The Stacks project

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. $\square$

Comments (2)

Comment #6456 by Jonas Ehrhard on

For the universality of the direct proof we use that being flat and being locally of finite presentation are both preserved under base change, right? I.e. 01U9 and01TS.

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  • 4 comment(s) on Section 29.25: Flat morphisms

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