## 59.8 The big crystalline site

We first define the big site. Given a divided power scheme $(S, \mathcal{I}, \gamma )$ we say $(T, \mathcal{J}, \delta )$ is a divided power scheme over $(S, \mathcal{I}, \gamma )$ if $T$ comes endowed with a morphism $T \to S$ of divided power schemes. Similarly, we say a divided power thickening $(U, T, \delta )$ is a divided power thickening over $(S, \mathcal{I}, \gamma )$ if $T$ comes endowed with a morphism $T \to S$ of divided power schemes.

Definition 59.8.1. In Situation 59.7.5.

A *divided power thickening of $X$ relative to $(S, \mathcal{I}, \gamma )$* is given by a divided power thickening $(U, T, \delta )$ over $(S, \mathcal{I}, \gamma )$ and an $S$-morphism $U \to X$.

A *morphism of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$* is defined in the obvious manner.

The category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$ is denoted $\text{CRIS}(X/S, \mathcal{I}, \gamma )$ or simply $\text{CRIS}(X/S)$.

For any $(U, T, \delta )$ in $\text{CRIS}(X/S)$ we have that $p$ is locally nilpotent on $T$, see discussion after Definition 59.7.3. A good way to visualize all the data associated to $(U, T, \delta )$ is the commutative diagram

\[ \xymatrix{ T \ar[dd] & U \ar[l] \ar[d] \\ & X \ar[d] \\ S & S_0 \ar[l] } \]

where $S_0 = V(\mathcal{I}) \subset S$. Morphisms of $\text{CRIS}(X/S)$ can be similarly visualized as huge commutative diagrams. In particular, there is a canonical forgetful functor

59.8.1.1
\begin{equation} \label{crystalline-equation-forget} \text{CRIS}(X/S) \longrightarrow \mathit{Sch}/X,\quad (U, T, \delta ) \longmapsto U \end{equation}

as well as its one sided inverse (and left adjoint)

59.8.1.2
\begin{equation} \label{crystalline-equation-endow-trivial} \mathit{Sch}/X \longrightarrow \text{CRIS}(X/S),\quad U \longmapsto (U, U, \emptyset ) \end{equation}

which is sometimes useful.

Lemma 59.8.2. In Situation 59.7.5. The category $\text{CRIS}(X/S)$ has all finite nonempty limits, in particular products of pairs and fibre products. The functor (59.8.1.1) commutes with limits.

**Proof.**
Omitted. Hint: See Lemma 59.5.3 for the affine case. See also Divided Power Algebra, Remark 23.3.5.
$\square$

Lemma 59.8.3. In Situation 59.7.5. Let

\[ \xymatrix{ (U_3, T_3, \delta _3) \ar[d] \ar[r] & (U_2, T_2, \delta _2) \ar[d] \\ (U_1, T_1, \delta _1) \ar[r] & (U, T, \delta ) } \]

be a fibre square in the category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$. If $T_2 \to T$ is flat and $U_2 = T_2 \times _ T U$, then $T_3 = T_1 \times _ T T_2$ (as schemes).

**Proof.**
This is true because a divided power structure extends uniquely along a flat ring map. See Divided Power Algebra, Lemma 23.4.2.
$\square$

The lemma above means that the base change of a flat morphism of divided power thickenings is another flat morphism, and in fact is the “usual” base change of the morphism. This implies that the following definition makes sense.

Definition 59.8.4. In Situation 59.7.5.

A family of morphisms $\{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\} $ of divided power thickenings of $X/S$ is a *Zariski, étale, smooth, syntomic, or fppf covering* if and only if

$U_ i = U \times _ T T_ i$ for all $i$ and

$\{ T_ i \to T\} $ is a Zariski, étale, smooth, syntomic, or fppf covering.

The *big crystalline site* of $X$ over $(S, \mathcal{I}, \gamma )$, is the category $\text{CRIS}(X/S)$ endowed with the Zariski topology.

The topos of sheaves on $\text{CRIS}(X/S)$ is denoted $(X/S)_{\text{CRIS}}$ or sometimes $(X/S, \mathcal{I}, \gamma )_{\text{CRIS}}$^{1}.

There are some obvious functorialities concerning these topoi.

## Comments (1)

Comment #6265 by Mingchen on