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 over a base $S$. The crystalline site $Cris(X/S)$ 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 $Cris(X/S)$ coincides with the infinitesimal site of $X$. (…details…).

Properties

Relation to de Rham space

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

Relation to de Rham cohomology

• comparison theorem (crystalline cohomology)

Relation to differential homotopy type theory

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

Related concepts

• p-adic cohomology

• de Rham cohomology

• Weil cohomology

  • étale 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

• 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 $\mathcal{D}$-modules (2010) [pdf]

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

• Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique $p \gt 0$, Lecture Notes in Mathematics, Vol. 407, Springer- Verlag, Berlin, 1974. (doi:10.1007/BFb0068636, MR 0384804)

• Pierre Berthelot, Arthur Ogus, $F$-Isocrystals and de Rham Cohomology, I, Invent. math. 72, 1983, pp. 159-199 (eudml:143016)

Review:

• Pierre Berthelot, Arthur Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243, (ISBN:0-691-08218-9)

Discussion of this in terms of Cech cohomology is in

• Bhargav Bhatt, Aise Johan de Jong, Crystalline cohomology and de Rham cohomology (pdf)

See also

• wikipedia crystalline cohomology