Lemma 76.31.4 (0D4Q)—The Stacks project

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Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.31: Dimension of fibres
Lemma 76.31.4 (cite)

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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. $\square$

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