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

以前，在USD-序列领域中，我们通常讨论的是系统由参数组成且理想是参数或极大理想的情形。如今，我们在这一领域的兴趣转向了更普遍的情况。因此，在这个研究项目中，我们关注的是“m-准素”理想，其中m是极大理想，我们的研究是从USD-序列的行为的研究开始的，假设它们形成一些m-准素理想的最小约化。关于滤子的论证，我们成功地将我们通常的方法应用于理想进滤子到更一般的滤子的分解。我们解决的问题，分析的环理论结构的里斯代数，说R，和相关的分次环，说G，相对于filtrations，而且计算其局部上同调更容易和明确的方式。然而，这似乎是很难找到答案，这些问题的一般过滤。因此，我们首先限制了我们的p 关于我们 我们把理想的性质推广到所谓的“等I不变”的情形，并讨论了由“m-准素”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，我们证明了在这种情况下G总是Buchsbaum环。对于R的Buchsbaum性，我们也得到了R是Buchsbaum的充分条件.首先，我们自然地推广了Shiro后藤教授（明治大学）将Cohen-Macaulay环中的m-准素理想归为Buchsbaum环范畴，若进一步假设m-准素理想的约化数至多为1，则Rees代数R必为Buchsbaum环.虽然这是一个非常特殊的情况，但我们现在认识到，这是以前给出的结果中可能最好的一个。因此，我们也知道m的Rees代数又是Buchsbaum环，其中m是Buchsbaum环的具有“最大嵌入维数”的极大理想.西田浩二教授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，Takesi川崎教授研究了构造Noether概型X的Cohen-Macaulayfication（Y）的问题，即Y被定义为一个Noether概型，它具有来自X的一个双有理态射，并且只有100个Cohen-Macaulay奇点，并且他已经成功地为相当一般的Noether概型构造了它。实际上，Y由Y = Proj R给出，其中R是一个合适理想的Rees代数，如果我们进一步假设R本身是一个Cohen-Macaulay环，我们称它为X = Spec A（resp. A简单）。他还阐明了一个Netherian（local）环A存在这样一个算术Macaulayfication的充分必要条件。少

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