The Stacks project

Lemma 59.46.2. Let $i : Z \to X$ be a closed immersion of schemes. Let $V \to Z$ be an étale morphism of schemes. There exist étale morphisms $U_ i \to X$ and morphisms $U_{i, Z} \to V$ such that $\{ U_{i, Z} \to V\} $ is a Zariski covering of $V$.

Proof. Since we only have to find a Zariski covering of $V$ consisting of schemes of the form $U_ Z$ with $U$ étale over $X$, we may Zariski localize on $X$ and $V$. Hence we may assume $X$ and $V$ affine. In the affine case this is Algebra, Lemma 10.143.10. $\square$

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