The multiplicity and the number of generators of an integrally closed ideal

TL;DR

This paper investigates an inequality relating generators, Loewy length, and multiplicity of integrally closed ideals in Noetherian local rings, exploring its validity across different singularity types and ring classes.

Contribution

It establishes the inequality for regular local rings, rational singularities, and cDV singularities, and classifies when it holds for Cohen-Macaulay rings of low dimension.

Findings

The inequality holds for regular local rings in all dimensions.

It is verified for rational singularities in dimension 2.

It is verified for cDV singularities in dimension 3.

Abstract

Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak m$-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 $R$ 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, multiplicity of the fiber cone, and the $h$-vector of the associated graded ring.