Lemma 70.11.1 (0870)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 70: Limits of Algebraic Spaces
Section 70.11: Finite type closed in finite presentation
Lemma 70.11.1 (cite)

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Lemma 70.11.1. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$ . If $Y$ quasi-compact and quasi-separated, then $X$ is a directed limit $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with each $X_ i$ affine and of finite presentation over $Y$ .

Proof. Consider the quasi-coherent $\mathcal{O}_ Y$ -module $\mathcal{A} = f_*\mathcal{O}_ X$ . By Lemma 70.9.4 we can write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ as a directed colimit of finitely presented $\mathcal{O}_ Y$ -algebras $\mathcal{A}_ i$ . Set $X_ i = \underline{\mathop{\mathrm{Spec}}}_ Y(\mathcal{A}_ i)$ , see Morphisms of Spaces, Definition 67.20.8. By construction $X_ i \to Y$ is affine and of finite presentation and $X = \mathop{\mathrm{lim}}\nolimits X_ i$ . $\square$

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