\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

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 ├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 10.148.11. $\square$


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