The Stacks project

Lemma 33.25.6. Let $k$ be a field. If $X$ is smooth over $\mathop{\mathrm{Spec}}(k)$ then the set

\[ \{ x \in X\text{ closed such that }k \subset \kappa (x) \text{ is finite separable}\} \]

is dense in $X$.

Proof. It suffices to show that given a nonempty smooth $X$ over $k$ there exists at least one closed point whose residue field is finite separable over $k$. To see this, choose a diagram

\[ \xymatrix{ X & U \ar[l] \ar[r]^-\pi & \mathbf{A}^ d_ k } \]

with $\pi $ ├ętale, see Morphisms, Lemma 29.36.20. The morphism $\pi : U \to \mathbf{A}^ d_ k$ is open, see Morphisms, Lemma 29.36.13. By Lemma 33.25.5 we may choose a closed point $w \in \pi (U)$ whose residue field is finite separable over $k$. Pick any $x \in U$ with $\pi (x) = w$. By Morphisms, Lemma 29.36.7 the field extension $\kappa (x)/\kappa (w)$ is finite separable. Hence $\kappa (x)/k$ is finite separable. The point $x$ is a closed point of $X$ by Morphisms, Lemma 29.20.2. $\square$

Comments (6)

Comment #1393 by Stulemeijer Thierry on

A very small typo : replace with , and with .

Comment #3949 by Rogozhin on

is just a Zariski covering of , but still it should probably say that you take the image of in at the end.

Comment #3955 by on

Well, I really think this is fine as is. If more people complain I will change it.

Comment #3961 by Laurent Moret-Bailly on

And of course should be nonempty.

Comment #4096 by on

OK, I still think this is fine as is because the lemma that is being used will produce a nonempty U.

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 056U. Beware of the difference between the letter 'O' and the digit '0'.