Lemma 32.2.1 (01YW)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.2: Directed limits of schemes with affine transition maps
Lemma 32.2.1 (cite)

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Lemma 32.2.1. Let $I$ be a directed set. Let $(S_ i, f_{ii'})$ be an inverse system of schemes over $I$ . If all the schemes $S_ i$ are affine, then the limit $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$ exists in the category of schemes. In fact $S$ is affine and $S = \mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits _ i R_ i)$ with $R_ i = \Gamma (S_ i, \mathcal{O})$ .

Proof. Just define $S = \mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits _ i R_ i)$ . It follows from Schemes, Lemma 26.6.4 that $S$ is the limit even in the category of locally ringed spaces. $\square$

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