Lemma 37.38.7. Let $S$ be a scheme. Let $\mathcal{U} = \{ S_ i \to S\} _{i \in I}$ be a smooth covering of $S$, see Topologies, Definition 34.5.1. Then there exists an étale covering $\mathcal{V} = \{ T_ j \to S\} _{j \in J}$ (see Topologies, Definition 34.4.1) which refines (see Sites, Definition 7.8.1) $\mathcal{U}$.

Proof. For every $s \in S$ there exists an $i \in I$ such that $s$ is in the image of $S_ i \to S$. By Lemma 37.38.6 we can find an étale morphism $g_ s : T_ s \to S$ such that $s \in g_ s(T_ s)$ and such that $g_ s$ factors through $S_ i \to S$. Hence $\{ T_ s \to S\}$ is an étale covering of $S$ that refines $\mathcal{U}$. $\square$

Comment #2539 by Anonymous on

Small typo: in the proof, $g_s(T)_s$ should be $g_s(T_s)$.

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