Research on the associated graded ring of a general filtration

通用过滤关联分级环的研究

• 批准号: 15540009
• 负责人: NISHIDA Koji
• 金额: $ 2.37万
• 依托单位: Chiba University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540009/
• 关键词: local ring; graded ring; filtration

In this research we aimed to generalize the classical theory on the Rees algebras of ideals so that it can be applied to the Rees algebras of filtrations, where the word "filtration" means a family F = {Fn}_<n∈N> of ideals such that A=F_0⊃F_1⊃F_2⊃・・・and FmFn⊂Fm+n for any m,n∈N.We first introduced a new notion that is called the reduction system of F. Under some good conditions on the reduction system of F, we could study the Cohen-Macaulay or the Gorenstein property of the associated graded ring of F. The good conditions stated above are the conditions concerning(i) the localization at prime ideals containing F_1 and(ii) the cohomological property on A/ Fn for finite number of n.Let l be the number of elements forming the reduction system of F and s be the height of F_1. If l-s【less than or equal】1, then our result is very practical. In fact, in that case, we found a lot of applications of the result. Furthermore we could establish satisfactory theory without assuming any conditions on l-s. Thus we may say that our purpose of this research has been almost achieved.

在这项研究中，我们的目的是推广理想里斯代数的经典理论，以便将其应用于过滤的里斯代数，其中“过滤”一词意味着理想族 F = {Fn}_<nεN> ，使得 A=F_0⊃F_1⊃F_2⊃・・・ 和 FmFn⊂Fm+n 对于任何 m,nεN。我们首先引入了一个新概念：被称为 F 的约简系统。在 F 约简系统的某些良好条件下，我们可以研究 F 的相关分级环的 Cohen-Macaulay 或 Gorenstein 性质。上述良好条件是以下条件：(i) 包含 F_1 的素理想的局域性和 (ii) 对于有限数量的 n，A/ Fn 上的上同调性质。令 l 为形成 F 的约简系统的元素数量，s 为高度 F_1 的。如果l-s【小于或等于】1，那么我们的结果就非常实用。事实上，在这种情况下，我们发现了很多结果的应用。此外，我们可以在不假设 l-s 任何条件的情况下建立令人满意的理论。可以说，我们本次研究的目的已经基本达到了。

项目成果

On Burch' s inequality and the reduction system of a filtration

关于伯奇不等式和过滤的约简系统

• 发表时间: 2006
• 期刊: Proceedings of the American Mathematical Society 134
• 作者: 崎谷眞也;他;崎谷眞也;Yayoi Kinoshita
• 通讯作者: Yayoi Kinoshita

Hilbert coefficients and Buchsbaumness of associated graded rings

相关分级环的希尔伯特系数和布克斯鲍姆度

• 发表时间: 2003
• 期刊: Journal of Pure and Applied Algebra 181
• 作者: 崎谷眞也;他;崎谷眞也;Yayoi Kinoshita;Koji Nishida;Koji Nishida;Shigeki Matsuda;Shigeki Matsuda;Shiro Goto
• 通讯作者: Shiro Goto

Gorenstein graded rings associated to ideals

Gorenstein 对与理想相关的戒指进行分级

• 发表时间: 2005
• 期刊: J. Algebra 294
• 作者: S.Goto;S.Goto
• 通讯作者: S.Goto

Broue's abelian defect group conjecture

布劳的阿贝尔缺陷群猜想

• 发表时间: 2014
• 作者: R.Kessar;S.Koshitani;M.Linckelmann;S. Koshitani;S. Koshitani;S. Koshitani;S. Koshitani;S. Koshitani;S. Koshitani
• 通讯作者: S. Koshitani

On the depth of the associated graded ring of a filtration

关于过滤相关分级环的深度

• 发表时间: 2005
• 期刊: Journal of Algebra 285
• 作者: 崎谷眞也;他;崎谷眞也;Yayoi Kinoshita;Koji Nishida
• 通讯作者: Koji Nishida