Lemma 76.31.4. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma 76.31.2. Then $n_{X/Y}$ is upper semi-continuous.
Proof. Let $Z_ d = \{ x \in |X| : \text{the fibre of }f\text{ at }x\text{ has dimension }> d\} $. Then $Z_ d$ is a closed subset of $|X|$ by Morphisms of Spaces, Lemma 67.34.4. Since $f$ is proper $f(Z_ d)$ is closed in $|Y|$. 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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)