The Stacks project

Lemma 69.5.16. 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. $Y$ quasi-compact and quasi-separated,

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

  3. the transition morphisms $X_{i'} \to X_ i$ are closed immersions,

  4. $X_ i \to Y$ locally of finite type

  5. $X \to Y$ is a closed immersion.

Then $X_ i \to Y$ is a closed immersion for $i$ large enough.

Proof. Choose an affine scheme $W$ and a surjective ├ętale morphism $W \to Y$. Then $X \times _ Y W$ is a closed subspace of $W$ and it suffices to check that $X_ i \times _ Y W$ is a closed subspace $W$ for some $i$ (Morphisms of Spaces, Lemma 66.12.1). By Lemma 69.5.11 this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma 32.4.20. $\square$

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