正则环中的迭代基元与积分依赖
Iterated socles and integral dependence in regular rings
摘要：
设$R$是域上的形式幂级数环，具有极大理想$\mathfrak m$，$I$是$R$的理想，使得$R/I$是Artin环.本文研究了$I$的迭代基元，即对于固定的正整数$s$，定义为最大理想$J$且具有$J\mathfrak m^s\subset I$的理想.我们感兴趣的是这些理想与理想的积分依赖性的概念。本文证明了在$I$上的迭代样条是积分的，约简数为1，条件是$s \leq \text{o}(I_1(φ_d))-1$，其中$\text{o}(I_1(φ_d))$是最后一个映射的元素在$R/I$的最小自由$R$ -分辨率中的理想的阶.在特征零点，我们还提供了每当$s\leq \text{o}(I_1(φ_d))$时迭代socles的生成元的公式。这一结果推广了Herzog以前的工作，他给出了任何${\mathfrak m}$ -主齐次理想$I$的柱脚生成元的公式，用最小齐次自由$R$ -$R/I$的分解中矩阵项的雅可比行列式表示。给出了具有一般高度的行列式理想的迭代基元的应用。特别是，我们给出了令人惊讶的简单公式的高度两个理想的幂级数环中的两个变量的迭代socles。这些生成元是从Hilbert-Burch矩阵中得到的合适的行列式。
英文摘要：
Let $R$ be a formal power series ring over a field, with maximal ideal $\mathfrak m$, and let $I$ be an ideal of $R$ such that $R/I$ is Artinian. We study the iterated socles of $I$, that is the ideals which are defined as the largest ideal $J$ with $J\mathfrak m^s\subset I$ for a fixed positive integer $s$. We are interested in these ideals in connection with the notion of integral dependence of ideals. In this article we show that the iterated socles are integral over $I$, with reduction number one, provided $s \leq \text{o}(I_1(φ_d))-1$, where $\text{o}(I_1(φ_d))$ is the order of the ideal of entries of the last map in a minimal free $R$-resolution of $R/I$. In characteristic zero, we also provide formulas for the generators of iterated socles whenever $s\leq \text{o}(I_1(φ_d))$. This result generalizes previous work of Herzog, who gave formulas for the socle generators of any ${\mathfrak m}$-primary homogeneous ideal $I$ in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free $R$-resolution of $R/I$. Applications are given to iterated socles of determinantal ideals with generic height. In particular, we give surprisingly simple formulas for iterated socles of height two ideals in a power series ring in two variables. These generators are suitable determinants obtained from the Hilbert-Burch matrix.
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