The Stacks project

Lemma 61.12.11. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. The functor $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{pro\text{-}\acute{e}tale})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$.

Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 7.29.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.29.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. Being cocontinuous just means that any pro-étale covering of $T/S$, $T$ affine, can be refined by a standard pro-étale covering of $T$. This is the content of Lemma 61.12.5. Hence (1) holds. We see $u$ is continuous simply because a standard pro-étale covering is a pro-étale covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering. $\square$

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