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

在本研究期间，Yves Andre和Bruno Kahn发现动机的有限维度概念可以得到极大的推广。Chow动机可能存在的有限维度是本研究的出发点。我们可以在任何张量范畴中，特别是在混合动机范畴中，表述有限维数。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（O'Sullivan），因此有限维的概念应该推广到舒尔有限的概念。这种推广提出了一些重大问题，例如舒尔零幂问题。在这种情况下，以下是本研究的主要结果。(1)重新认识Chow动机的有限维数概念(2)动机空间概念的相对性(3)Jacobian猜想的正特征逼近(4)Alexander格式的完备性(5)Chow动机是一维的当且仅当它们是可逆的(6)求出Schur维数(7)在光滑纤维变形下动机的有限维数是稳定的(8)动机的有限维数与合理性的关系动机Zeta函数在这个列表中，(7)可能在未来有很强的含义。这是与Vladimir Guletskii的联合研究，主要的限制是我们只能将这一结果应用于具有光滑纤维的家庭。如果可以将这一结果推广到非光滑纤维空间，那么这将是对所有Chow动机的有限维性证明的一个突破。

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.