Lemma 29.23.2 (01U1)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.23: Open morphisms
Lemma 29.23.2 (cite)

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Lemma 29.23.2. Let $f : X \to S$ be a morphism.

If $f$ is locally of finite presentation and generalizations lift along $f$ , then $f$ is open.
If $f$ is locally of finite presentation and generalizations lift along every base change of $f$ , then $f$ is universally open.

Proof. It suffices to prove the first assertion. This reduces to the case where both $X$ and $S$ are affine. In this case the result follows from Algebra, Lemma 10.41.3 and Proposition 10.41.8. $\square$

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