The Stacks project

Lemma 75.7.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T_ i \subset |X|$, $i = 1, \ldots , n$ be closed subsets. If $T_ i$, $i = 1, \ldots , n$ are proper over $Y$, then the same is true for $T_1 \cup \ldots \cup T_ n$.

Proof. Let $Z_ i$ be the reduced induced closed subscheme structure on $T_ i$. The morphism

\[ Z_1 \amalg \ldots \amalg Z_ n \longrightarrow X \]

is finite by Morphisms of Spaces, Lemmas 67.45.10 and 67.45.11. As finite morphisms are universally closed (Morphisms of Spaces, Lemma 67.45.9) and since $Z_1 \amalg \ldots \amalg Z_ n$ is proper over $S$ we conclude by Lemma 75.7.5 part (2) that the image $Z_1 \cup \ldots \cup Z_ n$ is proper over $S$. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CZH. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 75.7.6 as pdf