2007 Fiscal Year Final Research Report Summary
Singurality Theory and Frobenius Morphism
| Project/Area Number |
17540043
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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 | Nihon University |
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Principal Investigator |
WATANABE Keiichi Nihon University, College of Humanities and Sciences, Professor (10087083)
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| Co-Investigator(Kenkyū-buntansha) |
TOMARI Masataka Nihon University, College of Humanities and Sciences, Professor (60183878)
FUKUDA Takuo Nihon University, College of Humanities and Sciences, Professor (00009599)
KURANO Kazuhiko Meiji University, School of science and technology, Professor (90205188)
YOSHIDA Kenichi Nagoya University, Graduate School of Mathematics, Associate Professor (80240802)
TAKAGI Shunsuke Kyushu University, Faculty of Mathematics, Research Associate Professor (40380670)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Frobenius endomorphism / rational singularity / F-rational ring / tight closure / integral closure / totally reflexive module / F-threshold / multiplicity |
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| Research Abstract |
Frobenius endomorphism hi characteristic p > 0 is a very powerful tool in commutative ring theory as well as singularity theory or algebraic geometry iover a field of characteristic 0 via reduction mod p.
We applied the Frobenius endomorphism to various problems in commutative algebra and singularity theory. Inparticular, we showed the followings ;
1. F-thresholds ; F-threshold is defined in a Noetherian ring of characteristic p>0 to a pair (I,J) of two ideals of A.
This notion was originally introduced to describe multiplier ideals and jumping numbers in a regular local ring. But in our research, it turned out that this notion is closely related to tight closures and integral closures and also we have a nice conjecture concerning F-threshold and multiplicity of a parameter ideal.
2. Multi-graded rings, rational singularity and F-rational rings ;
The notion of multi-graded rings and their diagonal algebras is a very interesting object and very useful in making many interesting examples. In a joint paper with A Singh and E. Sato, K. Kurano and K. Watanabe made a new example with discrete divisor class group whose local cohomology modules shows very interesting feature. Also we showed a criterion for diagonal subalgebras of multi-graded hypersurfaces to be f-rational or F-regular in terms of the degree.
3. Totally reflexive modules ;
In the theory of totally reflexive modules, examples of non-trivial totally reflexive modules are very few. Watanabe and R. Takahashi constructed a family of non-trivial totally reflexive modules using geometry of curves of genus greater than 1. This is the first case that algebraic geometry is used in this theory.
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Research Products
(23 results)
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[Journal Article] F-thresholds, tight closure, integral closure, and multiplicity bounds2008
Author(s)
C., Huneke, M., Mustata, S., Takagi, K., Watanabe
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Journal Title
Michigan Mathematical Journal (to appear in)
Description
「研究成果報告書概要(欧文)」より
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