Lemma 58.8.2 (09ZS)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.8: Topological invariance of the fundamental group
Lemma 58.8.2 (cite)

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Lemma 58.8.2. Let $(A, I)$ be a henselian pair. Set $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/I)$ . The functor

$\textit{FÉt}_ X \longrightarrow \textit{FÉt}_ Z,\quad U \longmapsto U \times _ X Z$

is an equivalence of categories.

Proof. This is a translation of More on Algebra, Lemma 15.13.2. $\square$

Comments (2)

Comment #3036 by Brian Lawrence on December 13, 2017 at 20:45

Suggested slogan: Finite etale covers lift uniquely to henselian rings.

Comment #3145 by Johan on February 01, 2018 at 20:58

Need a pithier slogan.

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