| 研究課題/領域番号 |
19F19022
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| 研究機関 | 東京大学 |
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研究代表者 |
今井 直毅 東京大学, 大学院数理科学研究科, 准教授 (90597775)
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| 研究分担者 |
GAISIN ILDAR 東京大学, 大学院数理科学研究科, 外国人特別研究員
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| 研究期間 (年度) |
2019-04-25 – 2021-03-31
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| キーワード | Cohomology |
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| 研究実績の概要 |
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.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
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.
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| 今後の研究の推進方策 |
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.
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