Nagoya Mathematical Journal

New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension

Ian M. Aberbach and Florian Enescu

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We present results on the Watanabe–Yoshida conjecture for the Hilbert–Kunz multiplicity of a local ring of positive characteristic. By improving on a “volume estimate” giving a lower bound for Hilbert–Kunz multiplicity, we obtain the conjecture when the ring has either Hilbert–Samuel multiplicity less than or equal to 5 or dimension less than or equal to 6. For nonregular rings with fixed dimension, a new lower bound for the Hilbert–Kunz multiplicity is obtained.

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Nagoya Math. J., Volume 212 (2013), 59-85.

First available in Project Euclid: 1 August 2013

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Zentralblatt MATH identifier

Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]
Secondary: 13H15: Multiplicity theory and related topics [See also 14C17]


Aberbach, Ian M.; Enescu, Florian. New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension. Nagoya Math. J. 212 (2013), 59--85. doi:10.1215/00277630-2335204.

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