# The Stacks Project

## Tag 09ZS

Lemma 49.7.2. 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 15.8.12. $\square$

The code snippet corresponding to this tag is a part of the file pione.tex and is located in lines 1406–1415 (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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