Algebra structures on some canonical resolutions
Algebra structures on some canonical resolutions
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一些规范解析的代数结构
DOI：
10.1016/0021-8693(89)90244-5
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发表时间：
1989
影响因子：
0.9
通讯作者：
H. Srinivasan
中科院分区：
文献类型：
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作者：
H. Srinivasan
循环模有限自由分辨率的代数结构研究始于Buchsbaum和Eisenbud [B-El]。已知最小分辨率允许代数结构的循环模的类别包括剩余域[Gu]，完全交（Koszul复合体），同调维数最多为3的模[B-El]，余维数为4的Gorenstein [KM]和Herzog代数[K-M23]。Avramov [Al]给出的例子表明存在最小分辨率不允许代数结构的循环模。设R为具有极大理想m的诺etheran局部环。本文构造了两类循环模R/I的极小分辨率上的代数结构，即当Z为Jk形式时，其中J为正则序列生成的理想，当Z为一般n × m矩阵的极大次理想时，假设R包含有理数。在[Al]中，Avramov在模的最小分辨率上定义了代数结构存在的某些障碍，然后生成了具有非零障碍的模。一般的问题是，这些障碍的消失是否也足以证明循环模的最小代数解析的存在。当Z是由R和M= R/Z”中的正则序列对任意正整数k生成的理想时，则这些障碍物均为零。因此，Avramov和Schlessinger提出了R/Z '的最小分辨率是否承认代数结构的问题。本文推论3.6对这个问题给出了肯定的答案。
The study of algebra structures on finite free resolutions of cyclic modules begins with Buchsbaum and Eisenbud [B-El]. The classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field [Gu], complete intersections (Koszul complex), modules of homological dimension atmost 3 [B-El], Gorenstein of codimension four [KM], and Herzog algebras [K-M23. Avramov [Al] gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure. Let R be a noetherian local ring with maximal ideal m. In this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules R/I, namely, when Z is of the form Jk, where J is an ideal generated by a regular sequence, and when Z is the ideal of maximal minors of a generic n x m matrix, provided R contains the rationals. In [Al] Avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non-zero obstructions. The general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module. When Z is an ideal generated by a regular sequence in R and M= R/Z” for any positive integer k, then these obstructions are all zero. Therefore, Avramov and Schlessinger asked whether the minimal resolutions of R/Z’admit an algebra structure. Corollary 3.6 of this paper provides an affirmative answer to this question.