# The Stacks Project

## Tag 09ZS

Lemma 44.74.11. Let $(A, I)$ be a henselian pair. Set $X = \mathop{\rm Spec}(A)$ and $Z = \mathop{\rm Spec}(A/I)$. The functor $$U \longmapsto U \times_X Z$$ is an equivalence of categories between finite étale schemes over $X$ and finite étale schemes over $Z$.

Proof. This is a translation of More on Algebra, Lemma 15.7.11. $\square$

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 11728–11737 (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
$$U \longmapsto U \times_X Z$$
is an equivalence of categories between finite \'etale schemes over $X$
and finite \'etale schemes over $Z$.
\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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