Basic researchs on USD-sequences and its applications to improvements of singularities

以前，在美元序列领域中，我们通常讨论系统由参数组成且理想是参数或极大的情况。如今，我们对这一领域的兴趣转移到了更广泛的情况上。因此，在这个研究项目中，我们关注的是“m-初等”理想，其中m是极大理想，我们的研究是从研究U.S.-序列的行为开始的，假设它们构成了一些m-初等理想的极小约简。我们解决了Rees代数(如R)和相应的分次环(如G)关于滤子的环论结构的分析问题，以及更容易和更显式地计算它们的局部上同调的问题。然而，对于一般的过滤来说，似乎很难找到这些问题的答案。因此，我们首先限制了p…更多的问题转化为所谓的“等i-不变”情形，我们讨论了由“m-素数”理想定义的理想-进滤子。关于R和G的环论结构，特别是它们的Buchsbaum性，我们已经证明了在这种情况下G总是Buchsbaum环。对于R的Buchsbaum性，我们还得到了R为Buchsbaum的充分条件。Namwly，在我们自然地推广了由Meiji大学教授Shiro Goto(Meiji University.)提出的最小重数概念之后。在Cohen-Macaulay环的Buchsbaum环范畴中，如果进一步假设m-素数理想的约化次数至多为1，则Rees代数R一定是Buchsbaum环。虽然这是一个非常特殊的情况，但我们现在意识到，这是以前给出的结果中最好的一个。因此，我们还知道m的Rees代数也是Buchsbaum环，其中m是具有“最大嵌入维”的Buchsbaum环的极大理想。Nishida Koji在正则序列生成的理想的积分闭包方面得到了有趣的结果，此外，在引入滤子的“解析偏差”的新概念后，他成功地推广了Rees代数及其由适当滤子定义的相关分次环的Cohen-Macaulay性的类似判据。Takesi Kawasaki研究了Noether格式X的“Cohen-Macaulayation”构造问题，即Y被定义为具有来自X的二元态射且仅有有限多个Cohen-Macaulay奇点的Noether格式，并且他成功地构造了相当一般的Noether格式的Cohen-Macaulayation。实际上，Y由Y=Proj R给出，其中R是适当理想的Rees代数，如果我们进一步假设R本身是Cohen-Macaulay环，我们称它为X=Spec A(resp.A Simply)。他还阐明了Netherian(局部)环A存在这样一个算术Macaulay化的充要条件。

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 p … More roblems 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