Lemma 37.30.5. Let $f : X \to Y$ be a proper morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma 37.30.2. Then $n_{X/Y}$ is upper semi-continuous.

**Proof.**
Let $Z_ d = \{ x \in X \mid \dim _ x(X_{f(x)}) > d\} $. Then $Z_ d$ is a closed subset of $X$ by Morphisms, Lemma 29.28.4. Since $f$ is proper $f(Z_ d)$ is closed. Since $y \in f(Z_ d) \Leftrightarrow n_{X/Y}(y) > d$ we see that the lemma is true.
$\square$

## 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Your email address will not be published.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)