2006 Fiscal Year Final Research Report Summary
Research for multiplicities and tight closures on singular points of positive characteristic
| Project/Area Number |
16540021
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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 Professor, 大学院多元数理科学研究科, 助教授 (80240802)
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| Co-Investigator(Kenkyū-buntansha) |
HASHIMOO Mitsuyasu Magoya University, Graduate School of Mathematics, Assistant Professor, 大学院多元数理科学研究科, 助教授 (10208465)
ITO Yukari Nagoya University, Graduate School of Mathematics, Lecturer, 大学院多元数理科学研究科, 講師 (70285089)
WATANABE Kei-ichi Nihon University, Department of Mathematics, College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Hilbert-Kunz multiplicity / multiplicity / tight closure / multiplier ideal / singularity / F-regular / Buchsbaum rings / Stanley・Reisner rings |
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| Research Abstract |
1. On Lower bounds for Hilbert-Kunz multiplicities:
We have proved that any unmixed local ring with Hilbert-Kunz multiplicity one is a regular local ring before starting this research. Indeed, this theorem is a generalization of Nagata's classical theorem in positive characteristic. In this research, we considered a problem of finding a lower bound on Hilbert-Kunz multiplicities for non-regular local rings. As a result, we found a conjecture that such a lower bound is attained by quadratic hypersurface, and proved that it is true for local rings of at most dimension 4. The lower bound given by us is interesting in algebraic geometry, but we cannot obtain any sufficient theory. Moreover, the conjecture was proved by Enescu-Shimamoto in the case of complete intersections.
2. Minimal Hilbert-Kunz multiplicity
The notion of minimal Hilbert-Kunz multiplicities was introduced by us to estimate of badness of F-regular local rings. The invariant is a real number in the interval between 0 and 1. Recently, Aberbach etc. proved that a local ring is F-regular if and only if its minimal Hilbert-Kunz multiplicity is positive. We determined the minimal Hilbert-Kunz multiplicities for affine toric singularities and quotient singularities, which are typical F-regular rings.
3. Characterization of Buchsbaum Stanley-Reisner rings with minimal multiplicity.
We studied minimal free resolutions, multiplicities, h-vectors for Buchsbaum Stanley-Reisner rings together with Naoki Terai at Saga University. In particular, we gave a lower bound for multiplicities for those rings, and characterized such rings.
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