2006 Fiscal Year Final Research Report Summary
Study on Noetherian Local Rings in Commutative Algebra
| Project/Area Number |
16540047
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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 | Osaka Electro-Communication University |
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Principal Investigator |
NISHIMURA Jun-ichi Osaka Electro-Communication University, Faculty of Engineering, Professor, 工学部, 教授 (00025488)
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| Co-Investigator(Kenkyū-buntansha) |
SAKATA Sadahisa Osaka Electro-Communication University, Faculty of Biomedical Engineering, Professor, 医療福祉工学部, 教授 (60175362)
YAMAHARA Hideo Osaka Electro-Communication University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (30103344)
MIYAZAKI Mitsuhiro Kyoto University of Education, Faculty of Education, Associate Professor, 教育学部, 助教授 (90219767)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Noetherian local ring / big Cohen-Macaulay module / system of parameters / Homological Conjectures / monomial conjecture / Frobenius map / Witt expression / approximation theorem |
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| Research Abstract |
Construction of Big Cohen-Macaulay Modules and its applications :
Around 1970, H.Bass and M.Auslander et al. asked several problems on finitely generated modules over Noetherian local rings, known as Homological Conjectures. Because these questions are basic and important, they attracted many researchers in this field.
In 1973, C.Peskine-L.Szpiro showed that intersection conjecture on complexes of finitely generated free-modules over Noetherina local rings implies the problems above. And they solved the intersection conjecture for Noetherian local rings which contain fields of positive characteristic.
Soon after, M.Hochster remarked that the existence of Big Cohen-Macaulay Modules gives so-called monomial conjecture, direct-summand conjecture and new intersection conjecture, which induces Peskine-Szpiro's intersection conjecture. He showed that Noetherian local rings of equal characteristic have Big Cohen-Macaulay Modules, using Frobenius trick and M.Artin's approximation theorem.
Since then, almost all commutative algebraists have tried to construct Big Cohen-Macaulay Modules over Noetherian local rings of unequal characteristics.
We are studying the question above by using the structure theorem of completer local rings, Witt expression, Bertini theorem of Flenner, Jacobian criteria and generalized Frobenius map. Thanks to monomial conjecture for Noetherian local rings of equal characteristic, we are showing the existence of Big Cohen-Macaulay Modules over Noetherian local rings of unequal characteristics.
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