Lemma 29.39.7 (02NR)—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 29.39.7 (cite)

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Lemma 29.39.7. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$ . Assume $f$ is of finite type. The following are equivalent:

$\mathcal{L}$ is $f$ -relatively ample, and
there exist an open covering $S = \bigcup V_ j$ , for each $j$ an integers $d_ j \geq 1$ , $n_ j \geq 0$ , and immersions

$i_ j : X_ j = f^{-1}(V_ j) = V_ j \times _ S X \longrightarrow \mathbf{P}^{n_ j}_{V_ j}$

over $V_ j$ such that $\mathcal{L}^{\otimes d_ j}|_{X_ j} \cong i_ j^*\mathcal{O}_{\mathbf{P}^{n_ j}_{V_ j}}(1)$ .

Proof. We see that (1) implies (2) by taking an affine open covering of $S$ and applying Lemma 29.39.4 to each of the restrictions of $f$ and $\mathcal{L}$ . We see that (2) implies (1) by Lemma 29.37.4. $\square$

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