Lemma 100.3.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent

1. $f$ is locally separated, and

2. $\Delta _ f$ is an immersion.

Proof. The statements “$f$ is locally separated”, and “$\Delta _ f$ is an immersion” refer to the notions defined in Properties of Stacks, Section 99.3. Proof omitted. Hint: Argue as in the proofs of Lemmas 100.3.5 and 100.3.6. $\square$

Comment #816 by Matthew Emerton on

f is quasi-separated should read f is locally separated.

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• 4 comment(s) on Section 100.3: Properties of diagonals

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