nLab
crystalline cohomology

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 $X$ be a scheme overy a base $S$. The crystalline site $\mathrm{Cris}(X/S)$ of $X$ is

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

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

If $S$ is of characteristic 0, then $\mathrm{Cris}(X/S)$ coincides with the infinitesimal site of $X$. (…details…).

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

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

• Jacob Lurie, Notes on Crystals and algebraic D-modules (pdf)

• wikipedia crystalline cohomology

• P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243, ISBN0-691-08218-9

• P. Berthelot, A. Ogus, $F$-Isocrystals and de Rham Cohomology, I, Invent. math. 72, 1983, pp. 159-199