1997 Fiscal Year Final Research Report Summary
Applications of Frobenius map to Commutative Ring Theory and Algebraic Geometry
| Project/Area Number |
07454010
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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 (1997)
Tokai University (1995-1996) |
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Principal Investigator |
WATANABE Keiichi Nihon Univ., College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)
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| Co-Investigator(Kenkyū-buntansha) |
ITAI Masanori Tokai Univ., School of Sciences.Associate Professor, 理学部, 助教授 (80266361)
WATANABE Junzo Tokai Univ., School of Sciences, Professor, 理学部, 教授 (40022727)
MATSUURA Yutaka Nihon Univ., College of Humanities and Sciences, Associate Professor, 文理学部, 助教授 (50096905)
SUZUKI Masahiko Nihon Univ., College of Humanities and Sciences, Associate Professor, 文理学部, 助教授 (00171249)
MORI Makoto Nihon Univ., College of Humanities and Sciences, Professor, 文理学部, 教授 (60092532)
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| Project Period (FY) |
1995 – 1997
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| Keywords | Frobenius map / Algebraic Geometry / Singularities / rational singularity / F-rational ring / Terminal Singularity / regular ring / Hilbert-Kunz multiplicity |
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| Research Abstract |
1 Characterization of Singularities in Characteristic 0 via Frobenius endomorphism. We found that log-terminal singularity and F-regular rings are equivalent notions in the case the ring is Q-Gorenstein. The same is true for rational singularities and F-rational rings.
2 By definig F-terminal rings, the terminal singularities are characterized in 3-dimensional case. F-terminal and Q-Gorenstein imply terminal singularity in any dimension. But in dimension 4, unfortunately, there is a counterexample to the converse and F-rational ring is characterized by the property that its general hyperplane section is F-rational in Gorenstein case.
3 The characterization of regular local rings by Hilbert-Kunz multiplicity=1.Namely, an unmixed local ring of characteristic p is regular if and only if its H-K multiplicity is 1.
Also we succeeded to classify the 2-dimensional rings with Hibert-Kunz multiplicity less than 9/4.
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