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過去の記録 ～10/25｜次回の予定｜今後の予定 10/26～
| 開催情報 | 水曜日 17:00～18:00 数理科学研究科棟(駒場) 117号室 |
|---|---|
| 担当者 | 今井 直毅，ケリー シェーン |
次回の予定
2026年11月11日(水)
17:00-18:00 数理科学研究科棟(駒場) 123号室
斎藤 秀司 氏 (東京大学大学院数理科学研究科)
An integral Hyodo-Kato isomorphism
斎藤 秀司 氏 (東京大学大学院数理科学研究科)
An integral Hyodo-Kato isomorphism
[ 講演概要 ]
Let O_K be a complete mixed characteristic DVR with perfect residue field k and W=W(k) be the Witt ring and K_0=W[1/p]. Let X be a proper semistable scheme over O_K and X_0 and Y be its special and generic fibers respectively. The classical theorem of Hyodo and Kato provides a rational identification between the de Rham cohomology H^*_{DR}(Y/K) of Y over K and the rationalized log crystalline cohomology of X_0 relative to the log ring W^0 whose underlying ring is W and the log structure is associated to the map of monoids \mathbb{N} \to W sending 1 to 0.It gives a positive answer to a conjecture of Fontaine and Jannsen endowing H^*_{DR}(Y/K) with K_0[\phi,N]-module structure, where \phi is a Frobenius and N is a monodromy operator. A number of different proofs of the Hyodo-Kato theorem were given by Beilinson, Große-Kloenne and Ertl-Yamada.
The main result of this talk gives an integral version of the Hyodo-Kato theorem.
The proof is completely different from those given so far and relies on Yao’s log variant of the saturated de Rham complex of Bhatt, Lurie, and Mathew.
If time permits, I will explain a work in progress where the above result is upgraded by using derived log crystalline cohomology to a comparison theorem between categories of crystals on a log smooth scheme over the log point k^0 relative to W with different log structures lifting k^0 as well as its stacky reformulation via prismatization.
All results are joint works with Federico Binda (Milano).
Let O_K be a complete mixed characteristic DVR with perfect residue field k and W=W(k) be the Witt ring and K_0=W[1/p]. Let X be a proper semistable scheme over O_K and X_0 and Y be its special and generic fibers respectively. The classical theorem of Hyodo and Kato provides a rational identification between the de Rham cohomology H^*_{DR}(Y/K) of Y over K and the rationalized log crystalline cohomology of X_0 relative to the log ring W^0 whose underlying ring is W and the log structure is associated to the map of monoids \mathbb{N} \to W sending 1 to 0.It gives a positive answer to a conjecture of Fontaine and Jannsen endowing H^*_{DR}(Y/K) with K_0[\phi,N]-module structure, where \phi is a Frobenius and N is a monodromy operator. A number of different proofs of the Hyodo-Kato theorem were given by Beilinson, Große-Kloenne and Ertl-Yamada.
The main result of this talk gives an integral version of the Hyodo-Kato theorem.
The proof is completely different from those given so far and relies on Yao’s log variant of the saturated de Rham complex of Bhatt, Lurie, and Mathew.
If time permits, I will explain a work in progress where the above result is upgraded by using derived log crystalline cohomology to a comparison theorem between categories of crystals on a log smooth scheme over the log point k^0 relative to W with different log structures lifting k^0 as well as its stacky reformulation via prismatization.
All results are joint works with Federico Binda (Milano).
