The Stacks project

Lemma 66.4.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. If $Y$ is separated and $f$ is separated, then $X$ is separated.

  2. If $Y$ is quasi-separated and $f$ is quasi-separated, then $X$ is quasi-separated.

  3. If $Y$ is locally separated and $f$ is locally separated, then $X$ is locally separated.

  4. If $Y$ is separated over $S$ and $f$ is separated, then $X$ is separated over $S$.

  5. If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then $X$ is quasi-separated over $S$.

  6. If $Y$ is locally separated over $S$ and $f$ is locally separated, then $X$ is locally separated over $S$.

Proof. Parts (4), (5), and (6) follow immediately from Lemma 66.4.8 and Spaces, Definition 64.13.2. Parts (1), (2), and (3) reduce to 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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