Lemma 58.19.2 (0BLG)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.19: Finite étale covers of punctured spectra, I
Lemma 58.19.2 (cite)

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Lemma 58.19.2. In Situation 58.19.1. Assume one of the following holds

$\dim (A/\mathfrak p) \geq 2$ for every minimal prime $\mathfrak p \subset A$ with $f \not\in \mathfrak p$ , or
every connected component of $U$ meets $U_0$ .

Then

$\textit{FÉt}_ U \longrightarrow \textit{FÉt}_{U_0},\quad V \longmapsto V_0 = V \times _ U U_0$

is a faithful functor.

Proof. Case (2) is immediate from Lemma 58.17.5. Assumption (1) implies every irreducible component of $U$ meets $U_0$ , see Algebra, Lemma 10.60.13. Hence (1) follows from (2). $\square$

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