A solution to Itoh's conjecture for integral closure filtration

Abstract

Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring of dimension $d \geq 3$ 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 in 1992 that if third Hilbert coefficient of $G_{\mathfrak{a}}(A)^*$ , i.e., $e_3^{\mathfrak{a}^*}(A) = 0$ and $A$ is Gorenstein then $G_{\mathfrak{a}}(A)^*$ is Cohen-Macaulay. In this paper we prove Itoh's conjecture (more generally for analytically unramified Cohen-Macaulay local rings).