The Stacks project

Lemma 37.29.4. Let $f : X \to Y$ be a flat morphism of schemes of finite presentation. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma 37.29.2. Then $n_{X/Y}$ is lower semi-continuous.

Proof. Let $W \subset X$, $W = \coprod _{d \geq 0} U_ d$ be the open constructed in Lemmas 37.21.7 and 37.21.9. Let $y \in Y$ be a point. If $n_{X/Y}(y) = \dim (X_ y) = n$, then $y$ is in the image of $U_ n \to Y$. By Morphisms, Lemma 29.25.10 we see that $f(U_ n)$ is open in $Y$. Hence there is an open neighbourhoof of $y$ where $n_{X/Y}$ is $\geq n$. $\square$


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