The Stacks project

Lemma 67.4.13. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$.

1. The morphism $f$ is locally separated.

2. The morphism $f$ is (quasi-)separated in the sense of Definition 67.4.2 above if and only if $f$ is (quasi-)separated in the sense of Section 67.3.

In particular, if $f : X \to Y$ is a morphism of schemes over $S$, then $f$ is (quasi-)separated in the sense of Definition 67.4.2 if and only if $f$ is (quasi-)separated as a morphism of schemes.