Lemma 37.40.2. Let $f : U \to S$ be a surjective étale morphism of affine schemes. There exists a surjective, finite locally free morphism $\pi : T \to S$ and a finite open covering $T = T_1 \cup \ldots \cup T_ n$ such that each $T_ i \to S$ factors through $U \to S$. Diagram:
where the south-west arrow is a Zariski-covering.