On form rings which are Cohen-Macaulay
On form rings which are Cohen-Macaulay
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在 Cohen-Macaulay 成型环上
DOI：
10.1016/0021-8693(79)90159-5
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发表时间：
1979
期刊：
影响因子：
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通讯作者：
G. Valla
中科院分区：
文献类型：
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作者：
G. Valla
设m是d维局部Cohen-Macaulay环A的极大理想，则m的极小基上的生成元个数ZJ（m）由e（A）+ d-1有界，其中e（A）是A的重数（见[1]）.最近Sally证明了：若v（m）= e（A）+ d-1，则A关于m的形式环（记作GA（m））是Cohen-Macaulay环（见[5]）.本文将这个结果在两个方向上加以推广。首先，如果q是d维局部Cohen-Macaulay环A的m-准素理想和极大理想m，我们证明了G_i（q）是Cohen-Macaulay环，如果f_i（q/q”）= e（q）+（d-1）（1/q），其中e_i（M）表示A-模M的长度，e（q）表示q的重数（定理1）.另一方面，若p是局部Cohen-Macaulay环中高度为I的素理想，使得A/p是Cohen-Macaulay环，G_i（p）是自由A/p-模，则若v（p）= e（A_i）+ r-1，则G，（p）是Cohen-Macaulay（定理2）。本文假定所有的环都是交换的诺特环，并且有单位元。
If m is the maximal ideal of a d-dimensional local Cohen-Macaulay ring A, then ZJ (m), the number of generators in a minimal basis of m, is bounded above by e (A)+ d-1 where e (A) is the multiplicity of A (see [l]). Recently Sally proved that if v (m)= e (A)+ d-1, then the Form ring of A relative to m, denoted hereafter by GA (m), is Cohen-Macaulay (see [5]). In this note we extend this result in two directions. First, if q is an m-primary ideal of a local Cohen-Macaulay ring A of dimension d and maximal ideal m, we prove that G,(q) is Cohen-Macaulay if/,(9/q”)= e (q)+(d-1) & ‘&l/q), where e,(M) denotes the length of the A-module M and e (q) the multiplicity of q (Theorem 1).On the other hand, if p is a prime ideal of height I in a local Cohen-Macaulay ring, such that A/p is Cohen-Macaulay and G,(p) is a free A/p-module, we prove that if v (p)= e (A,)+ r-1 then G,(p) is Cohen-Macaulay(Theorem 2). In this paper all rings are supposed to be commutative noetherian and with identity.