The Stacks project

Lemma 66.4.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. If $X$ is separated then $X$ is separated over $S$.

  2. If $X$ is locally separated then $X$ is locally separated over $S$.

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

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

  1. If $X$ is separated over $S$ then $f$ is separated.

  2. If $X$ is locally separated over $S$ then $f$ is locally separated.

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

Proof. Parts (4), (5), and (6) follow immediately from Lemma 66.4.10 and Spaces, Definition 64.13.2. 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 65.3.1. $\square$

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