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Research of Motivic Geometry from the viewpoint of Non-commutative algebraic geometry

非交换代数几何视角下的动机几何研究

基本信息

批准号：
15540032
负责人：
KIMURA Shun-ichi
金额：
$ 1.92万
依托单位：
Hiroshima University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540032/
关键词：
Motives finite dimensionality Tensor Category Bloch's Conjecture Schur finite Jacobian Conjecture Alexander scheme Motivic zeta モチーフの有限次元性 Bivariant theory Jacobian conjecture Mixed Motive Nilpotence conjecture Bloch conjecture /

项目摘要

During the period of this research, it was found that the notion of finite dimensionality of motives can be greatly generalized, by Yves Andre and Bruno Kahn. The possible finite dimensionality of Chow motives was the starting point of this research. One can formulate the finite dimensionality in any tensor category, in particular in the category of Mixed motives. Unfortunately, we cannot expect that the category of mixed motives to be finite dimensional in my sense (O'Sullivan), and hence the notion of finite dimensionality should be generalized to the notion of Schur finiteness. This generalization posed major problems, for example, the problem of Schur Nilpotency.Under this circumstances, following is the list of major results of this research. (1)Brushing up the notion of finite dimensionality of Chow motives (2)Relativization of the notion of motivic spaces (3)Positive characteristic approach to the Jacobian conjecture (4)Etaleness property of Alexander schemes (5)Chow motives are 1 dimensional if and only if they are invertible (6)Finding the Schur dimension (7)The finite dimensionality of the motives is stable under the deformation with smooth fiber (8)The relation between the finite dimensionality of motives and the rationality of Motivic Zeta functionAmong this list, (7)may have a strong implication in the future. It is a joint work with Vladimir Guletskii, and the main limitation is that we can apply this result only for the family with the smooth fiber. If one can generalize this result to non-smooth fiber spaces, then that would be a breakthrough towards the proof of finite dimensionality of all Chow motives.

在本研究期间，我们发现，动机的有限维度的概念，可以大大推广，由安德烈和布鲁诺卡恩。周式动机可能的有限维性是本研究的出发点。人们可以在任何张量范畴中用公式表示有限维，特别是在混合动机范畴中。不幸的是，我们不能期望混合动机的范畴在我的意义上是有限维的（奥沙利文），因此有限维的概念应该推广到舒尔有限性的概念。这一推广提出了一些重大问题，例如Schur幂零性问题。在这种情况下，以下是本研究的主要结果。（1）提出Chow模的有限维概念（2）模空间概念的相对化（3）Jacobian猜想的正特征逼近（4）亚历山大格式的Etaleness性质（5）Chow模是1维的当且仅当它们可逆（6）求Schur维数（7）模的有限维在光滑纤维变形下是稳定的（8）动机的有限维性与动机Zeta函数的合理性之间的关系在这一列表中，（7）可能在未来有很强的含义。这是一个联合工作与弗拉基米尔Guletskii，和主要的限制是，我们可以应用这个结果只为家庭与光滑纤维。如果能将这个结果推广到非光滑纤维空间，那么这将是对证明所有Chow动机的有限维性的一个突破。