Lemma 15.45.6. Let R be a local ring. The following are equivalent: R is a normal domain, the henselization R^ h of R is a normal domain, and the strict henselization R^{sh} of R is a normal domain.
Proof. A preliminary remark is that a local ring is normal if and only if it is a normal domain (see Algebra, Definition 10.37.11). The ring maps R \to R^ h \to R^{sh} are faithfully flat. Hence one direction of the implications follows from Algebra, Lemma 10.164.3. Conversely, assume R is normal. Since R^ h and R^{sh} are filtered colimits of étale hence smooth R-algebras, the result follows from Algebra, Lemmas 10.163.9 and 10.37.17. \square
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