Lemma 110.44.4. 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 110.44.2. For the forward implication, note that we already know from Lemma 110.44.2 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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