Itoh’s conjecture for normal ideals

Let \((A,\mathfrak {m} )\) $(A,\mathfrak{m})$ be an analytically unramified Cohen–Macaulay local ring and let \(\mathfrak {a} \) $\mathfrak{a}$ be an \(\mathfrak {m} \) $\mathfrak{m}$ -primary ideal in A. If I is an ideal in A then let \(I^*\) $I^{\ast }$ be the integral closure of I in A. Let \(G_{\mathfrak {a}}(A) ^* = \bigoplus _{n\ge 0 }(\mathfrak {a} ^n)^*/(\mathfrak {a} ^{n+1})^*\) $G_{\mathfrak{a}}{(A)}^{\ast }={⨁}_{n\ge 0}{({\mathfrak{a}}^n)}^{\ast }/{({\mathfrak{a}}^{n+1})}^{\ast }$ be the associated graded ring of the integral closure filtration of \(\mathfrak {a} \) $\mathfrak{a}$ . Itoh conjectured in 1992 that if third Hilbert coefficient of \(G_{\mathfrak {a}}(A) ^*\) $G_{\mathfrak{a}}{(A)}^{\ast }$ , i.e., \(e_3^{\mathfrak {a} ^*}(A) = 0\) $e_3^{{\mathfrak{a}}^{\ast }}(A)=0$ and A is Gorenstein then \(G_{\mathfrak {a}}(A) ^*\) $G_{\mathfrak{a}}{(A)}^{\ast }$ 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 \(\mathfrak {a} \) $\mathfrak{a}$ is normal (i.e., \(\mathfrak {a} ^n\) ${\mathfrak{a}}^n$ is integrally closed for all \(n \ge 1\) $n\ge 1$ ) with \(e_3^\mathfrak {a} (A) = 0\) $e_3^{\mathfrak{a}}(A)=0$ then \(G_\mathfrak {a} (A)\) $G_{\mathfrak{a}}(A)$ is Cohen–Macaulay.