Lemma 29.39.5 (01VU)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.39: Ample and very ample sheaves relative to finite type morphisms
Lemma 29.39.5 (cite)

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Lemma 29.39.5. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. Assume $S$ quasi-compact and $f$ of finite type. The following are equivalent

$\mathcal{L}$ is $f$ -ample,
$\mathcal{L}^{\otimes d}$ is $f$ -very ample for some $d \geq 1$ ,
$\mathcal{L}^{\otimes d}$ is $f$ -very ample for all $d \gg 1$ .

Proof. Trivially (3) implies (2). Lemma 29.38.2 guarantees that (2) implies (1) since a morphism of finite type is quasi-compact by definition. Assume that $\mathcal{L}$ is $f$ -ample. Choose a finite affine open covering $S = V_1 \cup \ldots \cup V_ m$ . Write $X_ i = f^{-1}(V_ i)$ . By Lemma 29.39.4 above we see there exists a $d_0$ such that $\mathcal{L}^{\otimes d}$ is relatively very ample on $X_ i/V_ i$ for all $d \geq d_0$ . Hence we conclude (1) implies (3) by Lemma 29.38.7. $\square$

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