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

研究了线性极大Cohen-Macaulay模的泛化性和存在性。结果证明了线性极大Cohen-Macaulay模的一些推广定理，并给出了满射Buchsbaum模的几个性质，它的概念是线性Buchsbaum模的概念的推广。局部环A上的有限生成模M，如果M对应的分级模是具有分级线性分辨率的分级Cohen-Macaulay模，则称为线性Cohen-Macaulay A模。上述线性Cohen-Macaulay模的定义等价于以下条件：M的最小生成器数等于M的多重性。最后一个条件使我们能够定义线性Cohen-Macaulay模的推广。事实上，首席研究者和其他研究者已经将线性极大Cohen-Macaulay模推广到线性极大Buchsbaum模，其中i -不变量是Buchsbaum模的一个重要不变量。我们在这项研究中的主要结果之一是线性Buchsbaum模块（即线性Cohen-Macaulay模块）的推广定理；利用Vasconcelos引入的同调度概念，我们消除了上述障碍。另一方面，由于很难处理同调度，使用另一种不变量对线性Buchsbaum模块进行泛化的问题就留给我们了。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征的局部环的奇点。我们现在正准备与Kei-ichi Watanabe （Nihon university）合作发表有关这些研究的论文。

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.) .