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Bounds on the Hilbert-Kunz multiplicity

Published online by Cambridge University Press:  11 January 2016

Olgur Celikbas
Affiliation:
Department of Mathematics University of Kansas Lawrence, Kansas 66045-7523, USA ocelikbas@math.ku.edu
Hailong Dao
Affiliation:
Department of Mathematics University of Kansas Lawrence, Kansas 66045-7523, USA hdao@math.ku.edu
Craig Huneke
Affiliation:
Department of Mathematics University of Kansas Lawrence, Kansas 66045-7523, USA huneke@math.ku.edu
Yi Zhang
Affiliation:
Department of Mathematics University of Minnesota Minneapolis, Minnesota 55455, USA zhang397@umn.edu
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Abstract

In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed nonregular local rings, bounding them uniformly away from 1. Our results improve previous work of Aberbach and Enescu.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

References

[AE] Aberbach, I. M. and Enescu, F., Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension, Michigan Math. J. 57 (2008), 116.CrossRefGoogle Scholar
[AE1] Aberbach, I. M. and Enescu, F., New estimates of Hilbert-Kunz multiplicities for local rings of fixed dimension, preprint, arXiv:1101.5078v2 [math.AC] Google Scholar
[AL] Aberbach, I. M. and Leuschke, G., The F-signature and strong F-regularity, Math. Res. Lett. 10 (2003), 5156.CrossRefGoogle Scholar
[BE] Blickle, M. and Enescu, F., On rings with small Hilbert-Kunz multiplicity, Proc. Amer. Math. Soc. 132 (2004), 25052509.CrossRefGoogle Scholar
[BH] Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993.Google Scholar
[Ch] Choi, S., Betti numbers and the integral closure of ideals, Ph.D. dissertation, Purdue University, West Lafayette, Indiana, 1989.Google Scholar
[ES] Enescu, F. and Shimomoto, K., On the upper semi-continuity of the Hilbert-Kunz multiplicity, J. Algebra 285 (2005), 222237.CrossRefGoogle Scholar
[GM] Gessel, I. and Monsky, P., The limit as p—> ∞ of the Hilbert-Kunz multiplicity of Σxdi i , preprint, arXiv:1007.2004v1 [math.AC] +∞+of+the+Hilbert-Kunz+multiplicity+of+Σxdi+i+,+preprint,+arXiv:1007.2004v1+[math.AC]>Google Scholar
[HH] Hochster, M. and Huneke, C., Tight closure, invariant theory and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31116.Google Scholar
[HoY] Hochster, M. and Yao, Y., Second coefficients of Hilbert-Kunz functions for domains, preprint.Google Scholar
[H] Huneke, C., Tight Closure and its Applications, with an appendix by Melvin Hochster, CBMS Reg. Conf. Ser. Math. 88, Amer. Math. Soc, Providence, 1996.Google Scholar
[HL] Huneke, C. and Leuschke, G., Two theorems about maximal Cohen-Macaulay modules, Math. Ann. 324 (2002), 391404.CrossRefGoogle Scholar
[HMM] Huneke, C., McDermott, M. A., and Monsky, P., Hilbert-Kunz functions for normal rings, Math. Res. Lett. 11 (2004), 539546.CrossRefGoogle Scholar
[HY] Huneke, C. and Yao, Y., Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular, Proc. Amer. Math. Soc. 130 (2002), 661665.CrossRefGoogle Scholar
[Ku] Kurano, K., On Roberts rings, J. Math. Soc. Japan 53 (2001), 333355.CrossRefGoogle Scholar
[M] Monsky, P., The Hilbert-Kunz function, Math. Ann. 263 (1983), 4349.CrossRefGoogle Scholar
[SH] Swanson, I. and Huneke, C., Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge University Press, Cambridge, 2006.Google Scholar
[T] Tucker, K., F-signature exists, preprint, 2010.Google Scholar
[W] Watanabe, K.-i.Chains of integrally closed ideals” in Commutative Algebra (Grenoble/Lyon, 2001), Contemp. Math. 331, Amer. Math. Soc, Providence, 2003, 353358.Google Scholar
[WY1] Watanabe, K.-i. and Yoshida, K.-i., Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra 230 (2000), 295317.CrossRefGoogle Scholar
[WY2] Watanabe, K.-i. and Yoshida, K.-i., Minimal relative Hilbert-Kunz multiplicity, Illinois J. Math. 48 (2004), 273294.Google Scholar
[WY3] Watanabe, K.-i. and Yoshida, K.-i., Hilbert-Kunz multiplicity of three-dimensional local rings, Nagoya Math. J. 177 (2005), 4775.CrossRefGoogle Scholar
[Y] Yao, Y., Observations on the F-signature of local rings of characteristic p, J. Algebra 299 (2006), 198218.CrossRefGoogle Scholar

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