论文标题
关于系数理想
On Coefficient ideals
论文作者
论文摘要
令$（A，\ Mathfrak {M}）$为Cohen-Macaulay dimension $ d \ geq 2 $带有无限残留字段，让$ i $为$ \ mathfrak {m Mathfrak {m} $ - 主要理想。对于$ 0 \ leq i \ leq d $ let $ i_i $是$ i^{th} $ - 系数$ i $的理想。另外，令$ \ widetilde {i} = i_d $表示$ a $的ratliff-rush闭合。令$ g = g_i（a）$为$ i $的相关分级环。我们表明，如果$ \ dim h^j_ {g _+}（g）^\ vee \ leq j -1 $对于$ 1 \ 1 \ leq j \ leq j \ leq i \ leq i \ leq d-1 $，则$（i^n）_ {d-i} = \ widetilde {i^n} $ for All $ n \ geq 1 $。特别是，如果$ g $是概括的cohen-macaulay，则$（i^n）_1 = \ widetilde {i^n} $ for All $ n \ geq 1 $。结果，我们会发现，如果$ a $是一个具有$ g $ gentrized cohen-macaulay的分析不明的域，那么$ s_2 $ - rees algebra $ a [it] $ as $ \ bigoplus_ {n \ geq 0} \ geq 0} \ wideTilde {i^n} $。
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$ with infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. For $0 \leq i \leq d$ let $I_i$ be the $i^{th}$-coefficient ideal of $I$. Also let $\widetilde{I} = I_d$ denote the Ratliff-Rush closure of $A$. Let $G = G_I(A)$ be the associated graded ring of $I$. We show that if $\dim H^j_{G_+}(G)^\vee \leq j -1$ for $1 \leq j \leq i \leq d-1$ then $(I^n)_{d-i} = \widetilde{I^n}$ for all $n \geq 1$. In particular if $G$ is generalized Cohen-Macaulay then $(I^n)_1 = \widetilde{I^n}$ for all $n \geq 1$. As a consequence we get that if $A$ is an analytically unramified domain with $G$ generalized Cohen-Macaulay, then the $S_2$-ification of the Rees algebra $ A[It]$ is $\bigoplus_{n \geq 0} \widetilde{I^n}$.
