KAWASAKI Takesi (Tokyo Metropolitan University, Department of Mathematics, Assistant), 理学部数学科, 助手 (40301410)
NISHIDA Koji (Chiba University, Graduate school of Science and technology, Associate Professor), 大学院・自然科学研究科, 助教授 (60228187)
TODA Hiroshi (Himeji Dokkyo University, Faculty of Econoinformatics, Professor), 経済情報学部, 教授 (60025236)
|Budget Amount *help
¥2,000,000 (Direct Cost : ¥2,000,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥1,500,000 (Direct Cost : ¥1,500,000)
Before, in the field of USD-sequences, we had usually discussed the case where systems are consisted of parameters and ideals are parameter or maximal. Nowadays, our interests in this field move onto more general situations. Thus in this research project we were watching "m-primary" ideals, where m is maximal ideal, and our researches had began from the investigations of the behavior of USD-sequences under the assumption that they form minimal reductions of some m-primary ideals.
Concerning the argument on filtrations, we succeeded to apply our usual method on decompositions of ideal-adic filtrations into more general ones. We tackled the problems of analyzing the ring-theoretical structures of Rees algebras, say R, and associated graded rings, say G, with respect to filtrations and moreover of computing their local cohomology in more easy and explicit way. However, it seemed to be very difficult to find answers to these problems for general filtrations. Thus we firstly restricted our
problems into socalled "the equi-I-invariant" case and we dealed with the ideal-adic filtrations defined by "m-primary" ideals. Concerning the ring-theoretical structure of R and G, especially the Buchsbaumness of them, we had shown that G is always a Buchsbaum ring in this case. For the Buchsbaumness of R, we also got the sufficient conditions for R to be Buchsbaum. Namwly, after we naturally extended the notion "minimal multiplicity" introduced by Prof. Shiro Goto (Meiji Univ.) in Cohen-Macaulay rings into the category of Buchsbaum rings, Rees algebra R must be a Buchsbaum ring, if we further assume that the reduction numbers of m-primary ideals are at most one. Though this is a very special case, we now realize that this is the best possible one among results given before. Consequently, we also known that the Rees algebra of m is again a Buchsbaum ring, where m is the maximal ideal of a Buchsbaum ring with "maximal embedding dimension".
Prof. Koji Nishida obtained interesting results on the integral closures of ideals generated by regular sequences, moreover, after introducing the new notion of "analytic deviation" for a filtration, he had succeeded to generalize similar criterions for the Cohen-Macaulayness of Rees algebras and associated graded rings defined by suitable filtrations.
Prof. Takesi Kawasaki studied the problem of constructing a "Cohen-Macaulayfication", say Y, of a Noetherian scheme X, namely Y is defined as a Noetherian scheme having a birational morphism from X and only finitely many Cohen-Macaulay singularities, and he had succeeded to construct it for quite general Noetherian schemes. Actually, Y is given by Y = Proj R where R is a Rees algebra of a suitable ideal, and if we further assume R itself is a Cohen-Macaulay ring, we call it an "arithmetic" Macaulayfication of X = Spec A (resp. of A simply). He had also clarified the necessary and sufficient conditions in order to exist such an arithmetic Macaulayfication for a Netherian (local) ring A. Less