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

and the global sections

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. $\square$

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## Comments (2)

Comment #7420 by Laurent Moret-Bailly on

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