Lemma 29.25.10. A flat morphism locally of finite presentation is universally open.

**Proof.**
This follows from Lemmas 29.25.9 and Lemma 29.23.2 above. We can also argue directly as follows.

Let $f : X \to S$ be flat and locally of finite presentation. By Lemmas 29.25.8 and 29.21.4 any base change of $f$ is flat and locally of finite presentation. Hence it suffices to show $f$ is open. To show $f$ is open it suffices to show that we may cover $X$ by open affines $X = \bigcup U_ i$ such that $U_ i \to S$ is open. We may cover $X$ by affine opens $U_ i \subset X$ such that each $U_ i$ maps into an affine open $V_ i \subset S$ and such that the induced ring map $\mathcal{O}_ S(V_ i) \to \mathcal{O}_ X(U_ i)$ is flat and of finite presentation (Lemmas 29.25.3 and 29.21.2). Then $U_ i \to V_ i$ is open by Algebra, Proposition 10.41.8 and the proof is complete. $\square$

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