The Stacks project

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. $\square$