The Stacks project

Lemma 15.50.13. Let $(A, \mathfrak m)$ be a henselian local ring. Then $A$ is a filtered colimit of a system of henselian local G-rings with local transition maps.

Proof. Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of finite type $\mathbf{Z}$-algebras. Let $\mathfrak p_ i$ be the prime ideal of $A_ i$ lying under $\mathfrak m$. We may replace $A_ i$ by the localization of $A_ i$ at $\mathfrak p_ i$. Then $A_ i$ is a Noetherian local G-ring (Proposition 15.50.12). By Lemma 15.12.5 we see that $A = \mathop{\mathrm{colim}}\nolimits A_ i^ h$. By Lemma 15.50.8 the rings $A_ i^ h$ are G-rings. $\square$

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