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De Rham and $$p$$-adic etale cohomology [after G. Faltings, J.-M. Fontaine and others ]. (Cohomologie de de Rham et cohomologie étale $$p$$-adique [d’après G. Faltings, J.-M. Fontaine et al.].) (French) Zbl 0736.14005
Sémin. Bourbaki, Vol. 1989/90, 42ème année, Astérisque 189-190, Exp. No. 726, 325-374 (1990).
[For the entire collection see Zbl 0722.00001.]
Let $$X$$ be a proper and smooth scheme over a local field $$K$$ with residue field of characteristic $$p$$. This exposé is concerned with the “isomorphism” of $$p$$-adic periods between the De Rham cohomology of $$X/K$$ and the $$p$$-adic étale cohomology of $$X_{\overline K}$$ with values in $$\mathbb{Q}_ p$$, considered as a Galois module (the “mysterious functor”). The theory includes the Hodge-Tate conjecture, as well as several other now established conjectures.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F40 de Rham cohomology and algebraic geometry 32G20 Period matrices, variation of Hodge structure; degenerations
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