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Almost étale extensions. (English) Zbl 1027.14011
Berthelot, Pierre (ed.) et al., Cohomologies $$p$$-adiques et applications arithmétiques (II). Paris: Société Mathématique de France. Astérisque. 279, 185-270 (2002).
The theory of almost étale coverings allows to compare crystalline and $$p$$-adic étale cohomology, for schemes over a $$p$$-adic discrete valuation ring. Using Frobenius the main technical result (a purity theorem) is reproved and extended to all toroidal singularities. As a consequence one obtains Tsuji’s comparison theorem for schemes with such type of singularities, even for cohomology with coefficients in suitable local systems. On the way we have to establish some basic results on finiteness of crystalline cohomology with such coefficients.
For the entire collection see [Zbl 0990.00020].

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies 14A20 Generalizations (algebraic spaces, stacks) 14B05 Singularities in algebraic geometry