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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, [email protected]
Ken-ichi Yoshida
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan, [email protected]
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Abstract

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In this paper, we investigate the lower bound sHK(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 sHK (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]]/(X2 + Y2 + Z2 + W2) 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
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 177 , 2005 , pp. 47 - 75
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

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