Crystalline cohomology

In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by AlexanderGrothendieck (1966, 1968) and developed by PierreBerthelot (1974). Its values are modules over rings of Witt vectors over the base field.

Crystalline cohomology is partly inspired by the p-adic proof in Dwork (1960) of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety X in characteristic p is the de Rham cohomology of a smooth lift of X to characteristic 0, while de Rham cohomology of X is the crystalline cohomology reduced mod p (after taking into account higher Tors).