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).
$\square$
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