2005 Fiscal Year Final Research Report Summary
Study on minimal free resolution of Stanley-Reisner rings
| Project/Area Number |
16540028
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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, 文化教育学部, 助教授 (90259862)
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| Co-Investigator(Kenkyū-buntansha) |
TANAKA Tatsuji Saga University, Department of Science and Technology, Professor, 理工学部, 教授 (80039370)
NAKAHARA Tohru Saga University, Department of Science and Technology, Professor, 理工学部, 教授 (50039278)
ICHIKAWA Takashi Saga University, Department of Science and Technology, Professor, 理工学部, 教授 (20201923)
YOSHIDA Ken-ichi Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (80240802)
YANAGAWA Kohji Osaka University, Graduate School of Science, Assistant, 大学院・理学研究科, 助手 (40283006)
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| Project Period (FY) |
2004 – 2005
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| Keywords | Stanley-Reisner ring / Buchsbaum ring / Cohen-Macaulay ring / minimal free resolution / linear resolution / multiplicity / Castelnuovo-Mumford regularity |
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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 and to consider its combinatorial applications.
In the academic year 2004 we studied Buchsbaum Stanley-Reisner rings with linear resolution. We determined the lower bound for the multiplicity of Stanley-Reisner rings. And we showed that they have linear resolution if they possess the minimal multiplicity. We also showed a necessary and sufficient condition for Buchsbaum Stanley-Reisner rings to have linear resolution in terms of the reduced homology groups of the corresponding simplicial complex and its links.
In the academic year 2005 we mainly studied the relation between the multiplicity of Stanley-Reisner rings and their Castelnuovo-Mumford regularity. We proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to the dimension d 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
We also investigated Stanley-Reisner rings with d-linear resolution among those with linear resolution intensively. Using the above result we showed that a Stanley-Reisner ring has d-linear resolution if its multiplicity is less than or equal to d and if the degree of generators of the Stanley-Reisner ideal is more than or equal to d. Moreover we showed that a Stanley-Reisner ring has d-linear resolution if its multiplicity is less than or equal to 2d-1 and if the degree of generators of the Stanley-Reisner ideal is d. By Alexander duality, we also verified that a Stanley-Reisner ring is Cohen-Macaulay if its multiplicity is large enough.
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Research Products
(13 results)
