crystalline site

under construction

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

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

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

The abelian sheaf cohomology over $Cris(X)$ is the crystalline cohomology of $X$.

Let $k$ be a ring. Let $k \to R$ be a finitely presented $k$-algebra. Then the big infinitesimal topos of the $Spec(k)$-scheme $Spec(R)$ classifies the theory of commutative squares of ring homomorphisms

$\array{A & \rightarrow & B \\
\uparrow &&\uparrow \\
k & \rightarrow& R
}$

where the rings $A$ and $B$ are local, the top arrow $A \to B$ is surjective and has nilpotent kernel (i.e. every element of the kernel is nilpotent) This result is due to (Hutzler 2018). By the conditions on the top morphism, it is enough to require that $A$ or $B$ is local.

Routine arguments, to be made explicit in a further revision of this entry, allow to generalize this description to the non-affine case. Let $f \colon X \to S$ be a scheme over $S$. Assume that $X$ is locally of finite presentation over $S$. Then the big infinitesimal topos of the $S$-scheme $X$ classifies, as a $Sh(X)$-topos, the $Sh(X)$-theory of commutative squares of ring homomorphisms

$\array{A & \rightarrow & B \\
\uparrow &&\uparrow \\
f^{-1}\mathcal{O}_S & \rightarrow& \mathcal{O}_X
}$

where the rings $A$ and $B$ are local, the top arrow $A \to B$ is surjective and has nilpotent kernel (i.e. every element of the kernel is nilpotent), and both vertical arrows are local (i.e. reflect invertibility).

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 $\infty$-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 $W_n(k)$).

The result about the geometric theory classified by the big infinitesimal topos appears in

- Matthias Hutzler,
*Internal language and classified theories of toposes in algebraic geometry*, Master’s thesis at the University of Augsburg, 2018, GitLab, pdf download