Lemma 29.41.10 (0AH6)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.41: Proper morphisms
Lemma 29.41.10 (cite)

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Lemma 29.41.10. Suppose given a commutative diagram of schemes

$\xymatrix{ X \ar[rr]_ h \ar[rd]_ f & & Y \ar[ld]^ g \\ & S }$

Assume

$X \to S$ is a universally closed (for example proper) morphism, and
$Y \to S$ is separated and locally of finite type.

Then the scheme theoretic image $Z \subset Y$ of $h$ is proper over $S$ and $X \to Z$ is surjective.

Proof. The scheme theoretic image of $h$ is constructed in Section 29.6. Since $f$ is quasi-compact (Lemma 29.41.8) we find that $h$ is quasi-compact (Schemes, Lemma 26.21.14). Hence $h(X) \subset Z$ is dense (Lemma 29.6.3). On the other hand $h(X)$ is closed in $Y$ (Lemma 29.41.7) hence $X \to Z$ is surjective. Thus $Z \to S$ is a proper (Lemma 29.41.9). $\square$

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