Lemma 58.20.4. Let $(A, \mathfrak m)$ be a Noetherian local ring. Set $X = \mathop{\mathrm{Spec}}(A)$ and let $U = X \setminus \{ \mathfrak m\} $. Let $V$ be finite étale over $U$. Let $A^\wedge $ be the $\mathfrak m$-adic completion of $A$, let $X' = \mathop{\mathrm{Spec}}(A^\wedge )$ and let $U'$ and $V'$ be the base changes of $U$ and $V$ to $X'$. The following are equivalent

$V = Y \times _ X U$ for some $Y \to X$ finite étale, and

$V' = Y' \times _{X'} U'$ for some $Y' \to X'$ finite étale.

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