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

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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.) .