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arXiv 1502.05231math.AC
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等倍系数理想

Equimultiple Coefficient Ideals

P. H. Lima, V. H. Jorge Perez

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摘要：

设$(R,\mathfrak{m})$是拟纯局部环，$I$是解析扩展$s$的$R$的等重理想.本文引入了等重系数理想。修正$k\in \{1,...,s\}.$包含$I$的最大理想$L$，使得对于每个$i \in \{1,...,k\}$和$I$的每个最小素数$\mathfrak{p}$的$e_{i}(I_{\mathfrak{p}})=e_{i}(L_{\mathfrak{p}})$称为第$k$个等倍系数理想，用$I_{k}$表示。它是Shah \cite{S}首次引入的系数理想在$\mathfrak{m}$ -准素理想情形下的推广。我们也看到了这些理想的应用。例如，我们证明了分次环$G_{I}(R)$满足$S_{1}$条件当且仅当对所有$n$都有$I^{n}=(I^{n})_{1}$.

英文摘要：

Let $(R,\mathfrak{m})$ be a quasi-unmixed local ring and $I$ an equimultiple ideal of $R$ of analytic spread $s$. In this paper, we introduce the equimultiple coefficient ideals. Fix $k\in \{1,...,s\}.$ The largest ideal $L$ containing $I$ such that $e_{i}(I_{\mathfrak{p}})=e_{i}(L_{\mathfrak{p}})$ for each $i \in \{1,...,k\}$ and each minimal prime $\mathfrak{p}$ of $I$ is called the $k$-th equimultiple coefficient ideal denoted by $I_{k}$. It is a generalization of the coefficient ideals firstly introduced by Shah \cite{S} for the case of $\mathfrak{m}$-primary ideals. We also see applications of these ideals. For instance, we show that the associated graded ring $G_{I}(R)$ satisfies the $S_{1}$ condition if and only if $I^{n}=(I^{n})_{1}$ for all $n$.

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