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.

本文将理想Rees代数的经典理论推广到滤子Rees代数，其中滤子是指理想族F = {Fn}_<n∈N>，使得A=F_0 $> F_1 $> F_2 $>···，且对任意m，n∈N，FmFn <$Fm+n。在F的约化系的某些良好条件下，我们可以研究F的伴随分次环的Cohen-Macaulay性质或Gorenstein性质。上述条件是关于（i）在包含F_1的素理想上的局部化和（ii）有限个n时A/ Fn上的上同调性质的条件，设l为F的约化系的元素个数，s为F_1的高度。如果l-s[小于或等于]1，那么我们的结果是非常实用的。事实上，在这种情况下，我们发现了很多结果的应用。此外，我们可以建立令人满意的理论，而无需对l-s作任何假设。因此，我们可以说，我们的研究目的几乎已经达到。

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