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Algebraic Cycles and Higher Abel-Jacobi map

代数环和高级阿贝尔-雅可比图

基本信息

批准号：
14340009
负责人：
SAITO Shuji
金额：
$ 6.34万
依托单位：
Graduate School of Mathematical Sciences, University of Tokyo (2004-2005)Nagoya University (2002-2003)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

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.

动机上同是算术几何和代数几何中一个重要的研究对象。例如，设K是一个数字域，O_K是它的整数环。则K的理想类群与O_K中的单位群是方案Spec（O_K）的动机上同调。算术几何中的一个重要猜想是算术格式的动机上同调的有限性。这是对上述例子的有限结果的自然推广，是经典数论中的一个基本事实。迄今为止，除了Spec（O_K）或有限域上的曲线外，关于这个问题的结果很少。在我们的研究中，我们证明了动机上同的一个新的有限性结果。为了说明结果，设X是Spec（O_K）上的正则平射影（算术情况）或有限域F上的光滑射影（几何情况）。第一个重要的观察是，X的某个动机上同调的有限性是由Kato关于整数q = 1的KH_q(X)消失的猜想得出的。这里的KH_q(X)是附在X上的一个算术不变量。X=Spec（O_K）情况下的Kato猜想等价于数论中关于K的Brauer群的一个基本事实，它蕴涵了K上的中心简单代数的Hasse原理。更精确地说，我们得到了以下结论：定理设X是有限域上的一个射影光滑变分。设γ = 1为整数。假设在f上的光滑变化中嵌入尺寸为〓_K的子变化的奇异性的分辨率，则对于1〓q〓γ+2， KH_q(X)=0。我们还成功地证明了在γ=2的情况下奇点在上述意义上的分解。从而得到了1〓q〓4条件下的KH_q(X)=0，并由此得到了X的动机上同调的一个新的有限性结果。