Proposition 32.37.12. Let $X$ be a Noetherian separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf.

**Proof.**
Let $Z \subset X$ be the reduction of $X$. By Lemma 32.37.10 the scheme $Z$ has an ample invertible sheaf. Thus by Lemma 32.37.11 there exists an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ on $X$ whose restriction to $Z$ is ample. Then $\mathcal{L}$ is ample by an application of Cohomology of Schemes, Lemma 29.17.5.
$\square$

## 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Your email address will not be published.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)