nLab comparison theorem (crystalline cohomology)

Idea

A comparison between p-adic cohomology for varieties in characteristic $p \gt 0$ and crystalline cohomology was was discussed in

• Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique $g \gt 0$, Lecture Notes in Mathematics, Vol. 407, Springer- Verlag, Berlin, 1974. MR 0384804

The comparison theorem shown there (theorem V2.3.2) says that the crystalline cohomology over $\mathbb{Z}/p$ is canonically identified with the de Rham cohomology of a lift to p-adic geometry if one exists.

Discussion of this in terms of Cech cohomology is in

• Bhargav Bhatt, Aise Johan de Jong, Crystalline cohomology and de Rham cohomology (pdf)

Related concepts

• comparison theorem

• comparison theorem (étale cohomology)