Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-11T04:09:17.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Upper semi-continuity of the Hilbert–Kunz multiplicity

Part of: Commutative algebra: Homological methods Arithmetic rings and other special rings General commutative ring theory

Published online by Cambridge University Press:  02 February 2016

Ilya Smirnov
Ilya Smirnov*
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA email [email protected] Current address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the Hilbert–Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.

Keywords

Hilbert–Kunz multiplicitysemi-continuityF-finite rings

MSC classification

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 13F40: Excellent rings
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 3 , March 2016 , pp. 477 - 488
Copyright
© The Author 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aberbach, I. M. and Enescu, F., Lower bounds for Hilbert–Kunz multiplicities in local rings of fixed dimension, Michigan Math. J. 57 (2008), 1–16; Special volume in honor of Melvin Hochster; MR 2492437 (2010h:13028).CrossRefGoogle Scholar
Blickle, M. and Enescu, F., On rings with small Hilbert–Kunz multiplicity, Proc. Amer. Math. Soc. 132 (2004), 2505–2509 (electronic); MR 2054773 (2005b:13029).Google Scholar
Brenner, H. and Monsky, P., Tight closure does not commute with localization, Ann. of Math. (2) 171 (2010), 571–588; MR 2630050 (2011d:13005).Google Scholar
Enescu, F. and Shimomoto, K., On the upper semi-continuity of the Hilbert–Kunz multiplicity, J. Algebra 285 (2005), 222–237; MR 2119113 (2005j:13017).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 24 (1965); MR 0199181 (33 #7330).CrossRefGoogle Scholar
Hochster, M., Foundations of tight closure, Lecture Notes for Math 711, Fall 2007, University of Michigan (2007); available at http://www.math.lsa.umich.edu/∼hochster/711F07/711.html.Google Scholar
Hochster, M. and Huneke, C., F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 1–62; MR 1273534 (95d:13007).Google Scholar
Huneke, C. and Swanson, I., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336 (Cambridge University Press, Cambridge, 2006); MR 2266432 (2008m:13013).Google Scholar
Huneke, C. and Yao, Y., Unmixed local rings with minimal Hilbert–Kunz multiplicity are regular, Proc. Amer. Math. Soc. 130 (2002), 661–665.CrossRefGoogle Scholar
Kunz, E., Characterizations of regular local rings of characteristic p, Amer. J. Math. 91 (1969), 772–784; MR 0252389 (40 #5609).Google Scholar
Kunz, E., On Noetherian rings of characteristic p, Amer. J. Math. 98 (1976), 999–1013; MR 0432625 (55 #5612).Google Scholar
Matsumura, H., Commutative algebra, Mathematics Lecture Note Series, vol. 56, second edition (Benjamin/Cummings Publishing, Reading, MA, 1980); MR 575344 (82i:13003).Google Scholar
Monsky, P., The Hilbert–Kunz function, Math. Ann. 263 (1983), 43–49; MR 697329 (84k:13012).Google Scholar
Monsky, P., Hilbert–Kunz functions in a family: point-S 4 quartics, J. Algebra 208 (1998), 343–358; MR 1644019 (99k:13005).Google Scholar
Shepherd-Barron, N. I., On a problem of Ernst Kunz concerning certain characteristic functions of local rings, Arch. Math. (Basel) 31 (1978/79), 562–564; MR 531569 (81e:13012).CrossRefGoogle Scholar
Tucker, K., F-signature exists, Invent. Math. 190 (2012), 743–765; MR 2995185.Google Scholar
Watanabe, K. and Yoshida, K., Hilbert–Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra 230 (2000), 295–317; MR 1774769 (2001h:13032).Google Scholar