2005 Fiscal Year Final Research Report Summary
Research of Motivic Geometry from the viewpoint of Non-commutative algebraic geometry
| Project/Area Number |
15540032
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Hiroshima University |
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Principal Investigator |
KIMURA Shun-ichi Hiroshima University, Graduate School of Science, Associated Professor, 大学院・理学研究科, 助教授 (10284150)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Motives / finite dimensionality / Tensor Category / Bloch's Conjecture / Schur finite / Jacobian Conjecture / Alexander scheme / Motivic zeta |
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| Research Abstract |
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 function
Among 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.
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