The Stacks project

Lemma 109.43.4. Let $(A, \mathfrak m, \kappa )$ be a regular local ring of characteristic $p > 0$. Suppose $[\kappa : \kappa ^ p] < \infty $. Then $A$ is excellent if and only if $A \to A^\wedge $ is formally étale.

Proof. The backward implication follows from Lemma 109.43.2. For the forward implication, note that we already know from Lemma 109.43.2 that $A \to A^\wedge $ is formally unramified or equivalently that $\Omega _{A^\wedge /A}$ is zero. Thus, it suffices to show that the completion map is formally smooth when $A$ is excellent. By Néron-Popescu desingularization $A \to A^\wedge $ can be written as a filtered colimit of smooth $A$-algebras (Smoothing Ring Maps, Theorem 16.12.1). Hence $\mathop{N\! L}\nolimits _{A^\wedge /A}$ has vanishing cohomology in degree $-1$. Thus $A \to A^\wedge $ is formally smooth by Algebra, Proposition 10.138.8. $\square$

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