The Stacks project

Lemma 66.30.6. A flat morphism locally of finite presentation is universally open.

Proof. Let $f : X \to Y$ be a flat morphism locally of finite presentation of algebraic spaces over $S$. Choose a diagram

\[ \xymatrix{ U \ar[r]_\alpha \ar[d] & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are surjective and ├ętale, see Spaces, Lemma 64.11.6. By Lemmas 66.30.5 and 66.28.4 the morphism $\alpha $ is flat and locally of finite presentation. Hence by Morphisms, Lemma 29.25.10 we see that $\alpha $ is universally open. Hence $X \to Y$ is universally open according to Lemma 66.6.5. $\square$

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