2004 Fiscal Year Final Research Report Summary
Commutative ring theory and singularity theory
| Project/Area Number |
13440015
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Nihon University |
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Principal Investigator |
WATANABE Keiichi Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)
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| Co-Investigator(Kenkyū-buntansha) |
TOMARI Masataka Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (60183878)
FUKUDA Takuo Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (00009599)
MOTEGI Kamahi Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (40219978)
KURANO Kazuhiko Meiji University, School of Science and Technology, Professor, 理工学部, 教授 (90205188)
HARA Nobuo Tohoku University, Graduate School of Science, Assistant Lecturer, 大学院・理学研究科, 助教授 (90298167)
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| Project Period (FY) |
2001 – 2004
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| Keywords | multiplier ideal / lc threshold / F-pure / Hilbert-Kunz multiplicity / Frobenius endomorphism / log terminal singularity / tight closure / regular local ring |
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| Research Abstract |
The results are mainly concerning the followings 3 themes
1.Multiplier ideals ;
J.Lipman and K.Watanabe proved that every integrally closed ideal in 2 dimensional regular local rings is a multiplier ideal.
N.Hara and K.Yoshida defined a generalization of "tight closures" in characteristic p>0 and by using that concept, they Succeeded to calculate multiplier ideals by purely algebraic (by commutative ring theory) method.
S.Takagi and K.Watanabe established the notion of "F-pure thresholds", which corresponds to the notion of lc(=log canonical) threshold in characteristic 0, used in algebraic geometry. This concept has many interesting features in both singularity theory and commutative ring theory.
2.Hilbert-Kunz multiplicity ;
Hilbert-Kunz multiplicity is a kind of multiplicity defined for rings of positive characteristics. Watanabe and Yoshida proved before that a ring is regular if and only if the HK multiplicity of the ring is 1. This time we determined the rings whose HK multiplicity is smallest among non-regular rings in dimension 2 and 3.
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Research Products
(15 results)
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[Book] 環と体2002
Author(s)
渡辺 敬一
Total Pages
180
Publisher
朝倉書店
Description
「研究成果報告書概要(和文)」より
