Lemma 32.4.20. 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-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. $X_ i \to S$ locally of finite type

5. $X \to S$ a closed immersion.

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

Proof. By Lemma 32.4.18 we may assume $X_ i \to S$ is affine for all $i$. Choose a finite affine open covering $S = \bigcup _{j = 1, \ldots , n} V_ j$. Denote $f : X \to S$ and $f_ i : X_ i \to S$ the structure morphisms. It suffices to show that there exists an $i$ such that $f_ i^{-1}(V_ j)$ is a closed subscheme of $V_ j$ for $j = 1, \ldots , m$ (Morphisms, Lemma 29.2.1). This reduces us to the affine case: Let $R$ be a ring and $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $R \to A$ surjective and $A_ i \to A_{i'}$ surjective for all $i \leq i'$. Moreover $R \to A_ i$ is of finite type for all $i$. Goal: Show that $R \to A_ i$ is surjective for some $i$. To prove this choose an $i \in I$ and pick generators $x_1, \ldots , x_ m \in A_ i$ of $A_ i$ as an $R$-algebra. Since $R \to A$ is surjective we can find $r_ j \in R$ such that $r_ j$ maps to $x_ j$ in $A$. Thus there exists an $i' \geq i$ such that $r_ j$ maps to the image of $x_ j$ in $A_{i'}$ for $j = 1, \ldots , m$. Since $A_ i \to A_{i'}$ is surjective this implies that $R \to A_{i'}$ is surjective. $\square$

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