On rings with small Hilbert-Kunz multiplicity

Details On rings with small Hilbert-Kunz multiplicity On rings with small Hilbert-Kunz multiplicity

2003-08-04

arxiv.org/abs/math/0308022v1

Proc. Amer. Math. Soc. 132 (2004), no. 9, 2505--2509

Mathematics

Commutative Algebra

5 pages. to appear in Proc. AMS

Scientific paper

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 $p$ (characteristic) and $d$ (dimension), there exist a number $\epsilon(d,p) > 0$ such that any nonregular unmixed ring $R$ has Hilbert-Kunz multiplicity 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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