2019 Fiscal Year Research-status Report
Arithmetic cohomology over local fields
| Project/Area Number |
18K03258
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| Research Institution | Rikkyo University |
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Principal Investigator |
ガイサ トーマス 立教大学, 理学部, 教授 (30571963)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | Weil etale cohomology / Arithmetic cohomology / Brauer group / Duality |
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| Outline of Annual Research Achievements |
In the second year of the project I focused on applications of arithmetic cohomology). The goal of the project is to define Weil etale cohomology for regular flat and proper schemes over henselian discrete valuation rings. Morin and I gave a preliminary definition, however we have been unable to prove structure results of the theory. Instead we proved statements that our preliminary definition implies. There are two main results that we obtained:
1) Class field theory over local fields: S.Saito defined a reciprocity map from algebraic K-theory to the fundamental group of a smooth and proper scheme over a local field. The kernel of this map is the divisible subgroup, and its image is the torsion subgroup. Yoshida later calculated the cokernel, which is related to the combinatorics of the special fiber of a model. Our definition gives an integral model for the fundamental group of X, including the contribution from the special fiber. The proof of this results requires a proof of a duality result between Weil-etale cohomology in weights zero and d+1.
2) There are partial results on the Brauer group of a smooth and proper scheme over a local field by of Colliot-Thelene, S.Saito, and Sato. They focused on the Brauer group of the scheme module the Brauer group of the model. We were on the other hand able to prove results on the Brauer group of the model.
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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
The main goal of the project is still elusive, and we hope we will make more progress on this in the coming years. As mentioned above we were able to prove interesting results during the research. Currently we are fixing details in the proof of the results on class field theory and the Brauer group, and are preparing these results for publication.
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| Strategy for Future Research Activity |
After finishing the above mentioned papers, we plan to return to the original project on Weil etale cohomology. A question which arose during our work is the structure of the p-adic motivic cohomology of the model. There is a question of Flach and Morin relating it to de Rham cohomology, and we are thinking about this aspect. More generally, there seems to be some interesting relations to p-adic Hodge theory, and we hope to explore this new aspect of the theory.
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| Causes of Carryover |
緊急事態宣伝の影響でうまく使用額を使えませんでした．
残高で来年度の出張に利用するつもりです．
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Research Products
(3 results)
