Lemma 58.24.1 (0EK9)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.24: Finite étale covers of punctured spectra, IV
Lemma 58.24.1 (cite)

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Lemma 58.24.1. In Situation 58.19.1 assume

$A$ has a dualizing complex and is $f$-adically complete,
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 $\geq 4$, or
2. if $\mathfrak p \not\in V(f)$ and $V(\mathfrak p) \cap V(f) \not= \{ \mathfrak m\} $, then $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$.
for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$.

Then the restriction functor $\textit{FÉt}_ U \to \textit{FÉt}_{U_0}$ is essentially surjective.

Proof. Let $V_0 \to U_0$ be a finite étale morphism. By Lemma 58.23.1 there exists an open $U' \subset U$ containing $U_0$ and a finite étale morphism $V' \to U$ whose base change to $U_0$ is isomorphic to $V_0 \to U_0$. Since $U' \supset U_0$ we see that $U \setminus U'$ consists of points corresponding to prime ideals $\mathfrak p_1, \ldots , \mathfrak p_ n$ as in (3). By assumption we can find finite étale morphisms $V'_ i \to \mathop{\mathrm{Spec}}(A_{\mathfrak p_ i})$ agreeing with $V' \to U'$ over $U' \times _ U \mathop{\mathrm{Spec}}(A_{\mathfrak p_ i})$. By Limits, Lemma 32.20.1 applied $n$ times we see that $V' \to U'$ extends to a finite étale morphism $V \to U$. $\square$

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