## Tag `04GT`

Chapter 10: Commutative Algebra > Section 10.150: Henselization and strict henselization

Lemma 10.150.11. Let $\varphi : R \to S$ be a local map of local rings. Let $S/\mathfrak m_S \subset \kappa^{sep}$ be a separable algebraic closure. Let $S \to S^{sh}$ be the strict henselization of $S$ with respect to $S/\mathfrak m_S \subset \kappa^{sep}$. Let $R \to A$ be an étale ring map and let $\mathfrak q$ be a prime of $A$ lying over $\mathfrak m_R$. Given any commutative diagram $$ \xymatrix{ \kappa(\mathfrak q) \ar[r]_{\phi} & \kappa^{sep} \\ R/\mathfrak m_R \ar[r]^{\varphi} \ar[u] & S/\mathfrak m_S \ar[u] } $$ there exists a unique morphism of rings $f : A \to S^{sh}$ fitting into the commutative diagram $$ \xymatrix{ A \ar[r]_f & S^{sh} \\ R \ar[u] \ar[r]^{\varphi} & S \ar[u] } $$ such that $f^{-1}(\mathfrak m_{S^h}) = \mathfrak q$ and the induced map $\kappa(\mathfrak q) \to \kappa^{sep}$ is the given one.

Proof.This is a special case of Lemma 10.148.11. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 40971–40996 (see updates for more information).

```
\begin{lemma}
\label{lemma-strictly-henselian-functorial-prepare}
Let $\varphi : R \to S$ be a local map of local rings.
Let $S/\mathfrak m_S \subset \kappa^{sep}$ be a separable algebraic closure.
Let $S \to S^{sh}$ be the strict henselization of $S$
with respect to $S/\mathfrak m_S \subset \kappa^{sep}$.
Let $R \to A$ be an \'etale ring map and let $\mathfrak q$
be a prime of $A$ lying over $\mathfrak m_R$.
Given any commutative diagram
$$
\xymatrix{
\kappa(\mathfrak q) \ar[r]_{\phi} & \kappa^{sep} \\
R/\mathfrak m_R \ar[r]^{\varphi} \ar[u] & S/\mathfrak m_S \ar[u]
}
$$
there exists a unique morphism of rings
$f : A \to S^{sh}$ fitting into the commutative diagram
$$
\xymatrix{
A \ar[r]_f & S^{sh} \\
R \ar[u] \ar[r]^{\varphi} & S \ar[u]
}
$$
such that $f^{-1}(\mathfrak m_{S^h}) = \mathfrak q$ and the induced
map $\kappa(\mathfrak q) \to \kappa^{sep}$ is the given one.
\end{lemma}
\begin{proof}
This is a special case of Lemma \ref{lemma-map-into-henselian}.
\end{proof}
```

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