Study of Blowing-up rings

吹胀环的研究

• 批准号: 13640044
• 负责人: GOTO Shiro
• 金额: $ 2.43万
• 依托单位: Meiji University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640044/
• 关键词: Buchsbaum ring; Cohen-Macaulay ring; FLC ring; local cohomology; index of reducibility; multiplicity; system of parameters; Gorenstein ring; Gorenstein環; Buchsbaum環; a-invariant; homological dimension; Rees代数; 射影次元; 入射次元; 正則局所環; 優良イデアル; N

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

本研究的目的是研究以下问题：（1）找到Rees代数和相关的理想分级环成为Gorenstein环的实用准则。 (2)发展Gorenstein局部环中的美好理想理论。对于第二个问题，我们已经发展出令人满意的新理论。设 (A, m) 为维度 d 的 Gorenstein 局部环，并令 I 为 A 中的 m 初级理想。如果 I 的相关分级环 G(I) 是 Gorenstein 环且 a(G(I)) = 1-d，则我们说 I 是 A 中的好理想。虽然好的理想是仅次于A中的参数理想的最佳理想，但我们缺乏对此类理想的深入研究。该项目对它们进行了基础理论研究。当项目继续进行时，发现了关于整体封闭理想的意外结果，这导致我们对 Buchsbaurn（或更一般地说，FLC 局部环）中某些原初理想的约简数进行了研究。结果概括了 Corso-Polini-Huneke-Vasconcelos 给出的 Cohen-Macaulay 局部环上的结果。我们特别将这些研究总结为以下三篇论文：[1] S.Goto 和 H.Sakurai，Buchsbaum 环中的等式 $1^2 = Q1$，Rendiconti del Seminario Matematico dell'Universit di Padova 110(2003),25-56[2] S.Goto 和 H.Sakurai，与参数理想相关的 socle 理想的约简指数 在重数为 2 的 Buchsbaum 局部环中，J. Math。苏克。日本（即将出现）[3] S.Goto 和 H.Sakurai，何时等式 $1^2=Q1$ 在 Buchsbaum 环中成立？，Pieprint 2003

项目成果

S.Goto, S.Iai, K.Watanabe: "Good ideals in Gorenstein local rings"Trans.Amer.Math.Soc.. 353. 2309-2346 (2001)

S.Goto、S.Iai、K.Watanabe：“Gorenstein 局部环中的良好理想”Trans.Amer.Math.Soc.. 353. 2309-2346 (2001)

S.Goto, 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).

S.Goto，H.Sakurai：“与重数二的 Buchsbaum 局部环中的参数理想相关的底理想的约简指数”J.Math.Soc.Japan。

S.Goto, F.Hayasaka, S.Iai: "The $\roman{a}$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings"Proc.Amer.Math.Soc.. 131. 87-94 (2003)

S.Goto、F.Hayasaka、S.Iai：“与常规局部环中的理想过滤相关的分级环的 $ oman{a}$ 不变性和 Gorensteinness”Proc.Amer.Math.Soc.. 131. 87

S.Goto, K.Nishida: "Hilbert coefficients and Buchsbaumness of associated graded rings"J.Pure and Appi.Alg.. (to appear).

S.Goto、K.Nishida：“相关分级环的希尔伯特系数和 Buchsbaumness”J.Pure 和 Appi.Alg..（即将出现）。

後藤四郎, 櫻井秀人: "重複度2のBuchsbaum環と等式$I^2=QI$"明治大学理工学部研究報告. 27. 13-17 (2002)

后藤四郎、樱井秀人：“重数为 2 的 Buchsbaum 代数和方程 $I^2=QI$” 明治大学理工学院研究报告 27. 13-17 (2002)。