The Stacks project

Lemma 32.4.15. In Situation 32.4.5 let $\mathcal{L}_0$ be an invertible sheaf of modules on $S_0$. If the pullback $\mathcal{L}$ to $S$ is ample, then for some $i \in I$ the pullback $\mathcal{L}_ i$ to $S_ i$ is ample.

Proof. The assumption means there are finitely many sections $s_1, \ldots , s_ m \in \Gamma (S, \mathcal{L})$ such that $S_{s_ j}$ is affine and such that $S = \bigcup S_{s_ j}$, see Properties, Definition 28.26.1. By Lemma 32.4.7 we can find an $i \in I$ and sections $s_{i, j} \in \Gamma (S_ i, \mathcal{L}_ i)$ mapping to $s_ j$. By Lemma 32.4.13 we may, after increasing $i$, assume that $(S_ i)_{s_{i, j}}$ is affine for $j = 1, \ldots , m$. By Lemma 32.4.11 we may, after increasing $i$ a last time, assume that $S_ i = \bigcup (S_ i)_{s_{i, j}}$. Then $\mathcal{L}_ i$ is ample by definition. $\square$

Comments (0)

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Your email address will not be published.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 09MT. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 32.4.15 as pdf