Lemma 10.137.3. Let $R \to S$ be a smooth ring map. Any localization $S_ g$ is smooth over $R$. If $f \in R$ maps to an invertible element of $S$, then $R_ f \to S$ is smooth.
Proof. By Lemma 10.134.13 the naive cotangent complex for $S_ g$ over $R$ is the base change of the naive cotangent complex of $S$ over $R$. The assumption is that the naive cotangent complex of $S/R$ is $\Omega _{S/R}$ and that this is a finite projective $S$-module. Hence so is its base change. Thus $S_ g$ is smooth over $R$.
The second assertion follows in the same way from Lemma 10.134.11. $\square$
Comments (1)
Comment #718 by Keenan Kidwell on