The Stacks project

Lemma 70.5.13. Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume

  1. $Y$ is quasi-separated,

  2. $X_ i$ is quasi-compact and quasi-separated,

  3. the morphism $X \to Y$ is separated.

Then $X_ i \to Y$ is separated for all $i$ large enough.

Proof. Let $0 \in I$. Choose an affine scheme $W$ and an ├ętale morphism $W \to Y$ such that the image of $|W| \to |Y|$ contains the image of $|X_0| \to |Y|$. This is possible as $X_0$ is quasi-compact. It suffices to check that $W \times _ Y X_ i \to W$ is separated for some $i \geq 0$ because the diagonal of $W \times _ Y X_ i$ over $W$ is the base change of $X_ i \to X_ i \times _ Y X_ i$ by the surjective ├ętale morphism $(X_ i \times _ Y X_ i) \times _ Y W \to X_ i \times _ Y X_ i$. Since $Y$ is quasi-separated the algebraic spaces $W \times _ Y X_ i$ are quasi-compact (as well as quasi-separated). Thus we may base change to $W$ and assume $Y$ is an affine scheme. When $Y$ is an affine scheme, we have to show that $X_ i$ is a separated algebraic space for $i$ large enough and we are given that $X$ is a separated algebraic space. Thus this case follows from Lemma 70.5.9. $\square$

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