The Stacks project

Lemma 41.20.3. Let $f : X \to S$ be a morphism of schemes. In the following cases the functor (41.20.0.1) is fully faithful:

  1. $f$ is surjective and universally closed (e.g., finite, integral, or proper),

  2. $f$ is surjective and universally open (e.g., locally of finite presentation and flat, smooth, or etale),

  3. $f$ is surjective, quasi-compact, and flat.

Proof. This follows from Lemma 41.20.2. For example a closed surjective map of topological spaces is submersive (Topology, Lemma 5.6.5). Finite, integral, and proper morphisms are universally closed, see Morphisms, Lemmas 29.44.7 and 29.44.11 and Definition 29.41.1. On the other hand an open surjective map of topological spaces is submersive (Topology, Lemma 5.6.4). Flat locally finitely presented, smooth, and étale morphisms are universally open, see Morphisms, Lemmas 29.25.10, 29.34.10, and 29.36.13. The case of surjective, quasi-compact, flat morphisms follows from Morphisms, Lemma 29.25.12. $\square$

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