The Stacks project

Lemma 34.3.14. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$. The inclusion functor $S_{Zar} \to (\mathit{Sch}/S)_{Zar}$ satisfies the hypotheses of Sites, Lemma 7.21.8 and hence induces a morphism of sites

\[ \pi _ S : (\mathit{Sch}/S)_{Zar} \longrightarrow S_{Zar} \]

and a morphism of topoi

\[ i_ S : \mathop{\mathit{Sh}}\nolimits (S_{Zar}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar}) \]

such that $\pi _ S \circ i_ S = \text{id}$. Moreover, $i_ S = i_{\text{id}_ S}$ with $i_{\text{id}_ S}$ as in Lemma 34.3.13. In particular the functor $i_ S^{-1} = \pi _{S, *}$ is described by the rule $i_ S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.

Proof. In this case the functor $u : S_{Zar} \to (\mathit{Sch}/S)_{Zar}$, in addition to the properties seen in the proof of Lemma 34.3.13 above, also is fully faithful and transforms the final object into the final object. The lemma follows. $\square$

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