Lemma 37.50.2 (0B46)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.50: Projective schemes
Lemma 37.50.2 (cite)

Lemma 37.50.2. Let $S$ be a scheme which has an ample invertible sheaf. Let $\text{P}_ S$ be the full subcategory of the category of schemes over $S$ satisfying the equivalent conditions of Lemma 37.50.1.

if $S' \to S$ is a morphism of schemes and $S'$ has an ample invertible sheaf, then base change determines a functor $\text{P}_ S \to \text{P}_{S'}$,
if $X \in \text{P}_ S$ and $Y \in \text{P}_ X$, then $Y \in \text{P}_ S$,
the category $\text{P}_ S$ is closed under fibre products,
the category $\text{P}_ S$ is closed under finite disjoint unions,
if $X \to S$ is finite, then $X$ is in $\text{P}_ S$,
add more here.

Proof. Part (1) follows from Morphisms, Lemma 29.43.9.

Part (2) follows from the fifth characterization of Lemma 37.50.1 and the fact that compositions of proper morphisms are proper (Morphisms, Lemma 29.41.4).

If $X \to S$ and $Y \to S$ are projective, then $X \times _ S Y \to Y$ is projective by Morphisms, Lemma 29.43.9. 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 fifth characterization of Lemma 37.50.1.

Part (5) follows from Morphisms, Lemma 29.44.16. $\square$

The Stacks project

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