The Stacks project

Lemma 10.159.2. 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 10.159.1. 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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