The Stacks project

Lemma 59.21.2. Let $S$ be a scheme. The étale topos of $S$ is independent (up to canonical equivalence) of the construction of the small étale site in Definition 59.20.2.

Proof. We have to show, given two big étale sites $\mathit{Sch}_{\acute{e}tale}$ and $\mathit{Sch}_{\acute{e}tale}'$ containing $S$, then $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \cong \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}')$ with obvious notation. By Topologies, Lemma 34.12.1 we may assume $\mathit{Sch}_{\acute{e}tale}\subset \mathit{Sch}_{\acute{e}tale}'$. By Sets, Lemma 3.9.9 any affine scheme étale over $S$ is isomorphic to an object of both $\mathit{Sch}_{\acute{e}tale}$ and $\mathit{Sch}_{\acute{e}tale}'$. Thus the induced functor $S_{affine, {\acute{e}tale}} \to S_{affine, {\acute{e}tale}}'$ is an equivalence. Moreover, it is clear that both this functor and a quasi-inverse map transform standard étale coverings into standard étale coverings. Hence the result follows from Topologies, Lemma 34.4.12. $\square$

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