Lemma 32.13.4. 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. Let $Y \to X$ be a morphism of schemes over $S$. If $Y \to X$ is proper, $X_ i$ quasi-compact and quasi-separated, and $Y$ locally of finite type over $S$, then $Y \to X_ i$ is proper for $i$ large enough.
Proof. Choose a closed immersion $Y \to Y'$ with $Y'$ proper and of finite presentation over $X$, see Lemma 32.13.2. Then choose an $i$ and a proper morphism $Y'_ i \to X_ i$ such that $Y' = X \times _{X_ i} Y'_ i$. This is possible by Lemmas 32.10.1 and 32.13.1. Then after replacing $i$ by a larger index we have that $Y \to Y'_ i$ is a closed immersion, see Lemma 32.4.16. $\square$
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