The Stacks project

Lemma 15.45.2. Let $(R, \mathfrak m, \kappa )$ be a local ring. Then

  1. $R \to R^ h$, $R^ h \to R^{sh}$, and $R \to R^{sh}$ are formally étale,

  2. $R \to R^ h$, $R^ h \to R^{sh}$, resp. $R \to R^{sh}$ are formally smooth in the $\mathfrak m^ h$, $\mathfrak m^{sh}$, resp. $\mathfrak m^{sh}$-topology.

Proof. Part (1) follows from the fact that $R^ h$ and $R^{sh}$ are directed colimits of étale algebras (by construction), that étale algebras are formally étale (Algebra, Lemma 10.150.2), and that colimits of formally étale algebras are formally étale (Algebra, Lemma 10.150.3). Part (2) follows from the fact that a formally étale ring map is formally smooth and Lemma 15.37.2. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07QN. Beware of the difference between the letter 'O' and the digit '0'.