The Stacks project

Lemma 28.41.16. Let $S$ be a scheme which admits an ample invertible sheaf. Then

  1. any projective morphism $X \to S$ is H-projective, and

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

Proof. The assumptions on $S$ imply that $S$ is quasi-compact and separated, see Properties, Definition 27.26.1 and Lemma 27.26.11 and Constructions, Lemma 26.8.8. Hence Lemma 28.41.12 applies and we see that (1) implies (2). Let $\mathcal{E}$ be a finite type quasi-coherent $\mathcal{O}_ S$-module. By our definition of projective morphisms it suffices to show that $\mathbf{P}(\mathcal{E}) \to S$ is H-projective. If $\mathcal{E}$ is generated by finitely many global sections, then the corresponding surjection $\mathcal{O}_ S^{\oplus n} \to \mathcal{E}$ induces a closed immersion

\[ \mathbf{P}(\mathcal{E}) \longrightarrow \mathbf{P}(\mathcal{O}_ S^{\oplus n}) = \mathbf{P}^ n_ S \]

as desired. In general, let $\mathcal{L}$ be an invertible sheaf on $S$. By Properties, Proposition 27.26.13 there exists an integer $n$ such that $\mathcal{E} \otimes _{\mathcal{O}_ S} \mathcal{L}^{\otimes n}$ is globally generated by finitely many sections. Since $\mathbf{P}(\mathcal{E}) = \mathbf{P}(\mathcal{E} \otimes _{\mathcal{O}_ S} \mathcal{L}^{\otimes n})$ by Constructions, Lemma 26.20.1 this finishes the proof. $\square$


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