The Stacks project

Lemma 32.4.18. 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. $X \to S$ affine.

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

Proof. 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. For each $j$ the scheme $f^{-1}(V_ j) = \mathop{\mathrm{lim}}\nolimits _ i f_ i^{-1}(V_ j)$ is affine (as a finite morphism is affine by definition). Hence by Lemma 32.4.13 there exists an $i \in I$ such that each $f_ i^{-1}(V_ j)$ is affine. In other words, $f_ i : X_ i \to S$ is affine for $i$ large enough, see Morphisms, Lemma 29.11.3. $\square$

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