The Stacks project

Lemma 26.20.2. Let $f : X \to S$ be a morphism of schemes.

  1. If $f$ is universally closed then specializations lift along any base change of $f$, see Topology, Definition 5.19.4.

  2. If $f$ is quasi-compact and specializations lift along any base change of $f$, then $f$ is universally closed.

Proof. Part (1) is a direct consequence of Topology, Lemma 5.19.7. Part (2) follows from Lemmas 26.19.8 and 26.19.3. $\square$


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