The Stacks project

Lemma 32.4.21. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume

  1. $S$ quasi-separated,

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

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

  4. $X_ i \to S$ locally of finite type, and

  5. $X \to S$ an immersion.

Then $X_ i \to S$ is an immersion for $i$ large enough.

Proof. Choose an open subscheme $U \subset S$ such that $X \to S$ factors as a closed immersion $X \to U$ composed with the inclusion morphism $U \to S$. Since $X$ is quasi-compact, we may shrink $U$ and assume $U$ is quasi-compact. Denote $V_ i \subset X_ i$ the inverse image of $U$. Since $V_ i$ pulls back to $X$ we see that $V_ i = X_ i$ for all $i$ large enough by Lemma 32.4.11. Thus we may assume $X = \mathop{\mathrm{lim}}\nolimits X_ i$ in the category of schemes over $U$. Then we see that $X_ i \to U$ is a closed immersion for $i$ large enough by Lemma 32.4.20. This proves the lemma. $\square$

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