The Stacks project

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. $\square$

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