On the associated graded rings of parameter ideals in Buchsbaum rings
On the associated graded rings of parameter ideals in Buchsbaum rings
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布赫斯鲍姆环中参数理想的关联分级环
DOI：
10.1016/0021-8693(83)90109-6
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发表时间：
1983
影响因子：
0.9
通讯作者：
S. Goto
中科院分区：
文献类型：
--
作者：
S. Goto
本文的目的是仔细检查什么性质的参数系统享有内部的Buchsbaum环。设A是Noether局部环，dim A= d，m是A的极大理想.则A称为Buchsbaum，如果差是A的不变量1（A），不依赖于A的参数理想q的特定选择，其中I，（A/q）和e，（A）分别表示A-模A/q的长度和A相对于q的重数。这相当于说，每个系统a，a*，...，A的参数的a d是弱A序列，即等式（a，，...，ai）：ai，I=（ai，...，ai）：m对于所有0< i< d-1都成立[21]。因此每个Cohen-Macaulay局部环A都是Buchsbaum的，且I（A）= 0，反之亦然。在这个意义上，Buchsbaum环的概念是Cohen-Macaulay环的概念的扩展，并且起源于Vogel [29]对Buchsbaum [11]的一个问题的回答。注意，Buchsbaum环的概念可以规范地推广到Buchsbaum模的概念（参见，例如，[11，221]。设A的一个参数理想q，G表示伴随分次环G;（A）= ONa，q”/qn+“，q.回想一下，如果A是Cohen-Macaulay环，则G是Cohen-Macaulay环;实际上，在这种情况下，G是A/q上的多项式环。那么当A是布克斯鲍姆时会发生什么呢？这是本研究中的一个动机问题，并简要说明了我们的结论
The purpose of this paper is to closely examine what properties the systems of parameters enjoy inside of Buchsbaum rings. Let A be a Noetherian local ring with dim A= d and m the maximal ideal of A. Then A is called Buchsbaum if the difference is an invariant 1 (A) of A not depending on the particular choice of a parameter ideal q of A, where I,(A/q) and e,(A) denote the length of the A-module A/q and the multiplicity of A relative to q, respectively. This is equivalent to saying that every system a,, a*,..., ad of parameters for A is a weak A-sequence, ie, the equality (a,,..., ai): a,, I=(a,,..., ai): m holds for all 0< i< d-1 [21]. Thus every Cohen-Macaulay local ring A is clearly Buchsbaum with I (A)= 0 and vice versa. In this sense the notion of Buchsbaum rings is an extension of that of Cohen-Macaulay rings and is originated in an answer of Vogel [29] to a problem of Buchsbaum [11. Notice that the concept of Buchsbaum rings may be canonically generalized to that of Buchsbaum modules (cf., eg,[11, 221). Now let us fix a parameter ideal q of A and let G denote the associated graded ring G;(A)= Ona,, q”/qn+’of q. Recall that the ring G is Cohen-Macaulay if A is so; actually G is a polynomial ring over A/q in this case. Then what happens when A is Buchsbaum? This is a motive problem in the present research and our conclusion is briefly stated