group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A comparison between p-adic cohomology for varieties in characteristic and crystalline cohomology was was discussed in
The comparison theorem shown there (theorem V2.3.2) says that the crystalline cohomology over 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
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