Lemma 101.6.4 (04YZ)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.6: Higher diagonals
Lemma 101.6.4 (cite)

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Lemma 101.6.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then

$\Delta _{f, 1}$ separated $\Leftrightarrow$ $\Delta _{f, 2}$ closed immersion $\Leftrightarrow$ $\Delta _{f, 2}$ proper $\Leftrightarrow$ $\Delta _{f, 2}$ universally closed,
$\Delta _{f, 1}$ quasi-separated $\Leftrightarrow$ $\Delta _{f, 2}$ finite type $\Leftrightarrow$ $\Delta _{f, 2}$ quasi-compact, and
$\Delta _{f, 1}$ locally separated $\Leftrightarrow$ $\Delta _{f, 2}$ immersion.

Proof. Follows from Lemmas 101.3.5, 101.3.6, and 101.3.7 applied to $\Delta _{f, 1}$ . $\square$

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