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 10.153.2. Let $(R, \mathfrak m, k)$ be a local ring. If $k \subset K$ is a separable algebraic extension, then there exists a directed set $I$ and a system of finite étale extensions $R \subset R_ i$, $i \in I$ of local rings such that $R' = \mathop{\mathrm{colim}}\nolimits R_ i$ has residue field $K$ (as extension of $k$).

Proof. Let $R \subset R'$ be the extension constructed in the proof of Lemma 10.153.1. By construction $R' = \mathop{\mathrm{colim}}\nolimits _{\alpha \in A} R_\alpha $ where $A$ is a well-ordered set and the transition maps $R_\alpha \to R_{\alpha + 1}$ are finite étale and $R_\alpha = \mathop{\mathrm{colim}}\nolimits _{\beta < \alpha } R_\beta $ if $\alpha $ is not a successor. We will prove the result by transfinite induction.

Suppose the result holds for $R_\alpha $, i.e., $R_\alpha = \mathop{\mathrm{colim}}\nolimits R_ i$ with $R_ i$ finite étale over $R$. Since $R_\alpha \to R_{\alpha + 1}$ is finite étale there exists an $i$ and a finite étale extension $R_ i \to R_{i, 1}$ such that $R_{\alpha + 1} = R_\alpha \otimes _{R_ i} R_{i, 1}$. Thus $R_{\alpha + 1} = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} R_{i'} \otimes _{R_ i} R_{i, 1}$ and the result holds for $\alpha + 1$. Suppose $\alpha $ is not a successor and the result holds for $R_\beta $ for all $\beta < \alpha $. Since every finite subset $E \subset R_\alpha $ is contained in $R_\beta $ for some $\beta < \alpha $ and we see that $E$ is contained in a finite étale subextension by assumption. Thus the result holds for $R_\alpha $. $\square$


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