Lemma 29.37.3. Let $f : X \to S$ be a morphism of schemes. If there exists an $f$-ample invertible sheaf, then $f$ is separated.
Proof. Being separated is local on the base (see Schemes, Lemma 26.21.7 for example; it also follows easily from the definition). Hence we may assume $S$ is affine and $X$ has an ample invertible sheaf. In this case the result follows from Properties, Lemma 28.26.8. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.