The Stacks project

Lemma 37.48.2. Let $S$ be a scheme which has an ample invertible sheaf. Let $\text{QP}_ S$ be the full subcategory of the category of schemes over $S$ satisfying the equivalent conditions of Lemma 37.48.1.

  1. if $S' \to S$ is a morphism of schemes and $S'$ has an ample invertible sheaf, then base change determines a functor $\text{QP}_ S \to \text{QP}_{S'}$,

  2. if $X \in \text{QP}_ S$ and $Y \in \text{QP}_ X$, then $Y \in \text{QP}_ S$,

  3. the category $\text{QP}_ S$ is closed under fibre products,

  4. the category $\text{QP}_ S$ is closed under finite disjoint unions,

  5. if $X \to S$ is projective, then $X \in \text{QP}_ S$,

  6. if $X \to S$ is quasi-affine of finite type, then $X$ is in $\text{QP}_ S$,

  7. if $X \to S$ is quasi-finite and separated, then $X \in \text{QP}_ S$,

  8. if $X \to S$ is a quasi-compact immersion, then $X \in \text{QP}_ S$,

  9. add more here.

Proof. Part (1) follows from Morphisms, Lemma 29.40.2.

Part (2) follows from the fourth characterization of Lemma 37.48.1.

If $X \to S$ and $Y \to S$ are quasi-projective, then $X \times _ S Y \to Y$ is quasi-projective by Morphisms, Lemma 29.40.2. Hence (3) follows from (2).

If $X = Y \amalg Z$ is a disjoint union of schemes and $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module such that $\mathcal{L}|_ Y$ and $\mathcal{L}|_ Z$ are ample, then $\mathcal{L}$ is ample (details omitted). Thus part (4) follows from the fourth characterization of Lemma 37.48.1.

Part (5) follows from Morphisms, Lemma 29.43.10.

Part (6) follows from Morphisms, Lemma 29.40.7.

Part (7) follows from part (6) and Lemma 37.42.2.

Part (8) follows from part (7) and Morphisms, Lemma 29.20.16. $\square$


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