2005 Fiscal Year Final Research Report Summary
Cohomological study of arithmetic varieties
| Project/Area Number |
14340002
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | University of Tokyo |
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Principal Investigator |
SAITO Takeshi University of Tokyo, Graduate school of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70201506)
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| Co-Investigator(Kenkyū-buntansha) |
KATO Kazuya Kyoto University, Graduate school of Sciences, Professor, 大学院・理学研究科, 教授 (90111450)
TERASOMA Tomohide University of Tokyo, Graduate school of Mathematical Sciences, Assistant Professor, 大学院・数理科学研究科, 助教授 (50192654)
TSUJI Takeshi University of Tokyo, Graduate school of Mathematical Sciences, Assistant Professor, 大学院・数理科学研究科, 助教授 (40252530)
SHIHO Atsushi University of Tokyo, Graduate school of Mathematical Sciences, Assistant Professor, 大学院・数理科学研究科, 助教授 (30292204)
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| Project Period (FY) |
2002 – 2005
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| Keywords | etale cohomology / ramification / l-adic sheaf / conductor formula / Lefschetz trace formula / Euler number / characteristic class / local field |
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| Research Abstract |
As I wrote in the project proposal, I studied Riemann-Roch formulas for l-adic sheaves and filtration by ramification groups. The results obtained on Riemann-Roch formulas are much better than expected.
I formulated and proved a Grothendieck-Ogg-Shafarevich formula in an arbitrary dimension, that computes the Euler number of an l-adic sheaf in terms of its ramification along the boundary, in a joint research with one of the investigator, Kazuya Kato. The formula was proved in 60's for curves but remained to be generalized to higher dimension. First, we define the Swan class as a 0-cycle class supported on the boundary as a ramification invariant of an l-adic sheaf. By establishing a Lefschetz trace formula for open varieties, we prove that its degree computes the Euler number. The proof is written in a paper accepted for publication at the Annales of Mathematics.
We also obtained a conductor formula for a l-adic sheaf on a smooth variety over a local field, in a joint research with Kato. We use the K-theoretic localized intersection theory and a generalization to the open varieties of log Lefschetz trace formula, that we introduced in the study of the conductor formula of Bloch. We are now preparing a paper on the proof.
For an l-adic sheaf on a variety in positive characteristic, the characteristic class is defined. I studied it in a joint research with Ahmed Abbes who is a foreign collaborator of this research project. First, we established a relation with the Swan class. We also defined a refinement as a cohomology class supported on the closed fiber. Further, in the rank one case, we proved that it is the same as the 0-cycle class defined by Kato previously. In the course of proof, we have also established that the filtration by ramification groups defined for an arbitrary local field induces the filtration previously defined by Kato on the abelianized quotient in the equal characteristic case.
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