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

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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