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

1. $X \to S$ is quasi-projective,

2. $X \to S$ is H-quasi-projective,

3. there exists a quasi-compact open immersion $X \to X'$ of schemes over $S$ with $X' \to S$ projective,

4. $X \to S$ is of finite type and $X$ has an ample invertible sheaf, and

5. $X \to S$ 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.

Assume the equivalent conditions (1), (2), (3) hold. Choose an immersion $i : X \to \mathbf{P}^ n_ S$ over $S$. Let $\mathcal{L}$ be an ample invertible sheaf on $S$. To finish the proof we will show that $\mathcal{N} = f^*\mathcal{L} \otimes _{\mathcal{O}_ X} i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$ is ample on $X$. By Properties, Lemma 28.26.14 we reduce to the case $X = \mathbf{P}^ n_ S$. Let $s \in \Gamma (S, \mathcal{L}^{\otimes d})$ be a section such that the corresponding open $S_ s$ is affine. Say $S_ s = \mathop{\mathrm{Spec}}(A)$. Recall that $\mathbf{P}^ n_ S$ is the projective bundle associated to $\mathcal{O}_ S T_0 \oplus \ldots \oplus \mathcal{O}_ S T_ n$, see Constructions, Lemma 27.21.5 and its proof. Let $s_ i \in \Gamma (\mathbf{P}^ n_ S, \mathcal{O}(1))$ be the global section corresponding to the section $T_ i$ of $\mathcal{O}_ S T_0 \oplus \ldots \oplus \mathcal{O}_ S T_ n$. Then we see that $X_{f^*s \otimes s_ i^{\otimes n}}$ is affine because it is equal to $\mathop{\mathrm{Spec}}(A[T_0/T_ i, \ldots , T_ n/T_ i])$. This proves that $\mathcal{N}$ is ample by definition.

The equivalence of (1) and (5) follows from Morphisms, Lemmas 29.38.2 and 29.39.5. $\square$

Comment #7173 by Will Chen on

In the 4th paragraph of the proof, when you define $\mathcal{N}$, presumably the twisting sheaf should be over $\mathbb{P}^n_S$, not $\mathbb{P}^n_X$?

Comment #9454 by Branislav Sobot on

If one wishes, it is faster to conclude part (4) from part (5) via lemma 29.37.7

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).