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

针对可交换局部环上Cohen-Macaulay模的完全分类，我们在求解Cohen-Macaulay模的退化问题和g维0的模族问题方面取得了新的进展。(1) Cohen-Macaulay模的退化：利用退化关系可以在模的同构类集合上定义一个偏序。这个阶与Bongartz为有限维代数上的模定义的Horn阶有关。我们可以推测，只要Cohen-Macaulay模的范畴是fintie表示类型，那么这个序就是由Auslander-Reiten序列的退化产生的。这一猜想声称，这种几何上定义的顺序可能与奥斯兰德-雷滕颤振的组合性质有关。实际上我给出了这个猜想的完整证明在局部环是2维的情况下。我也证明了如果局部环是多维度为1的整域。这些结果发表在《代数杂志》（2002）上。(2) g维0的模：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的模是很重要的。曾有人认为g维0的范畴与Cohen-Macaulay模的范畴具有类似的性质。然而，我举了很多例子来反驳它。特别地，如果局部环的极大理想的立方为零，则成功地给出了环具有g维0的非平凡模的充分必要条件。实际上，我给出了一种构造这种连续参数不可分解模块的方法。利用这个构造，我证明了g维0的模族可能不是有限生成模的范畴中的逆变有限子范畴。这些结果发表在2002年罗马尼亚北约科学计划研讨会上，并将由Kluwer出版社出版。少

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