Study of graded rings associated to ideals and modules

点击翻译按钮获取中文摘要

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