1999 Fiscal Year Final Research Report Summary
STUDY OF NOETHERIAN LOCAL RINGS IN COMMUTAIVE ALGEBRA
| Project/Area Number |
10640002
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Hokkaido University of Education |
|---|
Principal Investigator |
NISHIMURA Jun-ichi Hokkaido University of Education, Sapporo Campus, Associate Professor, 教育学部・札幌校, 助教授 (00025488)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
OSADA Masayuki Sapporo Campus, Associate Professor, 教育学部・札幌校, 助教授 (10107229)
HASEGAWA Izumi Sapporo Campus, Professor, 教育学部・札幌校, 教授 (50002473)
OKUYAMA Tetsuro Asahikawa Campus, Professor, 教育学部・旭川校, 教授 (60128733)
KITAYAMA Masashi Kushiro Campus, Associate Professor, 教育学部・釧路校, 助教授 (80169888)
OKUBO Kazuyoshi Sapporo Campus, Professor, 教育学部・札幌校, 教授 (80113661)
|
|---|
| Project Period (FY) |
1998 – 1999
|
| Keywords | Noetherian local ring / Homological conjectures / Frobenius map / p-adic representation / Bertini Theorem / Witt ring / big Cohen-Macaulay modules / Structure theorem of complete local rings |
|---|
| Research Abstract |
Construction of big Cohen-Macaulay modules
Homological conjectures on finitely generated modules over Noetherian local rings are basic and deep problems in commutative algebra. M. Hochster has shown that the existence of a big Cohen-Macaulay module for a given system of parameters of a local ring implies the intersection conjecture and that any local ring which contains a field, or equal characteristic local ring, has such a module.
So, to prove the existence of a big Cohen-Macaulay module over unequal characteristic local ring is very important. We have shown that the following study is useful to solve the problem above :
1) p-adic representation of elements in an unequal characteristic complete local ring,
2) Flenner's Bertini Theorem,
3) a lifting of Frobenius map to the Henselization of an unequal characteristic complete local ring.
Construction of bad Noetherian local rings
From Akizuki's and Nagata's examples it is well-known that some bad examples of Noetherian local rings are meaningful.
We have constructed some such local rings, following C. Rotthaus, T. Ogoma and R.C. Heitmann, for example :
1) Three dimensional catenary factorial local domain, which is not universally catenary,
2) Two dimensional noraml local domain of characteristic 0, which is not analytically reduced,
3) Three dimensional local domain of characteristic 0, whose derived normal ring is not Noetherian.
|
