Lemma 71.16.2. Let $k$ be a field. Let $X$ be an algebraic space smooth over $\mathop{\mathrm{Spec}}(k)$. The set of $x \in |X|$ which are image of morphisms $\mathop{\mathrm{Spec}}(k') \to X$ with $k' \supset k$ finite separable is dense in $|X|$.

**Proof.**
Choose a scheme $U$ and a surjective étale morphism $U \to X$. The morphism $U \to \mathop{\mathrm{Spec}}(k)$ is smooth as a composition of an étale (hence smooth) morphism and a smooth morphism (see Morphisms of Spaces, Lemmas 66.39.6 and 66.37.2). Hence we can apply Varieties, Lemma 33.25.6 to see that the closed points of $U$ whose residue fields are finite separable over $k$ are dense. This implies the lemma by our definition of the topology on $|X|$.
$\square$

## Post a comment

Your email address will not be published.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)