Lemma 38.2.1. Let $i : Z \to X$ be a closed immersion of affine schemes. Let $Z' \to Z$ be an étale morphism with $Z'$ affine. Then there exists an étale morphism $X' \to X$ with $X'$ affine such that $Z' \cong Z \times _ X X'$ as schemes over $Z$.
Proof. See Algebra, Lemma 10.143.10. $\square$
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