The Stacks project

Lemma 33.38.5. Let $X$ be a quasi-compact scheme. If for every $x \in X$ there exists a pair $(\mathcal{L}, s)$ consisting of a globally generated invertible sheaf $\mathcal{L}$ and a global section $s$ such that $x \in X_ s$ and $X_ s$ is affine, then $X$ has an ample invertible sheaf.

Proof. Since $X$ is quasi-compact we can find a finite collection $(\mathcal{L}_ i, s_ i)$, $i = 1, \ldots , n$ of pairs such that $\mathcal{L}_ i$ is globally generated, $X_{s_ i}$ is affine and $X = \bigcup X_{s_ i}$. Again because $X$ is quasi-compact we can find, for each $i$, a finite collection of sections $t_{i, j}$ of $\mathcal{L}_ i$, $j = 1, \ldots , m_ i$ such that $X = \bigcup X_{t_{i, j}}$. Set $t_{i, 0} = s_ i$. Consider the invertible sheaf

\[ \mathcal{L} = \mathcal{L}_1 \otimes _{\mathcal{O}_ X} \ldots \otimes _{\mathcal{O}_ X} \mathcal{L}_ n \]

and the global sections

\[ \tau _ J = t_{1, j_1} \otimes \ldots \otimes t_{n, j_ n} \]

By Properties, Lemma 28.26.4 the open $X_{\tau _ J}$ is affine as soon as $j_ i = 0$ for some $i$. It is a simple matter to see that these opens cover $X$. Hence $\mathcal{L}$ is ample by definition. $\square$


Comments (1)

Comment #7420 by Laurent Moret-Bailly on

Beginning of proof: I would specify that the 's
are assumed globally generated.

In the second sentence, it should be "sections of ".


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Your email address will not be published.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