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28 March 2024
 
  » arxiv » math.AC/0307294

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Hilbert-Kunz multiplicity of three-dimensional local rings
Kei-ichi Watanabe ; Ken-ichi Yoshida ;
Date 22 Jul 2003
Subject Commutative Algebra MSC-class: Primary 13D40 Secondary 13A35, 13H05, 13H10, 13H15 | math.AC
AffiliationNihon University), Ken-ichi Yoshida (Nagoya University
AbstractIn this paper, we investigate a lower bound (say $s_{HK}(p,d)$) on Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension $d$ with characteristic $p>0$. Especially, we focus 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-degnerate quadric hyperplanes $k[[X,Y,Z,W]]/(X^2+Y^2+Z^2+W^2)$ 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 of $dim A = 4$ using the similar method as in the proof of the main theorem.
Source arXiv, math.AC/0307294
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