Itoh's conjecture for normal ideals

TL;DR

This paper proves a significant case of Itoh's conjecture, demonstrating that for Cohen-Macaulay rings with a normal ideal and vanishing third Hilbert coefficient, the associated graded ring is Cohen-Macaulay.

Contribution

It establishes that under certain conditions, the associated graded ring of a normal ideal with zero third Hilbert coefficient is Cohen-Macaulay, confirming a key case of Itoh's conjecture.

Findings

Proves Cohen-Macaulayness of the associated graded ring under specified conditions.

Confirms a key case of Itoh's conjecture for normal ideals.

Highlights the role of the third Hilbert coefficient in the Cohen-Macaulay property.

Abstract

Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring and let $\mathfrak{a}$ be an $\mathfrak{m}$-primary ideal in $A$. If $I$ is an ideal in $A$ then let $I^*$ be the integral closure of $I$ in $A$. Let $G_{\mathfrak{a}}(A)^* = \bigoplus_{n\geq 0 }(\mathfrak{a}^n)^*/(\mathfrak{a}^{n+1})^*$ be the associated graded ring of the integral closure filtration of $\mathfrak{a}$. Itoh conjectured that if $e_3^{\mathfrak{a}^*}(A) = 0$ and $A$ is Gorenstein then $G_{\mathfrak{a}}(A)^* $ is Cohen-Macaulay. In this paper we prove an important case of Itoh's conjecture: we show that if $A$ is Cohen-Macaulay and if $\mathfrak{a}$ is normal (i.e., $\mathfrak{a}^n$ is integrally closed for all $n \geq 1$) with $e_3^\mathfrak{a}(A) = 0$ then $G_\mathfrak{a}(A)$ is Cohen-Macaulay.