Lemma 29.37.9 (0893)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.37: Relatively ample sheaves
Lemma 29.37.9 (cite)

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Lemma 29.37.9. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. Let $S' \to S$ be a morphism of schemes. Let $f' : X' \to S'$ be the base change of $f$ and denote $\mathcal{L}'$ the pullback of $\mathcal{L}$ to $X'$ . If $\mathcal{L}$ is $f$ -ample, then $\mathcal{L}'$ is $f'$ -ample.

Proof. By Lemma 29.37.4 it suffices to find an affine open covering $S' = \bigcup U'_ i$ such that $\mathcal{L}'$ restricts to an ample invertible sheaf on $(f')^{-1}(U_ i')$ for all $i$ . We may choose $U'_ i$ mapping into an affine open $U_ i \subset S$ . In this case the morphism $(f')^{-1}(U'_ i) \to f^{-1}(U_ i)$ is affine as a base change of the affine morphism $U'_ i \to U_ i$ (Lemma 29.11.8). Thus $\mathcal{L}'|_{(f')^{-1}(U'_ i)}$ is ample by Lemma 29.37.7. $\square$

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