The Stacks project

Lemma 10.150.4. Let $R$ be a ring. Let $S \subset R$ be any multiplicative subset. Then the ring map $R \to S^{-1}R$ is formally étale.

Proof. Let $I \subset A$ be an ideal of square zero. What we are saying here is that given a ring map $\varphi : R \to A$ such that $\varphi (f) \mod I$ is invertible for all $f \in S$ we have also that $\varphi (f)$ is invertible in $A$ for all $f \in S$. This is true because $A^*$ is the inverse image of $(A/I)^*$ under the canonical map $A \to A/I$. $\square$

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