The Stacks project

Lemma 58.31.2. Let $X$ be a locally Noetherian scheme. Let $U \subset X$ be open and dense. Let $Y \to U$ be a finite étale morphism. Assume

  1. $Y$ is unramified over $X$ in codimension $1$, and

  2. $\mathcal{O}_{X, x}$ is regular for all $x \in X \setminus U$.

Then there exists a finite étale morphism $Y' \to X$ whose restriction to $X \setminus D$ is $Y$.

Proof. Let $\xi \in X \setminus U$ be a generic point of an irreducible component of $X \setminus U$ of codimension $1$. Then $\mathcal{O}_{X, \xi }$ is a discrete valuation ring. As in the discussion above, write $Y \times _ U \mathop{\mathrm{Spec}}(K_\xi ) = \mathop{\mathrm{Spec}}(L_\xi )$. Denote $B_\xi $ the integral closure of $\mathcal{O}_{X, \xi }$ in $L_\xi $. Our assumption that $Y$ is unramified over $X$ in codimension $1$ signifies that $\mathcal{O}_{X, \xi } \to B_\xi $ is finite étale. Thus we get $Y_\xi \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi })$ finite étale and an isomorphism

\[ Y \times _ U \mathop{\mathrm{Spec}}(K_\xi ) \cong Y_\xi \times _{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi })} \mathop{\mathrm{Spec}}(K_\xi ) \]

over $\mathop{\mathrm{Spec}}(K_\xi )$. By Limits, Lemma 32.20.3 we find an open subscheme $U \subset U' \subset X$ containing $\xi $ and a morphism $Y' \to U'$ of finite presentation whose restriction to $U$ recovers $Y$ and whose restriction to $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi })$ recovers $Y_\xi $. Finally, the morphism $Y' \to U'$ is finite étale after possible shrinking $U'$ to a smaller open by Limits, Lemma 32.20.4. Repeating the argument with the other generic points of $X \setminus U$ of codimension $1$ we may assume that we have a finite étale morphism $Y' \to U'$ extending $Y \to U$ to an open subscheme containing $U' \subset X$ containing $U$ and all codimension $1$ points of $X \setminus U$. We finish by applying Lemma 58.21.6 to $Y' \to U'$. Namely, all local rings $\mathcal{O}_{X, x}$ for $x \in X \setminus U'$ are regular and have $\dim (\mathcal{O}_{X, x}) \geq 2$. Hence we have purity for $\mathcal{O}_{X, x}$ by Lemma 58.21.3. $\square$

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