Hilbert–Kunz Multiplicity and the F-Signature



This paper is a much expanded version of two talks given in Ann Arbor in May of 2012 during the computational workshop on F-singularities. We survey some of the theory and results concerning the Hilbert-Kunz multiplicity and F-signature of positive characteristic local rings.


Filtration Hanes 



The author was partially supported by NSF grant DMS-1063538. I thank them for their support.


  1. 1.
    Aberbach, I.M.: Extensions of weakly and strongly F-rational rings by flat maps. J. Algebra 241, 799–807 (2001)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Aberbach, I.M.: The existence of the F-signature for rings with large Q-Gorenstein locus. J. Algebra 319, 2994–3005 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Aberbach, I.M., Leuschke, G.: The F-signature and strong F-regularity. Math. Res. Lett. 10, 51–56 (2003)MathSciNetMATHGoogle Scholar
  4. 4.
    Aberbach, I.M., Enescu, F.: The structure of F-pure rings. Math. Z. 250, 791–806 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Aberbach, I.M., Enescu, F.: When does the F-signature exist? Ann. Fac. Sci. Toulouse Math. 15, 195–201 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Aberbach, I.M., Enescu, F.: Lower bounds for the Hilbert–Kunz multiplicities in local rings of fixed dimension. Michigan Math. J. 57, 1–16 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Aberbach, I.M., Enescu, F.: New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension.Google Scholar
  8. 8.
    Blickle, M., Enescu, F.: On rings with small Hilbert–Kunz multiplicity. Proc. Amer. Math. Soc. 132, 2505–2509 (2004)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Blickle, M., Schwede, K., Tucker, K.: F-signature of pairs and the asymptotic behavior of Frobenius splittings. Adv. Math. 231, 3232–3258 (2012)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Blickle, M., Schwede, K., Tucker, K.: F-signature of pairs: Continuity, p-fractals and minimal log discrepancies. arXiv:1111.2762Google Scholar
  11. 11.
    Brenner, H.: A linear bound for Frobenius powers and an inclusion bound for tight closure. Michigan Math. J. 53, 585–596 (2005)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Brenner, H.: The rationality of the Hilbert–Kunz multiplicity in graded dimension two. Math. Ann. 334, 91–110 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Brenner, H.: The Hilbert–Kunz function in graded dimension two. Comm. Algebra 35, 3199–3213 (2007)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Brenner, H., Li, J., Miller, C.: A direct limit for limit Hilbert–Kunz multiplicity for smooth projective curves. arXiv:1104.2662Google Scholar
  15. 15.
    Brenner, H., Monsky, P.: Tight closure does not commute with localization. Ann. Math. 171, 571–588 (2010)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Bruns, W., Herzog, J.: Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  17. 17.
    Buchweitz, R.-O., Chen, Q.: Hilbert–Kunz functions of cubic curves and surfaces. J. Algebra 197, 246–267 (1997)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Celikbas, O., Dao, H., Huneke, C., Zhang, Y.: Bounds on the Hilbert–Kunz multiplicity. Nagoya Math. J. 205, 149–165 (2012)MathSciNetMATHGoogle Scholar
  19. 19.
    Conca, A.: Hilbert–Kunz function of monomial ideals and binomial hypersurfaces. Manuscripta Math. 90, 287–300 (1996)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Dutta, S.: Frobenius and multiplicities. J. Algebra 85, 424–448 (1983)MATHCrossRefGoogle Scholar
  21. 21.
    Enescu, F., Shimomoto, K.: On the upper semi-continuity of the Hilbert–Kunz multiplicity. J. Algebra 285, 222–237 (2005)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Epstein, N., Yao, Y.: Some extensions of Hilbert–Kunz multiplicity. arXiv:1103.4730Google Scholar
  23. 23.
    Eto, K.: Multiplicity and Hilbert–Kunz multiplicity of monoid rings. Tokyo J. Math. 25, 241–245 (2002)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Eto, K., Yoshida, K.: Notes on Hilbert–Kunz multiplicity of Rees algebras. Comm. Algebra 31, 5943–5976 (2003)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Fakhruddin, N., Trivedi, V.: Hilbert–Kunz functions and multiplicities for full flag varieties and elliptic curves. J. Pure Appl. Algebra 181, 23–52 (2003)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Gessel, I., Monsky, P.: The limit as p→ infinity of the Hilbert–Kunz multiplicity of sum(\(x_{i}^{d_{i}}\)). arXiv:1007.2004Google Scholar
  27. 27.
