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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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