Lemma 58.22.3 (0BLP)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.22: Finite étale covers of punctured spectra, II
Lemma 58.22.3 (cite)

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Lemma 58.22.3. In Situation 58.19.1 assume

$A$ is $f$ -adically complete,
$f$ is a nonzerodivisor.
$H^1_\mathfrak m(A/fA)$ is a finite $A$ -module.

Then the restriction functor

$\mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \textit{FÉt}_{U'} \longrightarrow \textit{FÉt}_{U_0}$

is fully faithful.

Proof. Follows from Lemma 58.17.3 and Algebraic and Formal Geometry, Lemma 52.15.8. $\square$

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