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New development of ring theory from the view point of representation theory and its application

从表示论的角度看环理论的新发展及其应用

基本信息

批准号：
13640024
负责人：
YOSHINO Yuji
金额：
$ 2.18万
依托单位：
OKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640024/
关键词：
local ring module Cohen-Macaulay ring degeneration representation type Gorenstein ring Auslander-Reiten quiver Cohen-Macaulay

项目摘要

Toward the complete classification of Cohen-Macaulay modules over a commutative local ring, we made new progress in solving the problems on degenerations of Cohen-Macaulay modules and the problems on the family of modules of G-dimension 0.(1) Degeneration of Cohen-Macaulay modules :One can define a partial order on the set of isomorphism classes of modules by using the degeneration relation. This order is related to the Horn order that has been defined by Bongartz for modules over finite dimensional algebras. One may conjecture that the order would be generated by the degenerations of Auslander-Reiten sequences whenever the cateogry of Cohen-Macaulay modules is of fintie representation type. This conjecture claims that such an order defined geometrically could be related to the combinatorial nature of Auslander-Reiten quiver. I actually gave a complete proof of this conjecture in the case that the local ring has dimension 2. I also proved this if the local ring is an integral domain of … More dimension 1. These results are published in Journal of Algebra (2002).(2) Modules of G-dimension 0 :As one of the generalizations of classification theory of Cohen-Macaulay modules over a Gorenstein ring, it is important to consider the modules of G-dimension 0 over a general local ring. It had been thought that the category of G-dimension 0 could have similar properties to the category of Cohen-Macaulay modules. However, I made a lot of examples that disprove it. In particular, if the cube of maximal ideal of the local ring is zero, then I succeeded to give a necessary and sufficient condition for the ring to have a nontrivial module of G-dimension 0. Actually I gave a way of construction of such indecomposable modules with continuous parameter. Using this construction I have shown that the family of modules of G-dimension 0 may not be a contravariantly finite subcategory in the cateogory of finitely generated modules. These results were reported in the Workshop of NATO Scientific Program in Romania (2002), and to be published from Kluwer Press. Less

在交换局部环上Cohen-Macaulay模的完全分类方面，我们在解决Cohen-Macaulay模的退化问题和G维0模族问题方面取得了新的进展. (1)Cohen-Macaulay模的退化：人们可以通过退化关系定义模的同构类集合上的偏序。这个顺序是有关的霍恩秩序已被定义为邦加茨模有限维代数。人们可以猜想，只要Cohen-Macaulay模的范畴是有限表示型的，则该阶将由Auslander-Reiten序列的退化产生.这个猜想声称这样一个几何定义的序可能与Auslander-Reiten的组合性质有关。实际上，我在局部环的维数为2的情况下，给出了这个猜想的完整证明。我还证明了这一点，如果局部环是一个整环的 ...更多信息维度1。这些结果发表在Journal of Algebra（2002）上。(2)G-维0模：作为Gorenstein环上Cohen-Macaulay模分类理论的推广之一，考虑一般局部环上G-维0模是很重要的。G维0的范畴被认为可以具有与Cohen-Macaulay模范畴相似的性质。特别是，当局部环的极大理想的立方为零时，我成功地给出了局部环存在G-维为0的非平凡模的一个充分必要条件。实际上，我给出了一种构造这种带连续参数的不可分解模的方法。利用这个构造，我证明了G维为0的模族在G维生成模范畴中可能不是一个逆变有限子范畴。这些结果已在罗马尼亚举行的北约科学方案讲习班（2002年）上报告，并将由Kluwer出版社出版。少