Integrally closed ideals of reduction number three

TL;DR

This paper investigates the Hilbert function of integrally closed ideals with reduction number three in Cohen-Macaulay rings, establishing a new inequality relating Hilbert coefficients and ideal length, with conditions for equality.

Contribution

It introduces a novel inequality for the Hilbert function of integrally closed ideals with reduction number three, extending previous bounds and characterizing equality cases.

Findings

Derived a new inequality involving Hilbert coefficients and ideal length.

Identified conditions under which the inequality becomes an equality.

Studied Cohen-Macaulayness of associated graded rings of determinantal rings.

Abstract

In a Cohen-Macaulay local ring $(A, \mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge \mathrm{e}_0(I) - \mathrm{e}_1(I) + \dfrac{\mathrm{e}_2(I) + \ell_A(I^2/QI)}{2}$, where $\mathrm{e}_i(I)$ denotes the $i$th Hilbert coefficients and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of $I$ is larger than or equal to $\dim A-1$. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.