nLab
crystalline site

under construction

Definition

Let $C_{\mathrm{et}}$ be the etale site of complex schemes of finite type. For $X$ a scheme, its infinitesimal site $\mathrm{Cris}(X)$ is the big site $C_{\mathrm{et}}/X_{\mathrm{dR}}$ of the de Rham space $X_{\mathrm{dR}}:C_{\mathrm{et}}\to \mathrm{Set}$:

the site whose objects are pairs $(\mathrm{Spec}A,(\mathrm{Spec}A{)}_{\mathrm{red}}\to X)$ of an affine $\mathrm{Spec}A$ and a morphism from its reduced part ($(\mathrm{Spec}A{)}_{\mathrm{red}}=\mathrm{Spec}(A/I)$ for $I$ the nilradical of $A$) into $X$.

More generally, for positive characteristic, the definition is more involved than that.

Properties

The abelian sheaf cohomology over $\mathrm{Cris}(X)$ is the crystalline cohomology of $X$.

Related concepts

• crystalline cohomology

• crystalline differential operator

References

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)

A review of some aspects is in

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

and on page 7 of

• Carlos Simpson, Constantin Teleman, deRham theorem for $\mathrm{\infty }$-stacks (pdf)

In the article

• Arthur Ogus, Cohomology of the infinitesimal site Annales scientifiques de l’École Normale Supérieure, Sér. 4, 8 no. 3 (1975), p. 295-318 (numdam)

it is shown that if $X$ is proper over an algebraically closed field $k$ of characteristic $p$, and embeds into a smooth scheme over $k$, then the infinitesimal cohomology of $X$ coincides with etale cohomology with coefficients in $k$ (or more generally $W_n(k)$ if we work with the infinitesimal site of $X$ over $W_n(k)$).