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

DOI:
https://doi.org/10.1090/S0002-9939-04-07469-6

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 and , there exists a number such that for any nonregular unmixed ring its Hilbert-Kunz multiplicity is at least . We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and -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:
https://doi.org/10.1090/S0002-9939-04-07469-6

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