| Project/Area Number |
11640049
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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 | MEIJI UNIVERSITY |
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Principal Investigator |
GOTO Shiro Meiji University, School of Science and Technology, Department of Mathematics, Professor, 理工学部, 教授 (50060091)
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| Co-Investigator(Kenkyū-buntansha) |
IAI Shin-ichiro Meiji University, School of Science and Technology, Assistant, 理工学部, 助手
NAKAMURA Yukio Meiji University, School of Science and Technology, Department of Mathematics, Lecturer, 理工学部, 講師 (00308066)
桂田 祐史 明治大学, 理工学部, 助教授 (80224484)
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| Project Period (FY) |
1999 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | Buchsbaum ring / Cohen-Macaulay ring / Gorenstein ring / canonical module / Rees algebra / associated graded ring / Buchsbaun環 / Cohen-Macaulay環 / Gorenstein環 / Rees代数 / Buchsbaum環 / S^1-manifold / ヒルベルト類体 |
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| Research Abstract |
Let I be an m-primary ideal in a Gorenstein local ring (A, m) with dim A = d and assume that I contains a parameter ideal Q in A as a reduction. Then we say that I is good ideal in A if G = 【symmetry】_n≧_0I^n/I^<n+1> is a Gorenstein ring with a(G) = 1 - d. The associated graded ring G of I is a Gorenstein ring with a(G) = -d if and only if I = Q.Therefore, good ideals in our sense are good ones next to the parameter ideals Q in A.A basic theory of good ideals is developed by this project. We have that I is a good ideal in A if and only if I^2= QI and I = Q : I.Firstly a criterion for finite-dimensional Gorenstein graded algebras A over fields k to have the nonempty sets X_A of good ideals shall be given. Secondly in the case where d=1 we will give a correspondence theorem between the set X_A and the set Y_A of certain overrings of A.A characterization of good ideals of the case where d = 2 will be given in terms of the goodness in their powers. Thanks to Kato's Rieman-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show the structure of the set X_A of good ideals in A heavily depends on d = dim A.The set X_A may be empty if d ≦ 2, while X_A is necessarily infinite if d ≧ 3. To analyze this phenomenon we shall lastly explore monomial good ideals in the polynomial ring k[X_1, X_2, X_3] in three variables over a field k. Let I be an ideal in a Gorenstein local ring A.Then I is said to be an equimultiple good ideal if I contains a reduction Q = (a_1, a_2, …, a_s) generated by s elements in A and if the associated graded ring G(I)=【symmetry】_n≧_0I^n/I^<n+1> of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = ht_AI.The structure of the sets X_<A,s> (s ≧ 0) of equimultiple good ideals I with ht_AI = s. Some of the results in the case where s = dim A are successfully generalized to those of equimultiple case with improvements.
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