Lemma 75.7.5. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces which are locally of finite type over $B$.

1. If $Y$ is separated over $B$ and $T \subset |X|$ is a closed subset proper over $B$, then $|f|(T)$ is a closed subset of $|Y|$ proper over $B$.

2. If $f$ is universally closed and $T \subset |X|$ is a closed subset proper over $B$, then $|f|(T)$ is a closed subset of $Y$ proper over $B$.

3. If $f$ is proper and $T \subset |Y|$ is a closed subset proper over $B$, then $|f|^{-1}(T)$ is a closed subset of $|X|$ proper over $B$.

Proof. Proof of (1). Assume $Y$ is separated over $B$ and $T \subset |X|$ is a closed subset proper over $B$. Let $Z$ be the reduced induced closed subspace structure on $T$ and apply Morphisms of Spaces, Lemma 67.40.8 to $Z \to Y$ over $B$ to conclude.

Proof of (2). Assume $f$ is universally closed and $T \subset |X|$ is a closed subset proper over $B$. Let $Z$ be the reduced induced closed subspace structure on $T$ and let $Z'$ be the reduced induced closed subspace structure on $|f|(T)$. We obtain an induced morphism $Z \to Z'$. Denote $Z'' = f^{-1}(Z')$ the scheme theoretic inverse image. Then $Z'' \to Z'$ is universally closed as a base change of $f$ (Morphisms of Spaces, Lemma 67.40.3). Hence $Z \to Z'$ is universally closed as a composition of the closed immersion $Z \to Z''$ and $Z'' \to Z'$ (Morphisms of Spaces, Lemmas 67.40.5 and 67.40.4). We conclude that $Z' \to B$ is separated by Morphisms of Spaces, Lemma 67.9.8. Since $Z \to B$ is quasi-compact and $Z \to Z'$ is surjective we see that $Z' \to B$ is quasi-compact. Since $Z' \to B$ is the composition of $Z' \to Y$ and $Y \to B$ we see that $Z' \to B$ is locally of finite type (Morphisms of Spaces, Lemmas 67.23.7 and 67.23.2). Finally, since $Z \to B$ is universally closed, we see that the same thing is true for $Z' \to B$ by Morphisms of Spaces, Lemma 67.40.7. This finishes the proof.

Proof of (3). Assume $f$ is proper and $T \subset |Y|$ is a closed subset proper over $B$. Let $Z$ be the reduced induced closed subspace structure on $T$. Denote $Z' = f^{-1}(Z)$ the scheme theoretic inverse image. Then $Z' \to Z$ is proper as a base change of $f$ (Morphisms of Spaces, Lemma 67.40.3). Whence $Z' \to B$ is proper as the composition of $Z' \to Z$ and $Z \to B$ (Morphisms of Spaces, Lemma 67.40.4). This finishes the proof. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).