Lemma 35.25.5. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. Let $Z$ be a closed subset of $X$. If there exists an fpqc covering $\{ Y_ i \to Y\} $ such that the inverse image $Z_ i \subset Y_ i \times _ Y X$ is proper over $Y_ i$ (Cohomology of Schemes, Definition 30.26.2) then $Z$ is proper over $Y$.

**Proof.**
Endow $Z$ with the reduced induced closed subscheme structure, see Schemes, Definition 26.12.5. For every $i$ the base change $Y_ i \times _ Y Z$ is a closed subscheme of $Y_ i \times _ Y X$ whose underlying closed subset is $Z_ i$. By definition (via Cohomology of Schemes, Lemma 30.26.1) we conclude that the projections $Y_ i \times _ Y Z \to Y_ i$ are proper morphisms. Hence $Z \to Y$ is a proper morphism by Lemma 35.23.14. Thus $Z$ is proper over $Y$ by definition.
$\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: