Study of graded rings associated to ideals and modules

本研究的目的是分析与理想和模相关的几种所谓的blow-up代数，以阐明给定理想和模的结构以及基环（局部环或分次环）的结构。发现了分次环的某些实用准则，如Rees代数R（I）、相关分次环G（I）和纤维锥F（I）是Gorenstein环、Cohen-Macaulay环和Buchsbaum环。特别地，本文研究了正则局部环中高度为2的完全交的首理想，这是与W. Heinzer和Mee-Kyoung Kim首席调查员对这些联合工作相当满意。另一方面，与R. Takahashi和N. Matsuoka关于Gorenstein局部环中的拟socle理想的研究开始了，这一主题现在是理想的约化数理论和相关分次环的环论性质的最有吸引力的研究之一。研究结果已被写入两篇论文（见下文），并提交给一些期刊发表。[1]S.后藤，N。Matsuoka和R.高，Gorenstein局部环中的拟柱理想，预印本2006[2] S。后藤、F. Hayasaka和R. Takahashi，关于某些Ext模块的消失，预印本2006

This research aims at the analysis of several kinds of so-called blow-up algebras associated to ideals and modules, in order to clarify the structure of given ideals and modules as well as that of the base(local or graded) rings. Certain practical criteria for graded rings, say the Rees algebra R(I), the associated graded ring G(I), and the fiber cone F(I), to be Gorenstein, Cohen-Macaulay, and Buchsbaum rings are discovered. Especially, the leading ideal of a complete intersection of height two in a regular local ring is closely studied, which are researches jointly with W. Heinzer and Mee-Kyoung Kim. The head investigator is quite satisfied by these joint works. On the other hand, a joint research with R. Takahashi and N. Matsuoka about quasi-socle ideals in a Gorenstein local ring was started and the theme is now a one of the most attractive studies of the theory of reduction numbers of ideals and the ring-theoretic properties of associated graded rings. The results are already written into two papers(see below) and were submitted to some journal for publication.[1] S. Goto, N. Matsuoka, and R. Takahashi, Quasi-socle ideals in Gorenstein local rings, Preprint 2006[2] S. Goto, F. Hayasaka, and R. Takahashi, On vanishing of certain Ext modules, Preprint 2006