The Stacks project

Lemma 60.9.5. In Situation 60.7.5. 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

\[ \xymatrix{ (X'/S')_{\text{cris}} \ar[r] \ar[d]_{u_{X'/S'}} & (X/S)_{\text{cris}} \ar[d]^{u_{X/S}} \\ \mathop{\mathit{Sh}}\nolimits (X'_{Zar}) \ar[r] & \mathop{\mathit{Sh}}\nolimits (X_{Zar}) } \]

Moreover, this diagram is an example of localization of morphisms of topoi as in Sites, Lemma 7.31.1.

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 $h_{X'} \in \mathop{\mathit{Sh}}\nolimits (X_{Zar})$ be the representable sheaf associated to $X'$ viewed as an object of $X_{Zar}$. It is clear that $\mathop{\mathit{Sh}}\nolimits (X'_{Zar})$ is the localization $\mathop{\mathit{Sh}}\nolimits (X_{Zar})/h_{X'}$. On the other hand, the category $\text{Cris}(X/S)/u_{X/S}^{-1}h_{X'}$ (see Sites, Lemma 7.30.3) is canonically identified with $\text{Cris}(X'/S')$ by the functor above. This finishes the proof. $\square$

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