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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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