Lemma 29.37.8 (0C4K)—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 29.37.8 (cite)

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Lemma 29.37.8. Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. Let $\mathcal{M}$ be an invertible $\mathcal{O}_ Y$ -module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. If $S$ is quasi-compact, $\mathcal{M}$ is $g$ -ample, and $\mathcal{L}$ is $f$ -ample, then $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is $g \circ f$ -ample for $a \gg 0$ .

Proof. Let $S = \bigcup _{i = 1, \ldots , n} V_ i$ be a finite affine open covering. By Lemma 29.37.4 it suffices to prove that $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample on $(g \circ f)^{-1}(V_ i)$ for $i = 1, \ldots , n$ . Thus the lemma follows from Lemma 29.37.7. $\square$

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