Title: p-Adic Hodge Theory in the Semi-Stable Reduction Case

We survey the statement and the proof (by K.~Kato and the author) of the semi-stable conjecture of Fontaine-Jannsen on $p$-adic \'etale cohomology and crystalline cohomology, generalizing it to truncated simplicial schemes. Thanks to the alteration of de Jong, this generalization especially implies that the $p$-adic \'etale cohomology of any proper variety (which may have singularity) is potentially semi-stable.

1991 Mathematics Subject Classification: 14F30, 14F20

Keywords and Phrases: $p$-adic \'etale cohomology, crystalline cohomology, semi-stable reduction, simplicial scheme

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