## Tag `09ZS`

Chapter 47: Étale Cohomology > Section 47.73: Affine analog of proper base change

Lemma 47.73.11. 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 15.8.12. $\square$

The code snippet corresponding to this tag is a part of the file `etale-cohomology.tex` and is located in lines 11670–11679 (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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