Lemma 33.38.5 (09NC)—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 (2)

Comment #7420 by Laurent Moret-Bailly on June 11, 2022 at 23:36

Beginning of proof: I would specify that the $\mathcal{L}_{i}$ 's
are assumed globally generated.

In the second sentence, it should be "sections $t_{i,j}$ of $\mathcal{L}_i$ ".

Comment #7589 by Stacks Project on August 12, 2022 at 15:38

Thanks, this was fixed a while back here.

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