A Note on Gorenstein Rings of Embedding Codimension Three
A Note on Gorenstein Rings of Embedding Codimension Three
复制标题
嵌入余维三Gorenstein环的注解
DOI：
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发表时间：
1973
影响因子：
0.8
通讯作者：
J. Watanabe
中科院分区：
文献类型：
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作者：
J. Watanabe
设A = R/，其中R是任意维数的正则局部环，是R的理想.如果A是Gorenstein环，且高度= 2，则很容易证明A是完全交，即，由两个元素生成（Serre [5]，命题3）。因此Gorenstein环不是完全相交的嵌入余维数至少为3。在Bass的论文[1]（第29页）中可以找到这些环的一个例子。这是作为三维正则局部环与由五个元素生成的理想的商获得的，即，由一个常规序列加上两个以上的元素生成。在本文中，建议通过这个例子，我们证明了，如果A是Gorenstein环，如果高度= 3，那么是极小生成的奇数个元素。如果A有一个更大的余维，那么就可能对的生成元的最小数目没有这样的限制，这将从证明中得到。
Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.