The Stacks project

Lemma 28.24.9. A flat morphism locally of finite presentation is universally open.

Proof. This follows from Lemmas 28.24.8 and Lemma 28.22.2 above. We can also argue directly as follows.

Let $f : X \to S$ be flat locally of finite presentation. 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. By definition 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 of finite presentation. Thus $U_ i \to V_ i$ is open by Algebra, Proposition 10.40.8. The lemma follows. $\square$


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