Nagoya Mathematical Journal

Hilbert-Kunz multiplicity of three-dimensional local rings

Kei-Ichi Watanabe and Ken-Ichi Yoshida

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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 $\dim A \ge 4$ 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.

Article information

Nagoya Math. J., Volume 177 (2005), 47-75.

First available in Project Euclid: 27 April 2005

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Zentralblatt MATH identifier

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]
Secondary: 13H05: Regular local rings 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13H15: Multiplicity theory and related topics [See also 14C17]


Watanabe, Kei-Ichi; Yoshida, Ken-Ichi. Hilbert-Kunz multiplicity of three-dimensional local rings. Nagoya Math. J. 177 (2005), 47--75.

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