The Stacks project

Lemma 10.148.5. 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 unramified, then $S = \mathop{\mathrm{colim}}\nolimits _{i \in I} S_ i$ is formally unramified over $R$

Proof. Consider a diagram as in Definition 10.148.1. By assumption there exists at most one $R$-algebra map $S_ i \to A$ lifting the compositions $S_ i \to S \to A/I$. Since every element of $S$ is in the image of one of the maps $S_ i \to S$ we see that there is at most one map $S \to A$ fitting into the diagram. $\square$

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