Lemma 37.49.1. Let $S$ be a scheme which has an ample invertible sheaf. Let $f : X \to S$ be a morphism of schemes. The following are equivalent
$X \to S$ is quasi-projective,
$X \to S$ is H-quasi-projective,
there exists a quasi-compact open immersion $X \to X'$ of schemes over $S$ with $X' \to S$ projective,
$X \to S$ is of finite type and $X$ has an ample invertible sheaf, and
$X \to S$ is of finite type and there exists an $f$-very ample invertible sheaf.
Proof. The implication (2) $\Rightarrow $ (1) is Morphisms, Lemma 29.40.5. The implication (1) $\Rightarrow $ (2) is Morphisms, Lemma 29.43.16. The implication (2) $\Rightarrow $ (3) is Morphisms, Lemma 29.43.11
Assume $X \subset X'$ is as in (3). In particular $X \to S$ is of finite type. By Morphisms, Lemma 29.43.11 the morphism $X \to S$ is H-projective. Thus there exists a quasi-compact immersion $i : X \to \mathbf{P}^ n_ S$. Hence $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$ is $f$-very ample. As $X \to S$ is quasi-compact we conclude from Morphisms, Lemma 29.38.2 that $\mathcal{L}$ is $f$-ample. Thus $X \to S$ is quasi-projective by definition.
The implication (4) $\Rightarrow $ (2) is Morphisms, Lemma 29.39.3.
The implication (5) $\Rightarrow $ (4) follows from Morphisms, Lemma 29.37.7.
The equivalence of (1) and (5) follows from Morphisms, Lemmas 29.38.2 and 29.39.5. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (4)
Comment #7173 by Will Chen on
Comment #7310 by Johan on
Comment #9454 by Branislav Sobot on
Comment #10204 by Stacks project on