Lemma 31.4.15. In Situation 31.4.5 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 27.26.1. By Lemma 31.4.7 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 31.4.13 we may, after increasing $i$, assume that $(S_ i)_{s_{i, j}}$ is affine for $j = 1, \ldots , m$. By Lemma 31.4.11 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. $\square$

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