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Tag 07KL

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

Lemma 54.9.5. In Situation 54.7.5. 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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