nLab crystalline cohomology





Special and general types

Special notions


Extra structure





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.


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

  • the category whose objects are all nilpotent SS-immersions UTU \hookrightarrow T, where UU is an open set of XX and and the ideal on TT defining this immersion being endowed with a nilpotent divided power structure (…details…).;

  • the Grothendieck topology on this category is the Zariski topology.

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


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.


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

  • Alexander Grothendieck, Crystals and de Rham cohomology of schemes , chapter IX in Dix Exposes sur la cohomologie des schemas (pdf)

A more recent account is

  • Luc Illusie, Crystalline cohomology, in Motives, Proc. Sympos. Pure Math., vol. 55, part 1, Amer. Math. Soc. Providence, RI, 1994, 43–70.

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

  • Jacob Lurie, Notes on crystals and algebraic 𝒟\mathcal{D}-modules (2010) [pdf]

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


Discussion of this in terms of Cech cohomology is in

See also

Last revised on May 4, 2023 at 12:06:34. See the history of this page for a list of all contributions to it.