Definition 10.155.3. Let $(R, \mathfrak m, \kappa )$ be a local ring.

The local ring map $R \to R^ h$ constructed in Lemma 10.155.1 is called the

*henselization*of $R$.Given a separable algebraic closure $\kappa \subset \kappa ^{sep}$ the local ring map $R \to R^{sh}$ constructed in Lemma 10.155.2 is called the

*strict henselization of $R$ with respect to $\kappa \subset \kappa ^{sep}$*.A local ring map $R \to R^{sh}$ is called a

*strict henselization*of $R$ if it is isomorphic to one of the local ring maps constructed in Lemma 10.155.2

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