Motivation
One idea for defining a Weil cohomology theory of a variety X over a field k of characteristic p is to 'lift' it to a variety X* over the ring of Witt vectors of k (that gives back X on reduction mod p), then take the de Rham cohomology of this lift. The problem is that it is not at all obvious that this cohomology is independent of the choice of lifting.
The idea of crystalline cohomology in characteristic 0 is to find a direct definition of a cohomology theory as the cohomology of constant sheaves on a suitable site
- Inf(X)
over X, called the infinitesimal site and then show it is the same as the de Rham cohomology of any lift.
The site Inf(X) is a category whose objects can be thought of as some sort of generalization of the conventional open sets of X. In characteristic 0 its objects are infinitesimal thickenings U→T of Zariski open subsets U of X. This means that U is the closed subscheme of a scheme T defined by a nilpotent sheaf of ideals on T; for example, Spec(k)→ Spec(k/(x2)).
Grothendieck showed that for smooth schemes X over C, the cohomology of the sheaf OX on Inf(X) is the same as the usual (smooth or algebraic) de Rham cohomology.
Read more about this topic: Crystalline Cohomology
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