The Stacks project

Theorem 59.32.8. Let $(R, \mathfrak m, \kappa )$ be a local ring and $\kappa \subset \kappa ^{sep}$ a separable algebraic closure. There exist canonical flat local ring maps $R \to R^ h \to R^{sh}$ where

  1. $R^ h$, $R^{sh}$ are filtered colimits of étale $R$-algebras,

  2. $R^ h$ is henselian, $R^{sh}$ is strictly henselian,

  3. $\mathfrak m R^ h$ (resp. $\mathfrak m R^{sh}$) is the maximal ideal of $R^ h$ (resp. $R^{sh}$), and

  4. $\kappa = R^ h/\mathfrak m R^ h$, and $\kappa ^{sep} = R^{sh}/\mathfrak m R^{sh}$ as extensions of $\kappa $.

Proof. The structure of $R^ h$ and $R^{sh}$ is described in Algebra, Lemmas 10.155.1 and 10.155.2. $\square$

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