The Stacks project

\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*}

Lemma 10.148.8. Let $(R, \mathfrak m, \kappa )$ be a strictly henselian local ring. Let $R \to S$ be an unramified ring map. Then

\[ S = A_1 \times \ldots \times A_ n \times B \]

with each $R \to A_ i$ surjective and no prime of $B$ lying over $\mathfrak m$.

Proof. First write $S = A_1 \times \ldots \times A_ n \times B$ as in Lemma 10.148.5. Now we see that $R \to A_ i$ is finite unramified and $A_ i$ local. Hence the maximal ideal of $A_ i$ is $\mathfrak mA_ i$ and its residue field $A_ i / \mathfrak m A_ i$ is a finite separable extension of $\kappa $, see Lemma 10.147.5. However, the condition that $R$ is strictly henselian means that $\kappa $ is separably algebraically closed, so $\kappa = A_ i / \mathfrak m A_ i$. By Nakayama's Lemma 10.19.1 we conclude that $R \to A_ i$ is surjective as desired. $\square$


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