On the roles of generalized linear CM modules in commutative ring theory

广义线性CM模在交换环理论中的作用

基本信息

批准号：
09640025
负责人：
YOSHIDA Ken-ichi
金额：
$ 1.92万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640025/
关键词：
Linear CM module Buchsbaum module multiplicity Linear Cohen-Macaulay module Linear Buchsbaum module homological degree

项目摘要

We have studied the generalization and the existence of linear maximal Cohen-Macaulay modules. As a result, we have proved some generalization theorem for linear maximal Cohen-Macaulay modules, and showed several properties of surjective Buchsbaum modules, the notion of which is a generalization of that of the linear Buchsbaum modules.A finitely generated module M over a local ring A is called a linear Cohen-Macaulay A-module if the associated graded module of M is a graded Cohen-Macaulay module which has a graded linear resolution. The above definition of linear Cohen-Macaulay module is equivalent to the following condition : the minimal number of generators of M is equal to the multiplicity of M.The last condition enables us to define a generalization of linear Cohen-Macaulay modules. In fact, the head-investigator and the other investigators have generalized of linear maximal Cohen-Macaulay modules to linear maximal Buchsbaum modules in terms of I-invariant, which is a important invariant for Buchsbaum modules. One of our main results in this investigation is a generalization theorem for linear Buchsbaum modules (thus linear Cohen-Macaulay modules) ; using the notion of homological degree introduced by Vasconcelos, we have removed the above obstruction. On the other hand, since it is hard to deal with homological degrees, the problem with generalization of linear Buchsbaum modules using another invariants is left us.Furthermore, throughout this investigation, we noticed that research of singularities is important and so that we began to study singularities of local rings with positive characteristic. We are now preparing papers about these research for publishing with Kei-ichi Watanabe (Nihon Univ.) .

我们研究了线性最大 Cohen-Macaulay 模的泛化和存在性。由此，我们证明了线性最大Cohen-Macaulay模的一些泛化定理，并展示了满射Buchsbaum模的几个性质，其概念是线性Buchsbaum模的泛化。如果 M 的关联分级模块是具有分级线性分辨率的分级 Cohen-Macaulay 模块，则环 A 称为线性 Cohen-Macaulay A 模块。上述线性 Cohen-Macaulay 模的定义等价于以下条件：M 的生成元的最小数量等于 M 的重数。最后一个条件使我们能够定义线性 Cohen-Macaulay 模的推广。事实上，首席研究员和其他研究员已经根据 I 不变量将线性最大 Cohen-Macaulay 模块推广到线性最大 Buchsbaum 模块，I 不变量是 Buchsbaum 模块的一个重要不变量。我们在这项研究中的主要结果之一是线性 Buchsbaum 模（即线性 Cohen-Macaulay 模）的泛化定理；利用Vasconcelos引入的同源度概念，我们消除了上述障碍。另一方面，由于很难处理同调度，因此使用另一个不变量泛化线性 Buchsbaum 模的问题就留给了我们。此外，在整个研究过程中，我们注意到奇点的研究很重要，因此我们开始研究具有正特性的局部环的奇点。我们现在正在准备有关这些研究的论文，以便与 Kei-ichi Watanabe（日本大学）一起发表。