Lemma 29.28.5. Let $f : X \to Y$ be a morphism of finite type with $Y$ quasi-compact. Then the dimension of the fibres of $f$ is bounded.
Proof. By Lemma 29.28.4 the set $U_ n \subset X$ of points where the dimension of the fibre is $\leq n$ is open. Since $f$ is of finite type, every point is contained in some $U_ n$ (because the dimension of a finite type algebra over a field is finite). Since $Y$ is quasi-compact and $f$ is of finite type, we see that $X$ is quasi-compact. Hence $X = U_ n$ for some $n$. $\square$
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