## Tag `09ZS`

Chapter 45: Étale Cohomology > Section 45.74: Affine analog of proper base change

Lemma 45.74.11. 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 15.7.12. $\square$

The code snippet corresponding to this tag is a part of the file `etale-cohomology.tex` and is located in lines 11762–11771 (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}
```

## Comments (0)

## Add a comment on tag `09ZS`

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.

There are no comments yet for this tag.