The Stacks project

Lemma 10.155.9. 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 ├ętale 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 10.153.11. $\square$

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