Study of Blowing-up rings

本文主要研究了以下几个问题：（1）给出了Rees代数及其分次理想环成为Gorenstein环的实用判据。(2)发展了Gorenstein局部环的好理想理论。至于第二个问题，我们已经发展了一个令人满意的新理论。设（A，m）是d维Gorenstein局部环，I是A中的m-准素理想.如果I的分次环G（I）是Gorenstein环且a（G（I））= 1-d，则称I是A中的好理想.虽然好理想是A中仅次于参数理想的最好理想，但我们对这类理想缺乏深入的研究。本课题对它们进行了基本的理论研究。在项目进行的过程中，发现了关于整闭理想的意想不到的结果，这使我们对Buchsbaurn中某些In-准素理想的约化数进行了其他研究，或者更一般地说，FLC局部环。推广了Corso-Polini-Huneke-Vasconcelos关于Cohen-Macaulay局部环的结果.我们特别将这些调查总结为以下三篇论文：[1] S.后藤和H.樱井，Buchsbaum环中的等式$1^2 = Q1$，帕多瓦110大学数学研讨会（2003），25 -56[2] S.后藤和H.樱井，柱理想的约化指数，与重数为2的Buchsbaum局部环中的参数理想相关联，J. Math. Soc.日本（即将出现）[3] S.后藤和H.樱井，等式$1^2=Q1$在Buchsbaum环中何时成立？，Pieprint 2003

The purpose of this research is to study the following problems: (1) Find practical criteria for Rees algebras and associated graded rings of ideals to be Gorenstein rings. (2) Develop the theory of good ideals in Gorenstein local rings. As for the second problem we have developed a satisfactory new theory. Let (A, m) be a Gorenstein local ring of dimension d and let I be an m-primary ideal in A. Then we say that I is a good ideal in A, if the associated graded ring G(I) of I is a Gorenstein ring and a(G(I)) = 1-d. Although good ideals are the best ones next to the parameter ideals in A, we lacked deep investigations of the ideals of this kind. The project has performed a basic theory of them. While the project proceeded, unexpected results about integrally closed ideals were discovered, which led us to the other research on the reduction numbers of certain In-primary ideals in Buchsbaurn, or more generally, FLC local rings. The results generalize those on Cohen-Macaulay local rings that were given by Corso-Polini-Huneke-Vasconcelos. We especially summarized these investigations into the following three papers:[1] S.Goto and H.Sakurai, The equality $1^2 = Q1$ in Buchsbaum rings, Rendiconti del Seminario Matematico dell'Universit di Padova 110(2003),25-56[2] S.Goto and H.Sakurai, The reduction exponent of socle ideals, associated to parameter ideals in a Buchsbaum local ring of multiplicity two, J. Math. Soc. Japan (to appear)[3] S.Goto and H.Sakurai, When does the equality $1^2=Q1$ hold true in Buchsbaum rings?, Pieprint 2003