Proof. (See also Topologies, Lemma 34.9.6.) Let $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ be an étale covering. Since an étale morphism is flat, and the elements of the covering should cover its target, the property fp (faithfully flat) is satisfied. To check the property qc (quasi-compact), let $V \subset U$ be an affine open, and write $\varphi _ i^{-1}(V) = \bigcup _{j \in J_ i} V_{ij}$ for some affine opens $V_{ij} \subset U_ i$. Since $\varphi _ i$ is open (as étale morphisms are open), we see that $V = \bigcup _{i\in I} \bigcup _{j \in J_ i} \varphi _ i(V_{ij})$ is an open covering of $V$. Further, since $V$ is quasi-compact, this covering has a finite refinement. $\square$

Comment #3235 by Dario Weißmann on

Typo in the proof: The $V$ is missing in $\varphi_i^{-1}(V)=...$

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