The Stacks project

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

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

2. 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 $\kappa \subset \kappa ^{sep}$.

3. 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