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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## Comments (2)

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