Lemma 58.31.3. Let X be a locally Noetherian scheme. Let D \subset X be an effective Cartier divisor such that D is a regular scheme. Let Y \to X \setminus D be a finite étale morphism. If Y is unramified over X in codimension 1, then there exists a finite étale morphism Y' \to X whose restriction to X \setminus D is Y.
Proof. This is a special case of Lemma 58.31.2. First, D is nowhere dense in X (see discussion in Divisors, Section 31.13) and hence X \setminus D is dense in X. Second, the ring \mathcal{O}_{X, x} is a regular local ring for all x \in D by Algebra, Lemma 10.106.7 and our assumption that \mathcal{O}_{D, x} is regular. \square
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