Lemma 10.150.3 (031N)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.150: Formally étale maps
Lemma 10.150.3 (cite)

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Lemma 10.150.3. Let $R$ be a ring. Let $I$ be a directed set. Let $(S_ i, \varphi _{ii'})$ be a system of $R$-algebras over $I$. If each $R \to S_ i$ is formally étale, then $S = \mathop{\mathrm{colim}}\nolimits _{i \in I} S_ i$ is formally étale over $R$

Proof. Consider a diagram as in Definition 10.150.1. By assumption we get unique $R$-algebra maps $S_ i \to A$ lifting the compositions $S_ i \to S \to A/I$. Hence these are compatible with the transition maps $\varphi _{ii'}$ and define a lift $S \to A$. This proves existence. The uniqueness is clear by restricting to each $S_ i$. $\square$

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