The Stacks project

Lemma 75.31.3. Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of finite presentation of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma 75.31.2. Then $n_{X/Y}$ is lower semi-continuous.

Proof. Let $V \to Y$ be a surjective ├ętale morphism where $V$ is a scheme. If we can show that the composition $n_{X/Y} \circ |g|$ is lower semi-continuous, then the lemma follows as $|g|$ is open. Hence we may assume $Y$ is a scheme. Working locally we may assume $V$ is an affine scheme. Then we can choose an affine scheme $U$ and a surjective ├ętale morphism $U \to X$. Then $n_{X/Y} = n_{U/Y}$. Hence we may assume $X$ and $Y$ are both schemes. In this case the lemma follows from More on Morphisms, Lemma 37.30.4. $\square$


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