The Stacks project

Lemma 15.44.2. 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 10.149.2), and that colimits of formally étale algebras are formally étale (Algebra, Lemma 10.149.3). Part (2) follows from the fact that a formally étale ring map is formally smooth and Lemma 15.36.2. $\square$


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