TITLE: Fontaine's crystalline conjecture revisited.

ABSTRACT (joint work with O. Brinon and A. Iovita):

Let $K$ be a local field with ring of integers $O_K$ such that $p$ generates the maximal ideal of $O_K$. Given a proper and smooth scheme $X$ over $O_K$, I will prove a comparison isomorphism between the \'etale cohomology of $X_\Kbar$ and the de Rham cohomology of $X_K$. Our approach is based both on Faltings' approach and on Fontaine's theory in the relative setting as developed by O. Brinon.