The Stacks project

Lemma 29.38.6. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. If $\mathcal{L}$ is relatively very ample on $X/S$ then $f$ is separated.

Proof. Being separated is local on the base (see Schemes, Section 26.21). An immersion is separated (see Schemes, Lemma 26.23.8). Hence the lemma follows since locally $X$ has an immersion into the homogeneous spectrum of a graded ring which is separated, see Constructions, Lemma 27.8.8. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 29.38: Very ample sheaves

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Your email address will not be published.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 01VQ. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 29.38.6 as pdf