Crystalline cohomology

In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ${\displaystyle X}$ over a base field ${\displaystyle k}$. Its values ${\displaystyle H^{n}(X/W)}$ are modules over the ring ${\displaystyle W}$ of Witt vectors over ${\displaystyle k}$. It was introduced by Alexander Grothendieck (1966, 1968) and developed by Pierre Berthelot (1974).

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 ${\displaystyle X}$ in characteristic ${\displaystyle p}$ is the de Rham cohomology of a smooth lift of ${\displaystyle X}$ to characteristic 0, while de Rham cohomology of ${\displaystyle X}$ is the crystalline cohomology reduced mod ${\displaystyle p}$ (after taking into account higher Tors).

The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking a local lifting of a scheme from characteristic ${\displaystyle p}$ to characteristic 0 and employing an appropriate version of algebraic de Rham cohomology.

Crystalline cohomology only works well for smooth proper schemes. Rigid cohomology extends it to more general schemes.