nLab
crystalline cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Crystalline cohomology is the abelian sheaf cohomology with respect to the crystalline site of a scheme. Hence, put more generally, it is the cohomology of de Rham spaces/coreduced objects.

Crystalline cohomology serves to refine the notion of de Rham cohomology for schemes.

Crystalline cohomology is in particular a Weil cohomology and is generalized by the notion of rigid cohomology.

Definition

Let XX be a scheme over a base SS. The crystalline site Cris(X/S)Cris(X/S) of XX is

If SS is of characteristic 0, then Cris(X/S)Cris(X/S) coincides with the infinitesimal site of XX. (…details…).

Properties

Relation to de Rham space

Crystalline cohomology of XX is the cohomology of the de Rham space of XX. See there for more.

Relation to de Rham cohomology

Relation to differential homotopy type theory

In differential homotopy type theory the infinitesimal flat modality sends coefficients to coefficients for crystalline cohomology.

References

Related entries: crystal, infinitesimal site, rigid cohomology, Dieudonné module, Monsky-Washnitzer cohomology, Grothendieck connection

An original account of the definition of the crystalline topos is section 7, page 299 of

A more recent account is

Discussion of this in the modern context of higher geometry/D-geometry is in

A p-adic cohomology for varieties in characteristic pp it was it was discussed in

Review:

Discussion of this in terms of Cech cohomology is in

See also