## Tag `09ZS`

Chapter 52: Fundamental Groups of Schemes > Section 52.7: Topological invariance of the fundamental group

Lemma 52.7.2. Let $(A, I)$ be a henselian pair. Set $X = \mathop{\rm Spec}(A)$ and $Z = \mathop{\rm Spec}(A/I)$. 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 15.11.2. $\square$

The code snippet corresponding to this tag is a part of the file `pione.tex` and is located in lines 1432–1441 (see updates for more information).

```
\begin{lemma}
\label{lemma-gabber}
Let $(A, I)$ be a henselian pair. Set $X = \Spec(A)$ and $Z = \Spec(A/I)$.
The functor
$$
\textit{F\'Et}_X \longrightarrow \textit{F\'Et}_Z,\quad
U \longmapsto U \times_X Z
$$
is an equivalence of categories.
\end{lemma}
\begin{proof}
This is a translation of
More on Algebra, Lemma \ref{more-algebra-lemma-finite-etale-equivalence}.
\end{proof}
```

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