Lemma 35.8.5. Let S be a scheme. Denote
\begin{matrix} \text{id}_{\tau , Zar}
& :
& (\mathit{Sch}/S)_\tau \to S_{Zar},
& \tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}
\\ \text{id}_{\tau , {\acute{e}tale}}
& :
& (\mathit{Sch}/S)_\tau \to S_{\acute{e}tale},
& \tau \in \{ {\acute{e}tale}, smooth, syntomic, fppf\}
\\ \text{id}_{small, {\acute{e}tale}, Zar}
& :
& S_{\acute{e}tale}\to S_{Zar},
\end{matrix}
the morphisms of ringed sites of Remark 35.8.4. Let \mathcal{F} be a sheaf of \mathcal{O}_ S-modules which we view a sheaf of \mathcal{O}-modules on S_{Zar}. Then
(\text{id}_{\tau , Zar})^*\mathcal{F} is the \tau -sheafification of the Zariski sheaf
(f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F})
on (\mathit{Sch}/S)_\tau , and
(\text{id}_{small, {\acute{e}tale}, Zar})^*\mathcal{F} is the étale sheafification of the Zariski sheaf
(f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F})
on S_{\acute{e}tale}.
Let \mathcal{G} be a sheaf of \mathcal{O}-modules on S_{\acute{e}tale}. Then
(\text{id}_{\tau , {\acute{e}tale}})^*\mathcal{G} is the \tau -sheafification of the étale sheaf
(f : T \to S) \longmapsto \Gamma (T, f_{small}^*\mathcal{G})
where f_{small} : T_{\acute{e}tale}\to S_{\acute{e}tale} is the morphism of ringed small étale sites of Remark 35.8.4.
Proof.
Proof of (1). We first note that the result is true when \tau = Zar because in that case we have the morphism of topoi i_ f : \mathop{\mathit{Sh}}\nolimits (T_{Zar}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar}) such that \text{id}_{\tau , Zar} \circ i_ f = f_{small} as morphisms T_{Zar} \to S_{Zar}, see Topologies, Lemmas 34.3.13 and 34.3.17. Since pullback is transitive (see Modules on Sites, Lemma 18.13.3) we see that i_ f^*(\text{id}_{\tau , Zar})^*\mathcal{F} = f_{small}^*\mathcal{F} as desired. Hence, by the remark preceding this lemma we see that (\text{id}_{\tau , Zar})^*\mathcal{F} is the \tau -sheafification of the presheaf T \mapsto \Gamma (T, f^*\mathcal{F}).
The proof of (3) is exactly the same as the proof of (1), except that it uses Topologies, Lemmas 34.4.13 and 34.4.17. We omit the proof of (2).
\square
Comments (2)
Comment #2959 by Ko Aoki on
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