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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On rings with small Hilbert-Kunz multiplicity


Authors: Manuel Blickle and Florian Enescu
Journal: Proc. Amer. Math. Soc. 132 (2004), 2505-2509
MSC (2000): Primary 13A35
Published electronically: April 8, 2004
MathSciNet review: 2054773
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Abstract: A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed $p$ and $d$, there exists a number $\epsilon(d,p) > 0$ such that for any nonregular unmixed ring $R$ its Hilbert-Kunz multiplicity is at least $1+\epsilon(d,p)$. We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and $F$-rational.


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Additional Information

Manuel Blickle
Affiliation: FB6 Mathematik, Universität Essen, 45117 Essen, Germany
Email: manuel.blickle@uni-essen.de

Florian Enescu
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112; Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Email: enescu@math.utah.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07469-6
PII: S 0002-9939(04)07469-6
Keywords: Regular rings, Hilbert-Kunz multiplicity, $F$-rational rings
Received by editor(s): October 31, 2002
Published electronically: April 8, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 by the authors