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Number Theory Seminar
Seminar information archive ～10/18｜Next seminar｜Future seminars 10/19～
| Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Naoki Imai, Shane Kelly |
2026/11/11
17:00-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Shuji Saito (University of Tokyo)
An integral Hyodo-Kato isomorphism
Shuji Saito (University of Tokyo)
An integral Hyodo-Kato isomorphism
[ Abstract ]
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 Yau’s log variant of the saturated de-Rham-Witt 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.
The 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 Yau’s log variant of the saturated de-Rham-Witt 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.
The results are joint works with Federico Binda (Milano).
