## Tag `07KL`

Chapter 51: Crystalline Cohomology > Section 51.9: The crystalline site

Lemma 51.9.5. In Situation 51.7.5. Let $X' \subset X$ and $S' \subset S$ be open subschemes such that $X'$ maps into $S'$. Then there is a fully faithful functor $\text{Cris}(X'/S') \to \text{Cris}(X/S)$ which gives rise to a morphism of topoi fitting into the commutative diagram $$ \xymatrix{ (X'/S')_{\text{cris}} \ar[r] \ar[d]_{u_{X'/S'}} & (X/S)_{\text{cris}} \ar[d]^{u_{X/S}} \\ \mathop{\textit{Sh}}\nolimits(X'_{Zar}) \ar[r] & \mathop{\textit{Sh}}\nolimits(X_{Zar}) } $$ Moreover, this diagram is an example of localization of morphisms of topoi as in Sites, Lemma 7.30.1.

Proof.The fully faithful functor comes from thinking of objects of $\text{Cris}(X'/S')$ as divided power thickenings $(U, T, \delta)$ of $X$ where $U \to X$ factors through $X' \subset X$ (since then automatically $T \to S$ will factor through $S'$). This functor is clearly cocontinuous hence we obtain a morphism of topoi as indicated. Let $h_{X'} \in \mathop{\textit{Sh}}\nolimits(X_{Zar})$ be the representable sheaf associated to $X'$ viewed as an object of $X_{Zar}$. It is clear that $\mathop{\textit{Sh}}\nolimits(X'_{Zar})$ is the localization $\mathop{\textit{Sh}}\nolimits(X_{Zar})/h_{X'}$. On the other hand, the category $\text{Cris}(X/S)/u_{X/S}^{-1}h_{X'}$ (see Sites, Lemma 7.29.3) is canonically identified with $\text{Cris}(X'/S')$ by the functor above. This finishes the proof. $\square$

The code snippet corresponding to this tag is a part of the file `crystalline.tex` and is located in lines 1948–1965 (see updates for more information).

```
\begin{lemma}
\label{lemma-localize}
In Situation \ref{situation-global}.
Let $X' \subset X$ and $S' \subset S$ be open subschemes such that
$X'$ maps into $S'$. Then there is a fully faithful functor
$\text{Cris}(X'/S') \to \text{Cris}(X/S)$
which gives rise to a morphism of topoi fitting into the commutative
diagram
$$
\xymatrix{
(X'/S')_{\text{cris}} \ar[r] \ar[d]_{u_{X'/S'}} &
(X/S)_{\text{cris}} \ar[d]^{u_{X/S}} \\
\Sh(X'_{Zar}) \ar[r] & \Sh(X_{Zar})
}
$$
Moreover, this diagram is an example of localization of morphisms of
topoi as in Sites, Lemma \ref{sites-lemma-localize-morphism-topoi}.
\end{lemma}
\begin{proof}
The fully faithful functor comes from thinking of
objects of $\text{Cris}(X'/S')$ as divided power
thickenings $(U, T, \delta)$ of $X$ where $U \to X$
factors through $X' \subset X$ (since then automatically $T \to S$
will factor through $S'$). This functor is clearly cocontinuous
hence we obtain a morphism of topoi as indicated.
Let $h_{X'} \in \Sh(X_{Zar})$ be the representable sheaf associated
to $X'$ viewed as an object of $X_{Zar}$. It is clear that
$\Sh(X'_{Zar})$ is the localization $\Sh(X_{Zar})/h_{X'}$.
On the other hand, the category $\text{Cris}(X/S)/u_{X/S}^{-1}h_{X'}$
(see Sites, Lemma \ref{sites-lemma-localize-topos-site})
is canonically identified with $\text{Cris}(X'/S')$ by the functor above.
This finishes the proof.
\end{proof}
```

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