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$$p$$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. (English) Zbl 0945.14008
The purpose of this paper is to prove in wide generality the $$C_{st}$$ conjecture (due to Fontaine and Jannsen) on $$p$$-adic étale and log. crystalline cohomology. Let $$K$$ be a complete discrete valuation field of characteristic $$0$$ with perfect residue field $$k$$ of characteristic $$p > 0$$ and $${\mathcal O}_{K}$$ be the ring of integers of $$K$$. There is a ring $$B_{st}$$ requiring heavy definitions defined by Fontaine depending on the choice of a prime element $$\pi$$ of $$K$$. The conjecture $$C_{st}$$ states that there is a canonical $$B_{st}$$-linear isomorphism $B_{st}\otimes_{{\mathbb{Q}}_{p}}H^{m}_{et}(X_{\overline{K}},{\mathbb{Q}}_{p})\simeq B_{st}\otimes_{K_{0}}H_{log.-crys}(X)$ preserving the action of the absolute Galois group $$\text{Gal}(\overline{K}/K)$$, associated linear endomorphisms $$\phi$$ and $$N$$ and the filtration induced after tensoring with $$B_{dR}$$ over $$B_{st}$$. Here, $$K_{0}$$ is the field of fractions of the ring $$W$$ of Witt-vectors in $$k$$. To prove this conjecture, the author develops and uses facts on the syntomic complex and $$p$$-adic vanishing cycles.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies
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