    Goto, S., Nakamura, Y.: Multiplicity and tight closures of parameters. J. Algebra 244, 302–311 (2001)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Han, C., Monsky, P.: Some surprising Hilbert–Kunz functions. Math. Z. 214, 119–135 (1993)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Hanes, D.: Notes on the Hilbert–Kunz function. J. Algebra 265, 619–630 (2003)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Hochster, M., Roberts, J.: Rings of invariants of reductive groups acting on regular rings are Cohen–Macaulay. Adv. Math. 13, 115–175 (1974)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Hochster, M., Huneke, C.: Tight closure, invariant theory and the Briançon–Skoda theorem. J. Amer. Math. Soc. 3, 31–116 (1990)MathSciNetMATHGoogle Scholar
  32. 32.
    Hochster, M., Huneke, C.: Phantom homology. Mem. Amer. Math. Soc. 103(490) (1993)Google Scholar
  33. 33.
    Hochster, M., Huneke, C.: F-regularity, test elements, and smooth base change. Trans. Amer. Math. Soc. 346, 1–62 (1994)MathSciNetMATHGoogle Scholar
  34. 34.
    Hochster, M., Yao, Y.: Second coefficients of Hilbert–Kunz functions for domains. PreprintGoogle Scholar
  35. 35.
    Huneke, C.: Tight closure and its applications. With an appendix by Melvin Hochster. CBMS Regional Conference Series in Mathematics, vol. 88 (1996)Google Scholar
  36. 36.
    Huneke, C., Leuschke, G.: Two theorems about maximal Cohen–Macaulay modules. Math. Ann. 324, 391–404 (2002)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Huneke, C., Yao, Y.: Unmixed local rings with minimal Hilbert–Kunz multiplicity are regular. Proc. Amer. Math. Soc. 130, 661–665 (2002)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Huneke, C., McDermott, M., Monsky, P.: Hilbert–Kunz functions for normal rings. Math. Res. Lett. 11, 539–546 (2004)MathSciNetMATHGoogle Scholar
  39. 39.
    Kreuzer, M.: Computing Hilbert–Kunz functions of 1-dimensional graded rings. Univ. Iagel. Acta Math. 45, 81–95 (2007)MathSciNetGoogle Scholar
  40. 40.
    Kunz, E.: Characterizations of regular local rings for characteristic p. Amer. J. Math. 91, 772–784 (1969)Google Scholar
  41. 41.
    Kunz, E.: On Noetherian rings of characteristic p. Amer. J. Math. 98, 999–1013 (1976)Google Scholar
  42. 42.
    Kurano, K.: On Roberts rings. J. Math. Soc. Japan 53, 333–355 (2001)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Kurano, K.: The singular Riemann–Roch theorem and Hilbert–Kunz functions. J. Algebra 304, 487–499 (2006)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Lyubeznik, G., Smith, K.E.: Strong and weak F-regularity are equivalent for graded rings. Amer. J. Math. 121, 1279–1290 (1999)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    McDonnell, L.: Hilbert–Samuel and Hilbert–Kunz functions of zero-dimensional ideals. Ph.D. Thesis, The University of Nebraska, Lincoln (2011)Google Scholar
  46. 46.
    Miller, L., Swanson, I.: Hilbert–Kunz functions of 2 ×2 determinantal rings. arXiv:1206.1015Google Scholar
  47. 47.
    Monsky, P.: The Hilbert–Kunz function. Math. Ann. 263, 43–49 (1983)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Monsky, P.: The Hilbert–Kunz function of a characteristic 2 cubic. J. Algebra 197, 268–277 (1997)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Monsky, P.: Hilbert–Kunz functions in a family: point-S 4 quartics. J. Algebra 208, 343–358 (1998)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Monsky, P.: Hilbert–Kunz functions in a family: line-S 4 quartics. J. Algebra 208, 359–371 (1998)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Monsky, P.: Hilbert–Kunz functions for irreducible plane curves. J. Algebra 316, 326–345 (2007)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Monsky, P.: Rationality of Hilbert–Kunz multiplicities: a likely counterexample. Special volume in honor of Melvin Hochster. Michigan Math. J. 57, 605–613 (2008)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Monsky, P.: Algebraicity of some Hilbert–Kunz multiplicities (modulo a conjecture). arXiv:0907.2470Google Scholar
  54. 54.
    Monsky, P.: Transcendence of some Hilbert–Kunz multiplicities (modulo a conjecture). arXiv:0908.0971Google Scholar
  55. 55.
