Lemma 58.9.3. Let $k'/k$ be an extension of algebraically closed fields. Let $X$ be a proper scheme over $k$. Then the functor

is an equivalence of categories between schemes finite étale over $X$ and schemes finite étale over $X_{k'}$.

Lemma 58.9.3. Let $k'/k$ be an extension of algebraically closed fields. Let $X$ be a proper scheme over $k$. Then the functor

\[ U \longmapsto U_{k'} \]

is an equivalence of categories between schemes finite étale over $X$ and schemes finite étale over $X_{k'}$.

**Proof.**
Let us prove the functor is essentially surjective. Let $U' \to X_{k'}$ be a finite étale morphism. Write $k' = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of finite type $k$-algebras. By Limits, Lemma 32.10.1 there exists an $i$ and a finitely presented morphism $U_ i \to X_{A_ i}$ whose base change to $X_{k'}$ is $U'$. After increasing $i$ we may assume that $U_ i \to X_{A_ i}$ is finite and étale (Limits, Lemmas 32.8.3 and 32.8.10). Since $k$ is algebraically closed we can find a $k$-valued point $t$ in $\mathop{\mathrm{Spec}}(A_ i)$. Let $U = (U_ i)_ t$ be the fibre of $U_ i$ over $t$. Let $A_ i^ h$ be the henselization of $(A_ i)_{\mathfrak m}$ where $\mathfrak m$ is the maximal ideal corresponding to the point $t$. By Lemma 58.9.1 we see that $(U_ i)_{A_ i^ h} = U \times \mathop{\mathrm{Spec}}(A_ i^ h)$ as schemes over $X_{A_ i^ h}$. Now since $A_ i^ h$ is algebraic over $A_ i$ (see for example discussion in Smoothing Ring Maps, Example 16.13.3) and since $k'$ is algebraically closed we can find a ring map $A_ i^ h \to k'$ extending the given inclusion $A_ i \subset k'$. Hence we conclude that $U'$ is isomorphic to the base change of $U$. The proof of fully faithfulness is exactly the same.
$\square$

Your email address will not be published.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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)