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摘要翻译：
给定一个Artinian局部环$R$，我们定义了它的Gorenstein colength$G(R)$，以度量Gorenstein Artin局部环对$R$的逼近程度。本文证明了在以下两种情况下，$R=T/I$满足不等式$G(R)\leq\lambda(R/\soc(R))$：(a)$T$是特征零点域上的幂级数环，$I$是参数系统的幂的理想；(b)$T$是具有无穷余域的二维正则局部环，$I$是$T$的极大理想的主次。在第一种情况下，我们通过构造到$R$上的Gorenstein Artin局部环映射来计算$G(R)$。我们进一步利用这个构造证明了一个参数系统的$n$次方的理想通过Gorenstein理想直接与$(n-1)$st次方相关联。一个类似的方法表明，这种理想也通过Gorenstein理想直接与自己联系起来。关键词：戈伦斯坦；戈伦斯坦联动。
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英文标题：
《Computing Gorenstein Colength》
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作者：
H. Ananthnarayan
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Commutative Algebra 交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Given an Artinian local ring $R$, we define its Gorenstein colength $g(R)$ to measure how closely we can approximate $R$ by a Gorenstein Artin local ring. In this paper, we show that $R = T/I$ satisfies the inequality $g(R) \leq \lambda(R/\soc(R))$ in the following two cases: (a) $T$ is a power series ring over a field of characteristic zero and $I$ an ideal that is the power of a system of parameters or (b) $T$ is a 2-dimensional regular local ring with infinite residue field and $I$ is primary to the maximal ideal of $T$. In the first case, we compute $g(R)$ by constructing a Gorenstein Artin local ring mapping onto $R$. We further use this construction to show that an ideal that is the $n$th power of a system of parameters is directly linked to the $(n-1)$st power via Gorenstein ideals. A similar method shows that such ideals are also directly linked to themselves via Gorenstein ideals. Keywords: Gorenstein colength; Gorenstein linkage.
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PDF链接：
https://arxiv.org/pdf/0810.4542
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