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.

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

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

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

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

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

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

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

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

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