Lemma 29.40.6 (01VZ)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.40: Quasi-projective morphisms
Lemma 29.40.6 (cite)

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Lemma 29.40.6. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

The morphism $f$ is locally quasi-projective.
There exists an open covering $S = \bigcup V_ j$ such that each $f^{-1}(V_ j) \to V_ j$ is H-quasi-projective.

Proof. By Lemma 29.40.5 we see that (2) implies (1). Assume (1). The question is local on $S$ and hence we may assume $S$ is affine, $X$ of finite type over $S$ and $\mathcal{L}$ is a relatively ample invertible sheaf on $X/S$ . By Lemma 29.39.4 we may assume $\mathcal{L}$ is ample on $X$ . By Lemma 29.39.3 we see that there exists an immersion of $X$ into a projective space over $S$ , i.e., $X$ is H-quasi-projective over $S$ as desired. $\square$

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