2013 Fiscal Year Annual Research Report
| Project/Area Number |
23340004
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| Research Institution | Nagoya University |
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Principal Investigator |
ガイサ トーマス 名古屋大学, 多元数理科学研究科, 教授 (30571963)
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| Project Period (FY) |
2011-04-01 – 2016-03-31
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| Keywords | ススリン・ホモロジー / 多様体の類体論 / Rojtmanの定理 |
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| Research Abstract |
In the previous year I was able to complete research on three papers. In the first one I proved that Suslin homology satisfies descent for proper hyperenvelopes, and for l-hyperenvelopes if l is invertible in the coefficients. I gave an axiomatic proof, which can also be applied to other theories in the future. This research is posted on the ArXiv, and submitted for publication.
In the second project I applied this method to prove Rojtman's theorem for normal schemes over algebraically closed fields. The theorem states that the torsion in the 0th Suslin homology is isormorphic to the torsion of the Albanese semi-abelian variety. The theorem is an improvement from previous results, which either used properness, smoothness, or characteristic 0. This result is also posted on the ArXiv and submitted for publication. A survey of this results is accepted for publication at the RIMS journal.
Finally, I concluded research on a joint project with Alexander Schmidt in Heidelberg, where we proved a version of the Hurewicz theorem for varieties (singular or not) over algebraically closed fields. The theorem states that the abelianized tame fundamental group mod m is isomorphic to Suslin homology with mod m coefficients. This paper is also posted on ArXiv, and submitted for publication.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I was able to prove several theorems which are related to the project, and required some time to write the research papers. But overall the project advances as expected.
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| Strategy for Future Research Activity |
At first I want to finish another project with Alexander Schmidt. We show that over a finite field, a modified version of Suslin homology (called Weil-Suslin homology) surjects onto the tame abelianized enlarged fundamental group, for any variety over a finite field. We also show that this fundamental group is a finitely generated abelian group, that the kernel of the surjection is the divisible subgroup of Weil-Suslin homology (under resolution of singularities), and that the map is an isormophism under Parshin's conjecture. The mathematics is in place, but it will require some time to write the paper.
After this I am planing to revisit a result of Grothendieck, where he shows that the finiteness of the Brauer group of a surface fibered over a curve over a finite field is equivalent to the finiteness of the Tate-Shafarevich group of the Jacobian of the special fiber of the curve. To prove this equivalence, one needs exactly to consider motivic cohomology over a stricly henselian valuation ring. I hope to obtain some results for different weights using the conjecture of Saito and Sato.
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