Lemma 15.45.4. Let R be a local ring. The following are equivalent: R is reduced, the henselization R^ h of R is reduced, and the strict henselization R^{sh} of R is reduced.
Reducedness passes to the (strict) henselization.
Proof. The ring maps R \to R^ h \to R^{sh} are faithfully flat. Hence one direction of the implications follows from Algebra, Lemma 10.164.2. Conversely, assume R is reduced. Since R^ h and R^{sh} are filtered colimits of étale, hence smooth R-algebras, the result follows from Algebra, Lemma 10.163.7. \square
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Comment #1605 by Johan on