Lemma 67.4.11 (04ZI)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.4: Separation axioms
Lemma 67.4.11 (cite)

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Lemma 67.4.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ .

If $X$ is separated then $X$ is separated over $S$ .
If $X$ is locally separated then $X$ is locally separated over $S$ .
If $X$ is quasi-separated then $X$ is quasi-separated over $S$ .

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

If $X$ is separated over $S$ then $f$ is separated.
If $X$ is locally separated over $S$ then $f$ is locally separated.
If $X$ is quasi-separated over $S$ then $f$ is quasi-separated.

Proof. Parts (4), (5), and (6) follow immediately from Lemma 67.4.10 and Spaces, Definition 65.13.2. Parts (1), (2), and (3) follow from parts (4), (5), and (6) by thinking of $X$ and $Y$ as algebraic spaces over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ , see Properties of Spaces, Definition 66.3.1. $\square$

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