The Stacks project

Lemma 65.8.10. Let $f : X \to Y$ be a morphism of algebraic spaces over a scheme $S$.

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

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

  3. A fibre product of quasi-compact and quasi-separated algebraic spaces is quasi-compact and quasi-separated.

Proof. Part (1) follows from Lemma 65.8.9 with $Z = S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. Part (2) follows from (1) and Lemma 65.4.10. For (3) let $X \to Y$ and $Z \to Y$ be morphisms of quasi-compact and quasi-separated algebraic spaces. Then $X \times _ Y Z \to Z$ is quasi-compact and quasi-separated as a base change of $X \to Y$ using (2) and Lemmas 65.8.4 and 65.4.4. Hence $X \times _ Y Z$ is quasi-compact and quasi-separated as an algebraic space quasi-compact and quasi-separated over $Z$, see Lemmas 65.4.9 and 65.8.5. $\square$


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