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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #718 by Keenan Kidwell on