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On rings with small Hilbert-Kunz multiplicity

Manuel Blickle·Florian Enescu
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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 $p$ (characteristic) and $d$ (dimension), there exist a number $ε(d,p) > 0$ such that any nonregular unmixed ring $R$ has Hilbert-Kunz multiplicity at least $1+ε(d,p)$. We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and $F$-rational.

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Manuel Blickle, Florian Enescu. 2003-08-04. On rings with small Hilbert-Kunz multiplicity. https://arxiv.org/abs/math/0308022

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