1999 Fiscal Year Final Research Report Summary
Classification of homogeneous ideals in terms of the degrees of generic Grobner bases
| Project/Area Number |
10640027
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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 | Hiroshima University |
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Principal Investigator |
AMASAKI Mutsumi Faculty of School Education, Hiroshima University, Associate Professor, 学校教育学部, 助教授 (10243536)
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| Co-Investigator(Kenkyū-buntansha) |
KAGEYAMA Sanpei Faculty of School Education, Hiroshima University, professor, 学校教育学部, 教授 (70033892)
ISHIBASHI Yasunori Faculty of School Education, Hiroshima University, professor, 学校教育学部, 教授 (30033848)
MIYAZAKI Chikashi University of the Ryukyus, Faculty of Science, associate professor, 理学部, 助教授 (90229831)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Bourbaki sequence / homogeneous ideal / graded module / Grobner basis / Weierstrass basis / basic sequence / local cohomology / free resolution |
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| Research Abstract |
Let R be a polynomial ring in r indeterminates over an infinite field and M a homogeneous submodule of a finitely generated graded free module over R.
We first made clear the relation between generic Grobner bases and Weierstrass bases of M. It can be summarized as follows. If one takes a Grobner basis of M with respect to the term over position order arising from the reverse lexicographic order in a suitable way, then it is also a Weierstrass basis of M.
Next, we have given a proof to a fundamental theorm which will constitute a part of the core of our study in the long run.
Theorem1 : : : Let p be an integer lying between 2 and r - 2. Suppose that M satisfies the following conditions (1) and (2).
(1) For each integer i lying between r - p + 1 and r - l, the ith local cohomology of M vanishes.
(2) M is reflexive over R.
Then, there exists a homogeneous prime ideal I of R which fits into a long Bourbaki sequence with M. Conversely, if such a prime ideal I exists, then M satisfies conditions (1) and (2).
Theorem2 : : : If M satisfies condition (1) above, then there is a homogeneous complete intersection f_1, . . . , f_{p - 2} of R satisfying the following conditions.
(3) Let A be the factor ring of R defined by these p - 2 polynomials. Then, A is normal.
(4) There are a finitely generated torsion-free graded module D over A and a homomorphism g from M to D over R such that g induces an isomorphism of the ith local cohomologies of M and D for each integer i lying between 0 and T - p + 1.
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Research Products
(8 results)
