The Stacks project

Lemma 66.40.8. Let $S$ be a scheme. Let

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

be a commutative diagram of morphism of algebraic spaces over $S$. Assume

  1. $X \to B$ is a proper morphism,

  2. $Y \to B$ is separated and locally of finite type,

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

Proof. The scheme theoretic image of $h$ is constructed in Section 66.16. Observe that $h$ is quasi-compact (Lemma 66.8.10) hence $|h|(|X|) \subset |Z|$ is dense (Lemma 66.16.3). On the other hand $|h|(|X|)$ is closed in $|Y|$ (Lemma 66.40.6) hence $X \to Z$ is surjective. Thus $Z \to B$ is a proper (Lemma 66.40.7). $\square$

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