekiguchi, J.: Three dimensional saito free divisors and singular curves. J. Sib. Fed. Univ. Math. Phys. 1, 33–41 (2008)

    Google Scholar 

  55. Stafford, J.T., Van den Bergh, M.: Noncommutative resolutions and rational singularities. Michigan Math. J. 57, 659–674 (2008). Special volume in honor of Melvin Hochster

    Article  MATH  MathSciNet  Google Scholar 

  56. Teissier, B.: The hunting of invariants in the geometry of discriminants. In: Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 565–678. Sijthoff and Noordhoff, Alphen aan den Rijn (1977)

  57. Terao, H.: Arrangements of hyperplanes and their freeness I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 293–312 (1980)

    MATH  MathSciNet  Google Scholar 

  58. Van den Bergh, M.: Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004)

  59. Van den Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3), 423–455 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  60. Vasconcelos, W.V.: Reflexive modules over Gorenstein rings. Proc. Amer. Math. Soc. 19, 1349–1355 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  61. Watanabe, K.: Rational singularities with k -action. In: Commutative algebra (Trento, 1981), Lecture Notes in Pure and Appl. Math., vol. 84, pp. 339–351. Dekker, New York (1983)

  62. Yoshino, Y.: Cohen–Macaulay modules over Cohen–Macaulay rings London Mathematical Society Lecture Note Series, vol. 146. Cambridge University Press, Cambridge (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Eleonore Faber.

Additional information

This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000, while the authors were in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the spring semester of 2013. H.D. was partially supported by NSF grant DMS 1104017. E.F. was supported by the Austrian Science Fund (FWF) in frame of project J3326. C.I. was supported by an NSERC Discovery grant.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dao, H., Faber, E. & Ingalls, C. Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings. Algebr Represent Theor 18, 633–664 (2015). https://doi.org/10.1007/s10468-014-9510-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-014-9510-y

Keywords

Mathematics Subject Classification (2010)

Navigation