The Stacks project

Lemma 61.25.4. Let $i : Z \to X$ be an integral universally injective and surjective morphism of schemes. Then $i_{{pro\text{-}\acute{e}tale}, *}$ and $i_{pro\text{-}\acute{e}tale}^{-1}$ are quasi-inverse equivalences of categories of pro-étale topoi.

Proof. There is an immediate reduction to the case that $X$ is affine. Then $Z$ is affine too. Set $A = \mathcal{O}(X)$ and $B = \mathcal{O}(Z)$. Then the categories of étale algebras over $A$ and $B$ are equivalent, see Étale Cohomology, Theorem 59.45.2 and Remark 59.45.3. Thus the categories of ind-étale algebras over $A$ and $B$ are equivalent. In other words the categories $X_{app}$ and $Z_{app}$ of Lemma 61.12.21 are equivalent. We omit the verification that this equivalence sends coverings to coverings and vice versa. Thus the result as Lemma 61.12.21 tells us the pro-étale topos is the topos of sheaves on $X_{app}$. $\square$

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