| Project/Area Number |
14340009
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Graduate School of Mathematical Sciences, University of Tokyo (2004-2005)
Nagoya University (2002-2003) |
|---|
Principal Investigator |
SAITO Shuji Graduate School of Mathematical Sciences, University of Tokyo, Graduate School of Mathematical Sciences, Professor (50153804)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
SAITO Takeshi University of Tokyo, Graduate School of Mathematical Sciences, Professor (70201506)
KATSURA Toshiyuki University of Tokyo, Graduate School of Mathematical Sciences, Professor (40108444)
MIYAOKA Yoichi University of Tokyo, Graduate School of Mathematical Sciences, Professor (50101077)
金銅 誠之 名古屋大学, 大学院・多元数理科学研究科, 教授 (50186847)
松本 耕二 名古屋大学, 大学院・多元数理科学研究科, 教授 (60192754)
藤原 一宏 名古屋大学, 大学院・多元数理科学研究科, 教授 (00229064)
佐藤 周友 名古屋大学, 大学院・多元数理科学研究科, 助手 (50324398)
斎藤 博 名古屋大学, 大学院・多元数理科学研究科, 助教授 (80135293)
小林 亮一 名古屋大学, 大学院・多元数理科学研究科, 教授 (20162034)
|
|---|
| Project Period (FY) |
2002 – 2005
|
| Project Status |
Completed (Fiscal Year 2005)
|
| Budget Amount *help |
¥9,900,000 (Direct Cost: ¥9,900,000)
Fiscal Year 2005: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2004: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2003: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2002: ¥2,700,000 (Direct Cost: ¥2,700,000)
|
|---|
| Keywords | motivic cohomology / finiteness conjecture / Kato conjecture / resolution of singularities / 代数的サイクル / 高次元類体論 / p進Hodge理論 / Chow群の有限性 / 高次Abel-Jacobi写像 / 高次Chow群 / Chow群 / 混合モチーフ / Hodge理論 / P進Hodge理論 / Bloch-加藤予想 / Beilinson予想 / P進ホッヂ理論 / ホッヂ理論 / 高次アーベルヤコビ写像 / ベイリンソン予想 / 高次チャウ群 |
|---|
| Research Abstract |
Motivic cohomology is one of the most significant objects to study in arithmetic and algebraic geometry. For example, let K be a number field and O_K be its ring of integers. Then the ideal class group of K and the group of units in O_K are motivic cohomology of the scheme Spec(O_K).
An important conjecture in arithmetic geometry is finiteness of motivic cohomology of arithmetic schemes. This is a natural generalization of the finiteness result for the above examples, which is a fundamental fact in classical number theory. There have been very few results on the problem so far except the case of Spec(O_K) or a curve over a finite field.
In our research we have proved a new finiteness result for motivic cohomology. To state a result, let X be either regular projective flat over Spec(O_K) (arithmetic case) or a projective smooth variety over a finite field F (geometric case). The first crucial observation is that the finiteness of a certain motivic cohomology of X follows from a conjecture of Kato on the vanishing of KH_q(X) for integers q〓1. Here KH_q(X) is a certain arithmetic invariant attached to X. The Kato conjecture in case X=Spec(O_K) is equivalent to a fundamental fact in number theory concerning the Brauer group of K, which implies the Hasse principle for central simple algebras over K.
We have shown the Kato conjecture in geometric case under the assumption of resolution of singu-larities. To be more precise we have obtain the following:
Theorem Let X be a projective smooth variety over a finite field. Let γ〓1 be an integer. Assume resolution of singularities for subvarieties of dimension〓_K embedded in a smooth variety over F. Then KH_q(X)=0 for 1〓q〓γ+2.
We have also succeeded to show the resolution of singularities in the above sense in case γ=2. Thus we get KH_q(X)=0 for 1〓q〓4 unconditionally and it gives rise to a new finiteness result for motivic cohomology of X.
|
