论文标题
ITOH对正常理想的猜想
Itoh's conjecture for normal ideals
论文作者
论文摘要
令$（a，\ mathfrak {m}）$为分析不明的cohen-macaulay本地环，让$ \ mathfrak {a} $为$ \ mathfrak {m mathfrak {m} $ - $ a $中的主要理想。如果$ i $是$ a $中的理想选择，那么让$ i^*$成为$ a $ a $的整体关闭。令$ g _ {\ mathfrak {a}}（a）^*= \ bigoplus_ {n \ geq 0}（\ Mathfrak {\ Mathfrak {a}^n）^*/（\ Mathfrak {\ mathfrak {a}^{a}^{n+1}） ITOH猜想，如果$ e_3^{\ m athfrak {a}^*}（a）= 0 $，而$ a $是gorenstein，则$ g _ {\ mathfrak {a}}}}（a）^* $ is cohen-macaulay。在本文中，我们证明了Itoh的猜想的重要情况：我们表明，如果$ a $是Cohen-Macaulay，并且如果$ \ Mathfrak {a} $是正常的（即$ \ Mathfrak {A} a}^n $，则在所有$ n \ geq 1 $上是$ e_3^^^\ mathfrak a的$ n \ egeq 1 $的正常关闭$ g_ \ mathfrak {a}（a）$是Cohen-Macaulay。
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.
