The Stacks project

Lemma 58.21.6. Let $j : U \to X$ be an open immersion of Noetherian schemes such that purity holds for $\mathcal{O}_{X, x}$ for all $x \not\in U$. Then

\[ \textit{FÉt}_ X \longrightarrow \textit{FÉt}_ U \]

is essentially surjective.

Proof. Let $V \to U$ be a finite étale morphism. By Noetherian induction it suffices to extend $V \to U$ to a finite étale morphism to a strictly larger open subset of $X$. Let $x \in X \setminus U$ be the generic point of an irreducible component of $X \setminus U$. Then the inverse image $U_ x$ of $U$ in $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is the punctured spectrum of $\mathcal{O}_{X, x}$. By assumption $V_ x = V \times _ U U_ x$ is the restriction of a finite étale morphism $Y_ x \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ to $U_ x$. By Limits, Lemma 32.19.3 we find an open subscheme $U \subset U' \subset X$ containing $x$ and a morphism $V' \to U'$ of finite presentation whose restriction to $U$ recovers $V \to U$ and whose restriction to $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ recovering $Y_ x$. Finally, the morphism $V' \to U'$ is finite étale after possible shrinking $U'$ to a smaller open by Limits, Lemma 32.19.4. $\square$


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