    Monksy, P., Teixeira, P.: p-fractals and power series. I. Some 2 variable results. J. Algebra 280, 505–536 (2004)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Monsky, P., Teixeira, P.: p-fractals and power series. II. Some applications to Hilbert–Kunz theory. J. Algebra 304, 237–255 (2006)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Roberts, P.: Le théorème d’intersection. C. R. Acad. Sci. Paris Sér. I Math. 304, 177–180 (1987)MATHGoogle Scholar
  58. 58.
    Schwede, K., Tucker, K.: A survey of test ideals. Progress in Commutative Algebra 2, Closures, Finiteness and Factorization, pp. 39–99. Walter de Gruyter GmbH & Co. KG, Berlin (2012)Google Scholar
  59. 59.
    Seibert, G.: The Hilbert–Kunz function of rings of finite Cohen–Macaulay type. Arch. Math. 69, 286–296 (1997)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Shepherd-Barron, N.I.: On a problem of Ernst Kunz concerning certain characteristic functions of local rings. Arch. Math. (Basel) 31, 562–564 (1978/79)Google Scholar
  61. 61.
    Singh, A.K.: The F-signature of an affine semigroup ring. J. Pure Appl. Algebra 196, 313–321 (2005)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Smith, K.E., Van den Bergh, M.: Simplicity of rings of differential operators in prime characteristic. Proc. London Math. Soc. 75, 32–62 (1997)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules. Cambridge University Press, Cambridge (2006)MATHGoogle Scholar
  64. 64.
    Teixeira, P.: Syzygy gap fractals I. Some structural results and an upper bound. J. Algebra 350, 132–162 (2012)MathSciNetMATHGoogle Scholar
  65. 65.
    Trivedi, V.: Semistability and Hilbert–Kunz multiplicities for curves. J. Algebra 284, 627–644 (2005)MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Trivedi, V.: Strong semistability and Hilbert–Kunz multiplicity for singular plane curves. Commutative algebra and algebraic geometry, pp. 165–173. Contemporary Mathematics, vol. 390. American Mathematical Society, Providence (2005)Google Scholar
  67. 67.
    Trivedi, V.: Hilbert–Kunz multiplicity and reduction mod p. Nagoya Math. J. 185, 123–141 (2007)Google Scholar
  68. 68.
    Tucker, K.: F-signature exists. InventionesGoogle Scholar
  69. 69.
    Upadhyay, S.: The Hilbert–Kunz function for binomial hypersurfaces. arXiv:1101.5936Google Scholar
  70. 70.
    Von Korff, M.: F-Signature of affine toric varieties. arXiv:1110.0552Google Scholar
  71. 71.
    Vraciu, A.: Drops in joint Hilbert–Kunz multiplicities and projective equivalence of ideals. Preprint (2012)Google Scholar
  72. 72.
    Watanabe, K.-i.: Chains of integrally closed ideals. In: Commutative algebra (Grenoble/Lyon, 2001). Contemporary Mathematics, vol. 331, pp. 353–358. American Mathematical Society, Providence (2003)Google Scholar
  73. 73.
    Watanabe, K.-i., Yoshida, K.-i.: Hilbert–Kunz multiplicity and an inequality between multiplicity and colength. J. Algebra 230, 295–317 (2000)Google Scholar
  74. 74.
    Watanabe, K.-i., Yoshida, K.: Hilbert–Kunz multiplicity of two-dimensional local rings. Nagoya Math. J. 162, 87–110 (2001)Google Scholar
  75. 75.
    Watanabe, K.-i., Yoshida, K.: Hilbert–Kunz multiplicity, McKay correspondence and good ideals in two-dimensional rational singularities. Manuscripta Math. 104, 275–294 (2001)Google Scholar
  76. 76.
    Watanabe, K.-i., Yoshida, K.-i.: Minimal relative Hilbert–Kunz multiplicity. Illinois J. Math. 48, 273–294 (2004)Google Scholar
  77. 77.
    Watanabe, K.-i., Yoshida, K.-i.: Hilbert–Kunz multiplicity of three-dimensional local rings. Nagoya Math. J. 177, 47–75 (2005)Google Scholar
  78. 78.
    Yao, Y.: Modules with finite F-representation type. J. London Math. Soc. 72, 53–72 (2005)MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    Yao, Y.: Observations on the F-signature of local rings of characteristic p. J. Algebra 299, 198–218 (2006)MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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