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 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.
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