The multiplicity and the number of generators of integrally closed ideals

Journal of Singularities
volume 19 (2019), 61-75

Received: 27 August 2018.

DOI: 10.5427/jsing.2019.19e

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Abstract:

Let (R, m) be a Noetherian local ring and I an m-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of I. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if Spec R has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension 2, and cDV singularities in dimension 3. In addition, we can classify when the inequality always hold for a Cohen-Macaulay R of dimension at most two. We also discuss relations to various topics: classical results on rings with minimal multiplicity and rational singularities, the recent work on p_g ideals by Okuma-Watanabe-Yoshida, a conjecture of Huneke, Mustaţă, and Watanabe on F-threshold, multiplicity of the fiber cone, and the h-vector of the associated graded ring.

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