2001 Fiscal Year Final Research Report Summary
On ring-theoretical invariants of singular points in positive characteristic
| Project/Area Number |
11640021
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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 | Nagoya University |
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Principal Investigator |
YOSHIDA Kenichi Nagoya University Graduate School of Mathematics, Assistant, 大学院・多元数理科学研究科, 助手 (80240802)
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| Co-Investigator(Kenkyū-buntansha) |
MUKAI Shigeru Kyoto University, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (80115641)
HASHIMOTO Mitsuyasu Nagoya University Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (10208465)
OKADA Soichi Nagoya University Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (20224016)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Hilbert-Kunz multiplicity / regular / Cohen- Macaulay / F-rational / rational singularity / multiplicity / tight closure / integral closure |
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| Research Abstract |
We have studied Hilbert-Kunz multiplicity as an invariant of singular points in positive char acteristic for three years. The most important result in our work is to give a characterization of regular local rings in terms of Hilbert-Kunz multiplicity. Actually, many researchers tried to gener alize our theorem. After this research, we have studied Hilbert-Kunz multiplicity of ideals defined by the dual graph of the resolution of singularities. Note that Hilbert-Kunz multiplicity for such an ideal is a ring-theoretical invariant associated to isolated singularity in positive characteristic. As one of our results, for integrally closed ideals in a rational double point, we obtained algorithm for calculating their Hilbert-Kunz multiplicities in terms of the dual graph. On the other hand, we have tried calculation of Hilbert-Kunz multiplicity for blow-up rings, but we could not get complete algorithm. As a partial result, we get some inequalities with respect to blow-up rings and the basering.
Also, we introduced the notion of the minimal Hilbert-Kunz multiplicity and gave several method for calculation. This invariant can be described as the difference of the Hilbert-Kunz multiplicities of some pairs of ideals. Furthermore, we found that this invariant is equal to the invariant which is defined by other researchers. We gave a presentation of our results as above at Symposium on Commutative algebra and at Symposium on Algebra on Summer in 2001. Also, we have a project to study blow-up rings in positive characteristic.
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