2006 Fiscal Year Final Research Report Summary
Research on the associated graded ring of a general filtration
| Project/Area Number |
15540009
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Chiba University |
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Principal Investigator |
NISHIDA Koji Chiba Univ., Graduate School of Science and Technology, Associate Professor, 大学院自然科学研究科, 助教授 (60228187)
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| Co-Investigator(Kenkyū-buntansha) |
IAI Shin-Ichiro Hokkaido University of Education, Faculty of Education, Associate Professor, 教育学部, 助教授 (50333125)
KOSHITANI Shigeo Chiba University, Faculty of Science, Professor, 理学部, 教授 (30125926)
MATSUDA Shigeki Chiba University, Faculty of Science, Associate Professor, 理学部, 助教授 (90272301)
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| Project Period (FY) |
2003 – 2006
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| Keywords | local ring / graded ring / filtration |
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| Research Abstract |
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.
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Research Products
(11 results)
