The Stacks project

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

  1. If $f$ is locally of finite presentation and generalizations lift along $f$, then $f$ is open.

  2. 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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