Lemma 67.40.8 (0AGD)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.40: Proper morphisms
Lemma 67.40.8 (cite)

numberstags

Lemma 67.40.8. 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

$X \to B$ is a proper morphism,
$Y \to B$ 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 67.16. Observe that $h$ is quasi-compact (Lemma 67.8.10) hence $|h|(|X|) \subset |Z|$ is dense (Lemma 67.16.3). On the other hand $|h|(|X|)$ is closed in $|Y|$ (Lemma 67.40.6) hence $X \to Z$ is surjective. Thus $Z \to B$ is a proper (Lemma 67.40.7). $\square$

Comments (0)

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).

Comments (0)

Post a comment