The Stacks project

Lemma 10.155.5. Let $R \to S$ be a local map of local rings. Let $S \to S^ h$ be the henselization. Let $R \to A$ be an ├ętale ring map and let $\mathfrak q$ be a prime of $A$ lying over $\mathfrak m_ R$ such that $R/\mathfrak m_ R \cong \kappa (\mathfrak q)$. Then there exists a unique morphism of rings $f : A \to S^ h$ fitting into the commutative diagram

\[ \xymatrix{ A \ar[r]_ f & S^ h \\ R \ar[u] \ar[r] & S \ar[u] } \]

such that $f^{-1}(\mathfrak m_{S^ h}) = \mathfrak q$.

Proof. This is a special case of Lemma 10.153.11. $\square$

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