| Project/Area Number |
09440013
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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 | Osaka University |
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Principal Investigator |
HIBI Takayuki Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80181113)
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| Co-Investigator(Kenkyū-buntansha) |
YANAGAWA Koji Graduate School of Science, Assistant, 大学院・理学研究科, 助手 (40283006)
NAMIKAWA Yoshinori Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (80228080)
MIYANISHI Masayoshi Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80025311)
SUZUKI Takashi Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40114516)
KAWANAKA Noriaki Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10028219)
小川 裕之 大阪大学, 大学院・理学研究科, 助手 (70243160)
磯崎 洋 大阪大学, 大学院・理学研究科, 助教授 (90111913)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 1998: ¥3,400,000 (Direct Cost: ¥3,400,000)
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| Keywords | componentwise linear / generic initial ideal / Cohen-Macaulay ring / squaretree inonomial / lexsegment ideal / graded Betti number / polynomial ring / Kruskal-Katona thearem / Componentwise linear / h-三角形 / 単体的複体 / Stanley-Reisner ideal / Cohen-Macaulay / 次数付ベッチ数列 / 双対複体 / 凸多面体 / 有限自由分解 / 外積代数 / グレブナ-基底 / 三角形分割 / Hilbert函数 / Gorenstein環 |
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| Research Abstract |
The important activity during the period of the present research project is, first, to present the concept of componentwise linear ideals and to establish its fundamental theory and, second, to study generic initial ideals of simplicial complexes and to discuss their concrete and effective applications to combinatorics. First of all, we obtained the theorem that the squarefree monomial ideal associated with a simplicial complex is componentwise linear if and only if its dual complex is sequentially Cohen-Macaulay, and explained the algebraic aspect of sequentially Cohen-Macaulay complexes and their h-triangles. Second, based on fundamental study about generic initial ideals of coruponentwise linear ideals, the important result that a homogeneous ideal of the polynomial ring possesses the stable Betti numbers if and only if the ideal is componentwise linear was established. Such the theorem guarantees that componentwise linear ideals will play an important role in computational commutative algebra. Third, in order to obtain sophisticated generalization of Kruskal-Katona theorem in classical combinatorics on finite sets, via the discussion on the existence of a squarefree strongly stable ideal having the same graded Betti numbers as those of the generic initial ideal of a squarefree ideal in the polynomial ring, we did succeed in obtaining the affirmative answer to the outstanding conjecture that the graded Betti numbers of a squarefree ideal with a fixed Hubert function are less than or equal to those of the lexsegment ideal.
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