The Stacks project

Lemma 85.4.9. Let $A$ be a weakly admissible topological ring. Let $I \subset A$ be a weak ideal of definition. Then $(A, I)$ is a henselian pair.

Proof. Let $A \to A'$ be an ├ętale ring map and let $\sigma : A' \to A/I$ be an $A$-algebra map. By More on Algebra, Lemma 15.11.6 it suffices to lift $\sigma $ to an $A$-algebra map $A' \to A$. To do this, as $A$ is complete, it suffices to find, for every open ideal $J \subset I$, a unique $A$-algebra map $A' \to A/J$ lifting $\sigma $. Since $I$ is a weak ideal of definition, the ideal $I/J$ is locally nilpotent. We conclude by More on Algebra, Lemma 15.11.2. $\square$

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