Hostname: page-component-84b7d79bbc-7nlkj Total loading time: 0 Render date: 2024-07-26T04:04:00.535Z Has data issue: false hasContentIssue false
  • English
  • Français

New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension

Published online by Cambridge University Press:  11 January 2016

Ian M. Aberbach  and
Florian Enescu
Ian M. Aberbach
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA, aberbachi@missouri.edu
Florian Enescu
Affiliation:
Department of Mathematics and Statistics, Georgia State University, Atlanta, Georgia 30303, USA, fenescu@gsu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present results on the Watanabe–Yoshida conjecture for the Hilbert–Kunz multiplicity of a local ring of positive characteristic. By improving on a “volume estimate” giving a lower bound for Hilbert–Kunz multiplicity, we obtain the conjecture when the ring has either Hilbert–Samuel multiplicity less than or equal to 5 or dimension less than or equal to 6. For nonregular rings with fixed dimension, a new lower bound for the Hilbert–Kunz multiplicity is obtained.

Keywords

13A3513H15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 212 , December 2013 , pp. 59 - 85
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[1] 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. MR 2492437. DOI 10. 1307/mmj/1220879393.CrossRefGoogle Scholar
[2] Blickle, M. and Enescu, F., On rings with small Hilbert-Kunz multiplicity, Proc. Amer. Math. Soc. 132 (2004), 2505–2509. MR 2054773. DOI 10.1090/ S0002-9939-04-07469-6.CrossRefGoogle Scholar
[3] Celikbas, O., Dao, H., Huneke, C., and Zhang, Y., Bounds on the Hilbert-Kunz multiplicity, Nagoya Math. J. 205 (2012), 149-165. MR 2891167.CrossRefGoogle Scholar
[4] Chakerian, D. and Logothetti, D., Cube slices, pictorial triangles, and probability, Math. Mag. 64 (1991), 219–241. MR 1131009. DOI 10.2307/2690829.CrossRefGoogle Scholar
[5] Enescu, F. and Shimomoto, K., On the upper semi-continuity of the Hilbert-Kunz multiplicity, J. Algebra 285 (2005), 222–237. MR 2119113. DOI 10.1016/j.jalgebra. 2004.11.014.CrossRefGoogle Scholar
[6] Eto, K. and Yoshida, K.-i., Notes on Hilbert-Kunz multiplicity of Rees Algebra, Comm. Algebra 31 (2003), 5943-5976. MR 2014910. DOI 10.1081/AGB-120024861.Google Scholar
[7] Goto, S. and Nakamura, Y., Multiplicity and tight closures of parameters, J. Algebra 244 (2001), 302–311. MR 1856539. DOI 10.1006/jabr.2001.8907.CrossRefGoogle Scholar
[8] Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Brian¸con-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31–116. MR 1017784. DOI 10.2307/1990984.Google Scholar
[9] Huneke, C. and Swanson, I., Integral closure of ideals, rings, and modules, London Math. Soc. Lecture Note Ser. 336, Cambridge University Press, Cambridge, 2006. MR 2266432.Google Scholar
[10] Monsky, P., The Hilbert-Kunz function, Math. Ann. 263 (1983), 43–49. MR 0697329. DOI 10.1007/BF01457082.CrossRefGoogle Scholar
[11] Monsky, P. and Gessel, I., The limit as p → ∞ of the Hilbert-Kunz multiplicity of , preprint, 2010, arXiv:1007.2004 [math.AC].Google Scholar
[12] Nagata, M., Local Rings, Pure Appl. Math. (Hoboken) 13, Wiley, New York, 1962. MR 0155856.Google Scholar
[13] Northcott, D. G. and Rees, D., Reductions of ideals in local rings, Math. Proc. Cambridge Philos. Soc. 50 (1954), 145–158. MR 0059889.Google Scholar
[14] Rees, D., Valuations associated with ideals, II, J. London Math. Soc. 31 (1956), 221–228. MR 0078971.CrossRefGoogle Scholar
[15] Sally, J. D., Numbers of Generators of Ideals in Local Rings, Marcel Dekker, New York, 1978. MR 0485852.Google Scholar
[16] Sally, J. D., Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra 56 (1979), 168–183. MR 0527163. DOI 10.1016/0021-8693(79)90331-4.CrossRefGoogle Scholar
[17] Sally, J. D., Tangent cones at Gorenstein singularities, Compos. Math. 40 (1980), 167–175. MR 0563540.Google Scholar
[18] Watanabe, K.-i. and Yoshida, K.-i., Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra 230 (2000), 295–317. MR 1774769. DOI 10. 1006/jabr.1999.7956.CrossRefGoogle Scholar
[19] Watanabe, K.-i. and Yoshida, K.-i., Hilbert-Kunz multiplicity of two-dimensional local rings, Nagoya Math. J. 162 (2001), 87–110. MR 1836134.CrossRefGoogle Scholar
[20] Watanabe, K.-i. and Yoshida, K.-i., Hilbert-Kunz multiplicity of three-dimensional local rings, Nagoya Math. J. 177 (2005), 47–75. MR 2124547.CrossRefGoogle Scholar
[21] Wolfram Research, Mathematica, Version 7.0, Champaign, IL, 2008.Google Scholar
[22] Yoshida, K.-i., “Small Hilbert-Kunz multiplicity and (A1 )-type singularity” in Proceedings of the 4th Japan-Vietnam Joint Seminar on Commutative Algebra by and for Young Mathematicians, Meiji University, Japan, 2009.Google Scholar