The Stacks project

Reducedness passes to the (strict) henselization.

Lemma 15.44.4. 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.

Proof. The ring maps $R \to R^ h \to R^{sh}$ are faithfully flat. Hence one direction of the implications follows from Algebra, Lemma 10.162.2. Conversely, assume $R$ is reduced. Since $R^ h$ and $R^{sh}$ are filtered colimits of ├ętale, hence smooth $R$-algebras, the result follows from Algebra, Lemma 10.161.7. $\square$


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