计算Gorenstein色长
Computing Gorenstein Colength
摘要：
给定一个Artin局部环$R$，我们定义了它的GorensteinColength $g(R)$来度量用GorensteinArtin局部环逼近$R$的程度.本文证明了$R = T/I$在以下两种情况下满足不等式$g(R) \leq λ(R/\soc(R))$：（a）$T$是特征为零的域上的幂级数环，$I$是参数系的幂;（b）$T$是具有无穷剩余域的2维正则局部环，$I$是$T$的极大理想的素性.在第一种情况下，我们计算$g(R)$通过构建GorensteinArtin局部环映射到$R$。我们进一步使用这种结构表明，一个理想，是$n$次幂的参数系统是直接链接到$(n-1)$第一权力通过Gorenstein理想。类似的方法表明，此类理想也通过Gorenstein理想直接与自身联系。关键词：Gorenstein长度; Gorenstein连锁
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 $g(R) \leq λ(R/\soc(R))$ 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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