The Stacks project

Lemma 66.23.1. Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$. Any algebraic space locally of finite type over $X$ is Jacobson.

Proof. Let $U \to X$ be a surjective ├ętale morphism where $U$ is a scheme. Then $U$ is Jacobson (by definition) and for a morphism of schemes $V \to U$ which is locally of finite type we see that $V$ is Jacobson by the corresponding result for schemes (Morphisms, Lemma 29.16.9). Thus if $Y \to X$ is a morphism of algebraic spaces which is locally of finite type, then setting $V = U \times _ X Y$ we see that $Y$ is Jacobson by definition. $\square$


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