Lemma 10.150.2. Let $R \to S$ be a ring map of finite presentation. The following are equivalent:

$R \to S$ is formally étale,

$R \to S$ is étale.

Lemma 10.150.2. Let $R \to S$ be a ring map of finite presentation. The following are equivalent:

$R \to S$ is formally étale,

$R \to S$ is étale.

**Proof.**
Assume that $R \to S$ is formally étale. Then $R \to S$ is smooth by Proposition 10.138.13. By Lemma 10.148.2 we have $\Omega _{S/R} = 0$. Hence $R \to S$ is étale by definition.

Assume that $R \to S$ is étale. Then $R \to S$ is formally smooth by Proposition 10.138.13. By Lemma 10.148.2 it is formally unramified. Hence $R \to S$ is formally étale. $\square$

Your email address will not be published.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 (0)

There are also: