Lemma 58.20.5. In Situation 58.19.1. Let $V$ be finite étale over $U$. Assume

$A$ has a dualizing complex,

the pair $(A, (f))$ is henselian,

one of the following is true

$A_ f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 3$, or

for every prime $\mathfrak p \subset A$, $f \not\in \mathfrak p$ we have $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 2$.

$V_0 = V \times _ U U_0$ is equal to $Y_0 \times _{X_0} U_0$ for some $Y_0 \to X_0$ finite étale.

Then $V = Y \times _ X U$ for some $Y \to X$ finite étale.

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