2019 Fiscal Year Annual Research Report
Deligne-Lusztig 多様体と Fargues-Fontaine 曲線
| Project/Area Number |
19F19022
|
|---|
| Research Institution | The University of Tokyo |
|---|
Principal Investigator |
今井 直毅 東京大学, 大学院数理科学研究科, 准教授 (90597775)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
GAISIN ILDAR 東京大学, 大学院数理科学研究科, 外国人特別研究員
|
|---|
| Project Period (FY) |
2019-04-25 – 2021-03-31
|
| Keywords | Cohomology |
|---|
| Outline of Annual Research Achievements |
Let f: X -> Y be a proper smooth morphism of p-adic formal schemes over O_C (the ring of integers of a complete algebraically closed non-Archimedean extension of Qp). Recently Bhatt-Morrow-Scholze constructed a so called A_inf-cohomology (over a point) which captures various p-adic cohomology theories (in the process reproving the important crystalline conjecture). The current project jwith Teruhisa Koshikawa has two objectives. Firstly, we construct a relative version of A_inf-cohomology for f and relate it to the theory of coefficients recently developed by Morrow-Tsuji. Secondly, we compare this relative A_inf-cohomology with the pushforward of the structural sheaf on the prismatic site. The main theorem of the original A_inf-cohomology paper by Bhatt-Morrow-Scholze is the so called absolute crystalline comparison isomorphism. However, now with the intervention of the prismatic site (by Bhatt-Scholze), the comparison with the latter should be considered the main task. The principal novelty for constructing the relative A_inf-cohomology is to use an idea recently developed by Abbes-Gros in their recent work on the relative Hodge-Tate spectral sequence. The idea to pushforward to a fiber product of topoi containing the proetale and etale sites of X and Y. Currently we have proved the relative p-adic etale comparison and completed local calculations via Faltings’ almost purity.
|
| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We was able to construct a relative version of A_inf-cohomology and relate it to the theory of coefficients recently developed by Morrow-Tsuji. Further, we was able to compare this relative A_inf-cohomology with the pushforward of the structural sheaf on the prismatic site. Therefore the research is going well as expected.
|
| Strategy for Future Research Activity |
In recent work of Colmez-Dospinescu-Niziol, the authors calculate the integral p-adic etale cohomology of Drinfeld symmetric space. It is therefore natural to now understand the integral p-adic etale cohomology of the tower sitting above Drinfeld space. In this work, we consider just the first level appearing in the tower. This is a wildly ramified covering of Drinfeld upper half space. In previous work, Haoran Wang calculated the etale cohomology of this covering with rational l-adic coefficients. We plan to use the strategy developed by Haoran Wang, in particular using the (formal) open affinoids that he constructed and combining this with the strategy of Colmez-Dospinescu-Niziol.
|
