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

研究了线性极大Cohen-Macaulay模的存在性和推广。证明了线性极大Cohen-Macaulay模的一些推广定理，并给出了满射Buchsbaum模的一些性质，满射Buchsbaum模的概念是线性Buchsbaum模的推广.局部环A上的一个n阶生成模M称为线性Cohen-Macaulay A-模，如果M的伴随分次模是一个具有分次线性分解的分次Cohen-Macaulay模.线性Cohen-Macaulay模的上述定义等价于以下条件：M的最小生成元数等于M的重数。最后一个条件使我们能够定义线性Cohen-Macaulay模的一个推广。实际上，主要研究者和其他研究者已经将线性极大Cohen-Macaulay模推广到了线性极大Buchsbaum模上，其中I-不变量是Buchsbaum模的一个重要不变量。我们的主要结果之一是线性Buchsbaum模（因此线性Cohen-Macaulay模）的推广定理;使用Vasconcelos引入的同调度概念，我们消除了上述障碍。另一方面，由于同调度的处理比较困难，使得线性Buchsbaum模用另一个不变量进行推广的问题成为了一个遗留问题。此外，在整个研究过程中，我们注意到奇异性的研究是非常重要的，于是我们开始研究具有正特征的局部环的奇异性。我们正在准备与日本大学渡边惠一合作发表的论文。.

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.) .