## Tag `07IM`

Chapter 54: Crystalline Cohomology > Section 54.9: The crystalline site

Remark 54.9.6 (Structure morphism). In Situation 54.7.5. Consider the closed subscheme $S_0 = V(\mathcal{I}) \subset S$. If we assume that $p$ is locally nilpotent on $S_0$ (which is always the case in practice) then we obtain a situation as in Definition 54.8.1 with $S_0$ instead of $X$. Hence we get a site $\text{Cris}(S_0/S)$. If $f : X \to S_0$ is the structure morphism of $X$ over $S$, then we get a commutative diagram of ringed topoi $$ \xymatrix{ (X/S)_{\text{cris}} \ar[r]_{f_{\text{cris}}} \ar[d]_{u_{X/S}} & (S_0/S)_{\text{cris}} \ar[d]^{u_{S_0/S}} \\ \mathop{\textit{Sh}}\nolimits(X_{Zar}) \ar[r]^{f_{small}} & \mathop{\textit{Sh}}\nolimits(S_{0, Zar}) \ar[rd] \\ & & \mathop{\textit{Sh}}\nolimits(S_{Zar}) } $$ see Remark 54.9.3. We think of the composition $(X/S)_{\text{cris}} \to \mathop{\textit{Sh}}\nolimits(S_{Zar})$ as the structure morphism of the crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism $$ \tau_{X/S} : (X/S)_{\text{cris}} \longrightarrow \mathop{\textit{Sh}}\nolimits(S_{Zar}) $$ is defined as we can take the lower route through the diagram above.

The code snippet corresponding to this tag is a part of the file `crystalline.tex` and is located in lines 1983–2010 (see updates for more information).

```
\begin{remark}[Structure morphism]
\label{remark-structure-morphism}
In Situation \ref{situation-global}.
Consider the closed subscheme $S_0 = V(\mathcal{I}) \subset S$.
If we assume that $p$ is locally nilpotent on $S_0$ (which is always
the case in practice) then we obtain a situation as in
Definition \ref{definition-divided-power-thickening-X} with $S_0$ instead
of $X$. Hence we get a site $\text{Cris}(S_0/S)$. If $f : X \to S_0$
is the structure morphism of $X$ over $S$, then we get a
commutative diagram of ringed topoi
$$
\xymatrix{
(X/S)_{\text{cris}}
\ar[r]_{f_{\text{cris}}} \ar[d]_{u_{X/S}} &
(S_0/S)_{\text{cris}} \ar[d]^{u_{S_0/S}} \\
\Sh(X_{Zar}) \ar[r]^{f_{small}} & \Sh(S_{0, Zar}) \ar[rd] \\
& & \Sh(S_{Zar})
}
$$
see Remark \ref{remark-functoriality-cris}. We think of the composition
$(X/S)_{\text{cris}} \to \Sh(S_{Zar})$ as the structure morphism of the
crystalline site. Even if $p$ is not locally nilpotent on $S_0$
the structure morphism
$$
\tau_{X/S} : (X/S)_{\text{cris}} \longrightarrow \Sh(S_{Zar})
$$
is defined as we can take the lower route through the diagram above.
\end{remark}
```

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