Lemma 58.20.5 (0EK6)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.20: Purity in local case, I
Lemma 58.20.5 (cite)

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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
1. $A_ f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 3$ , or
2. 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.

Proof. We reduce to the complete case using Lemma 58.20.4. (The assumptions carry over; see proof of Lemma 58.19.5.)

In the complete case we can lift $Y_0 \to X_0$ to a finite étale morphism $Y \to X$ by More on Algebra, Lemma 15.13.2; observe that $(A, fA)$ is a henselian pair by More on Algebra, Lemma 15.11.4. Then we can use Lemma 58.19.5 to see that $V$ is isomorphic to $Y \times _ X U$ and the proof is complete. $\square$

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