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:

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

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

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

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