Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 245–278 | Cite as

Equimultiplicity in Hilbert–Kunz theory

  • Ilya SmirnovEmail author


We study further the properties of Hilbert–Kunz multiplicity as a measure of singularity. This paper develops a theory of equimultiplicity for Hilbert–Kunz multiplicity and applies it to study the behavior of Hilbert–Kunz multiplicity on the Brenner–Monsky hypersurface. A number of applications follows, in particular we show that Hilbert–Kunz multiplicity attains infinitely many values and that equimultiple strata may not be locally closed.


Hilbert–Kunz multiplicity Tight closure Equimultiplicity 

Mathematics Subject Classification

13D40 13A35 13H15 14B05 



The results of this paper are a part of the author’s thesis written under Craig Huneke in the University of Virginia. I am indebted to Craig for his support and guidance. This project would not be possible without his constant encouragement. I also want to thank Mel Hochster and the anonymous referee who carefully read and helped to improve this manuscript.


  1. 1.
    Aberbach, I.M., Enescu, F.: Lower bounds for Hilbert–Kunz multiplicities in local rings of fixed dimension. Mich. Math. J. 57, 1–16 (2008). Special volume in honor of Melvin HochsterMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aberbach, I.M., Hochster, M., Huneke, C.: Localization of tight closure and modules of finite phantom projective dimension. J. Reine Angew. Math. 434, 67–114 (1993)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bierstone, E., Milman, P.D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blickle, M., Enescu, F.: On rings with small Hilbert–Kunz multiplicity. Proc. Am. Math. Soc. 132(9), 2505–2509 (2004). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brenner, H., Monsky, P.: Tight closure does not commute with localization. Ann. Math. (2) 171(1), 571–588 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, H., Li, J., Miller, C.: A direct limit for limit Hilbert–Kunz multiplicity for smooth projective curves. J. Algebra 372, 488–504 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dinh, T.T.: Associated primes of the example of Brenner and Monsky. Commun. Algebra 41(1), 109–123 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Epstein, N.: A tight closure analogue of analytic spread. Math. Proc. Camb. Philos. Soc. 139(2), 371–383 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Epstein, N., Vraciu, A.: A length characterization of \(*\)-spread. Osaka J. Math. 45(2), 445–456 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hochster, M., Huneke, C.: Tight closure, invariant theory, and the Briançon–Skoda theorem. J. Am. Math. Soc. 3(1), 31–116 (1990)zbMATHGoogle Scholar
  11. 11.
    Hochster, M., Huneke, C.: \(F\)-regularity, test elements, and smooth base change. Trans. Am. Math. Soc. 346(1), 1–62 (1994)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hochster, M., Huneke, C.: Localization and test exponents for tight closure. Mich. Math. J. 48, 305–329 (2000). Dedicated to William Fulton on the occasion of his 60th birthdayMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huneke, C., Yao, Y.: Unmixed local rings with minimal Hilbert–Kunz multiplicity are regular. Proc. Am. Math. Soc. 130(3), 661–665 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kunz, E.: Characterizations of regular local rings of characteristic \(p\). Am. J. Math. 91, 772–784 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kunz, E.: On Noetherian rings of characteristic \(p\). Am. J. Math. 98(4), 999–1013 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lech, C.: On the associativity formula for multiplicities. Ark. Mat. 3, 301–314 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lipman, J.: Equimultiplicity, reduction, and blowing up. In: Commutative algebra (Fairfax, Va., 1979), Lecture Notes in Pure and Appl. Math., vol. 68, pp. 111–147. Dekker, New York, (1982)Google Scholar
  18. 18.
    Ma, L.: Lech’s conjecture in dimension three. Adv. Math. 322, 940–970 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Matsumura, H.: Commutative algebra, Mathematics Lecture Note Series, vol. 56, 2nd edn. Benjamin Cummings Publishing Co., Inc., Reading (1980)Google Scholar
  20. 20.
    Monsky, P.: The Hilbert–Kunz function. Math. Ann. 263(1), 43–49 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Monsky, P.: Hilbert–Kunz functions in a family: line-\(S_4\) quartics. J. Algebra 208(1), 359–371 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Monsky, P.: Hilbert–Kunz functions in a family: point-\(S_4\) quartics. J. Algebra 208(1), 343–358 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nagata, M.: The theory of multiplicity in general local rings. In: Proceedings of the international symposium on algebraic number theory, Tokyo & Nikko, 1955, pp. 191–226. Science Council of Japan, Tokyo (1956)Google Scholar
  24. 24.
    Rees, D.: Rings associated with ideals and analytic spread. Math. Proc. Camb. Philos. Soc. 89(3), 423–432 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shepherd-Barron, N.I.: On a problem of Ernst Kunz concerning certain characteristic functions of local rings. Arch. Math. (Basel), 31(6), 562–564 (1978/1979)Google Scholar
  26. 26.
    Singh, A.K., Swanson, I.: Associated primes of local cohomology modules and of Frobenius powers. Int. Math. Res. Not. 33, 1703–1733 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Smirnov, I.: Upper semi-continuity of the Hilbert-Kunz multiplicity. Compos. Math. 152(3), 477–488 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Trivedi, V.: Hilbert–Kunz multiplicity and reduction mod \(p\). Nagoya Math. J. 185, 123–141 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tucker, K.: \(F\)-signature exists. Invent. Math. 190(3), 743–765 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Watanabe, K., Yoshida, K.: Hilbert–Kunz multiplicity and an inequality between multiplicity and colength. J. Algebra 230(1), 295–317 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations