Strict Henselization

Definition

Let $(R, \mathfrak{m}, k)$ be a local ring with residue field $k$, and let $k^{s}$ be a separable closure of $k$. A local ring is strictly Henselian if it is a Henselian local ring and its residue field is separably closed. A strict Henselization of $R$ is a local ring $R^{sh}$ with a local homomorphism $i \colon R \to R^{sh}$. The residue field of $R^{sh}$ is identified with $k^{s}$, and the following hold:

1. $R^{sh}$ is strictly Henselian.
2. The map of residue fields induced by $i$ is the inclusion $k \subseteq k^{s}$.
3. Let $S$ be a strictly Henselian local ring with residue field $\kappa_S$, let $f \colon R \to S$ be a local homomorphism, and let $\sigma \colon k^{s} \to \kappa_S$ be a $k$-embedding. Then there is exactly one local homomorphism $g \colon R^{sh} \to S$ with $g \circ i = f$ that induces $\sigma$ on residue fields.