Lemma 15.45.10. Let R be a Noetherian local ring. The following are equivalent: R is a regular local ring, the henselization R^ h of R is a regular local ring, and the strict henselization R^{sh} of R is a regular local ring.
Proof. By Lemma 15.45.3 we know that R^ h and R^{sh} are Noetherian, hence the lemma makes sense. Let \mathfrak m be the maximal ideal of R. Let x_1, \ldots , x_ t \in \mathfrak m be a minimal system of generators of \mathfrak m, i.e., such that the images in \mathfrak m/\mathfrak m^2 form a basis over \kappa = R/\mathfrak m. Because R \to R^ h and R \to R^{sh} are faithfully flat, it follows that the images x_1^ h, \ldots , x_ t^ h in R^ h, resp. x_1^{sh}, \ldots , x_ t^{sh} in R^{sh} are a minimal system of generators for \mathfrak m^ h = \mathfrak mR^ h, resp. \mathfrak m^{sh} = \mathfrak mR^{sh}. Regularity of R by definition means t = \dim (R) and similarly for R^ h and R^{sh}. Hence the lemma follows from the equality of dimensions \dim (R) = \dim (R^ h) = \dim (R^{sh}) of Lemma 15.45.7 \square
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