The Stacks project

Lemma 66.4.13. 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 66.4.2 above if and only if $f$ is (quasi-)separated in the sense of Section 66.3.

In particular, if $f : X \to Y$ is a morphism of schemes over $S$, then $f$ is (quasi-)separated in the sense of Definition 66.4.2 if and only if $f$ is (quasi-)separated as a morphism of schemes.

Proof. This is the equivalence of (1) and (2) of Lemma 66.4.12 combined with the fact that any morphism of schemes is locally separated, see Schemes, Lemma 26.21.2. $\square$


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