Lemma 58.20.7 (0BLM)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.20: Purity in local case, I
Lemma 58.20.7 (cite)

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Lemma 58.20.7. In Situation 58.19.1. Let $V$ be finite étale over $U$. Assume

$A$ has depth $\geq 3$,
$V_0 = V \times _ U U_0$ 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.

Proof. The assumption of depth forces $H^1_\mathfrak m(A) = H^2_\mathfrak m(A) = 0$, see Dualizing Complexes, Lemma 47.11.1. Hence Lemma 58.20.6 applies. $\square$

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