2007 Fiscal Year Final Research Report Summary
Study on minimal free resolution of Stanley-Reisner rings
| Project/Area Number |
18540041
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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 | Saga University |
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Principal Investigator |
TERAI Naoki Saga University, Department of Culture and Education, Associate Professor (90259862)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAHARA Tohru Saga University, Department of Science and Technology, Professor (50039278)
UEHARA Tsuyoshi Saga University, Department of Science and Technology, Professor (80093970)
ICHIKAWA Takashi Saga University, Department of Science and Technology, Professor (20201923)
YOSHIDA Ken-ichi Nagoya University, Graduate School of Mathematics, Associate Professor (80240802)
YANAGAWA Kohji Kansai University, Department of Science and Technology, Associate Professor (40283006)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Stanley-Reisner ring / minimal free resolution / multiplicity / Castelnuovo-Mumford reaularity |
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| Research Abstract |
The purpose of this research is to study algebraic and combinatorial properties of minimal free resolution of Stanley-Reisner rings. We focused on the relation between the multiplicity and the Castelnuovo-Mumford regularity of Stanley-Reisner rings.
Before the academic year 2005 we proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to the dimension d of the Stanley-Reisner rings if its multiplicity is less than or equal to d. Moreover we verified that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 2d-1, and if the degree of generators of the Stanley-Reisner ideal is less than or equal to d
In the academic year 2006, developing these results, we proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 3d-2, and if the degree of generators of the first syzygy module of the Stanley-Reisner ideal is less than or equal to d+1. From this result we conjectured that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to (p+2)d- (p-1), and if the degree of generators of the p-th syzygy module of the Stanley-Reisner ideal is less than or equal to d+p. In the academic year 2007 we proved that the above conjecture holds if the dimension of the Stanley-Reisner rings is 2 or 3. We also found that this conjecture is a generalization of the lower bound theorem, that is famous in convex polytope theory, in the facet case.
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[Presentation] A Note on Schmitt-Vogel lemma2008
Author(s)
Kyohko, Kimura, Naoki, Terai, Ken-ichi, Yoshida
Organizer
RIMS Symposium "algorithm in algebra, language, and computation theory"
Place of Presentation
Kyoto
Year and Date
2008-02-22
Description
「研究成果報告書概要(欧文)」より
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