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Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings

Published online by Cambridge University Press:  11 January 2016

Kei-ichi Watanabe  and
Ken-ichi Yoshida
Kei-ichi Watanabe
Affiliation:
Department of Mathematics, College of Humanities and Sciences, Nihon University, Setagaya-ku, Tokyo 156-0045, Japan, watanabe@math.chs.nihon-u.ac.jp
Ken-ichi Yoshida
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan, yoshida@math.nagoya-u.ac.jp
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Abstract

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In this paper, we investigate the lower bound s HK(p, d) of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that s HK (p, 3) = 4/3 and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degenerate quadric hypersurface k[[X, Y, Z,W]]/(X 2 + Y 2 + Z 2 + W 2) under mild conditions.

Furthermore, we pose a generalization of the main theorem to the case of dim A ≥ 4 as a conjecture, and show that it is also true in case dim A = 4 using the similar method as in the proof of the main theorem.

Keywords

13D4013A3513H0513H1013H15

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 177 , 2005 , pp. 47 - 75
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

References

[1] Buchweitz, R. O. and Chen, Q., Hilbert-Kunz functions of cubic curves and surfaces, J. Algebra, 197 (1997), 246–267.CrossRefGoogle Scholar
[2] Buchweitz, R. O., Chen, Q. and Pardue, K., Hilbert-Kunz functions, preprint.Google Scholar
[3] Blickle, M. and Enescu, F., On rings with small Hilbert-Kunz multiplicity, Proc. Amer. Math. Soc., 132 (2004), 2505–2509.CrossRefGoogle Scholar
[4] Conca, A., Hilbert-Kunz functions of monomials and binomial hypersurfaces, Manu-scripta Math., 90 (1996), 287–300.CrossRefGoogle Scholar
[5] Eto, K. and Yoshida, K., Notes on Hilbert-Kunz multiplicity of Rees algebras, Comm. Algebra, 31 (2003), 5943–5976.CrossRefGoogle Scholar
[6] Fedder, R. and Watanabe, K.-i., A characterization of F-regularity in terms of F -purity, Commutative algebra (Berkeley, CA, 1987), Math. Sci. Research Inst. Publ., vol. 15, Springer-Verlag, New York (1989), pp. 227–245.Google Scholar
[7] Fakhruddin, N. and Trivedi, V., Hilbert-Kunz functions and multiplicities for full flag varieties and elliptic curves, J. Pure Appl. Algebra, 181 (2003), 23–52.CrossRefGoogle Scholar
[8] Goto, S. and Nakamura, Y., Multiplicity and tight closures of parameters, J. Algebra, 244 (2001), 302–311.CrossRefGoogle Scholar
[9] Hanes, D., Notes on the Hilbert-Kunz function, Comm. Algebra, 30 (2002), 3789–3812.CrossRefGoogle Scholar
[10] Han, C. and Monsky, P., Some surprising Hilbert-Kunz functions, Math. Z., 214 (1993), 119–135.CrossRefGoogle Scholar
[11] Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Brian¸con-Skoda theorem, J. Amer. Math. Soc., 3 (1990), 31–116.Google Scholar
[12] Hochster, M. and Huneke, C., F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346 (1994), 1–62.Google Scholar
[13] Huneke, C., Tight closure and its applications, American Mathematical Society, 1996.CrossRefGoogle Scholar
[14] 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
[15] Kunz, E., Characterizations of regular local rings of characteristic p, Amer. J. Math., 41 (1969), 772–784.CrossRefGoogle Scholar
[16] Kunz, E., On Noetherian rings of characteristic p, Amer. J. Math., 88 (1976), 999–1013.CrossRefGoogle Scholar
[17] Matsumura, H., Commutative ring theory, Cambridge University Press, 1986.Google Scholar
[18] Monsky, P., The Hilbert-Kunz function, Math. Ann., 263 (1983), 43–49.CrossRefGoogle Scholar
[19] Monsky, P., A personal letter from Monsky to K.-i. Watanabe.Google Scholar
[20] Nagata, M., Local rings, Interscience, 1962.Google Scholar
[21] Rees, D., A note on analytically unramified local rings, J. London Math. Soc., 36 (1961), 24–28.CrossRefGoogle Scholar
[22] Watanabe, K.-i. and Yoshida, K., Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra., 230 (2000), 295–317.CrossRefGoogle Scholar
[23] Watanabe, K.-i. and Yoshida, K., Hilbert-Kunz multiplicity of two-dimensional local rings, Nagoya Math. J., 162 (2001), 87–110.CrossRefGoogle Scholar
[24] Watanabe, K.-i. and Yoshida, K., Hilbert-Kunz multiplicity, McKay correspondence and good ideals in two-dimensional rational singularities, Manuscripta Math., 104 (2001), 275–294.CrossRefGoogle Scholar