The Stacks project

Lemma 15.93.8. Let $A$ be a ring derived complete with respect to an ideal $I$. Then $(A, I)$ is a henselian pair.

Proof. Let $f \in I$. By Lemma 15.11.15 it suffices to show that $(A, fA)$ is a henselian pair. Observe that $A$ is derived complete with respect to $fA$ (follows immediately from Definition 15.91.4). By Lemma 15.91.3 the map from $A$ to the $f$-adic completion $A'$ of $A$ is surjective. By Lemma 15.11.4 the pair $(A', fA')$ is henselian. Thus it suffices to show that $(A, \bigcap f^ nA)$ is a henselian pair, see Lemma 15.11.9. This follows from Lemmas 15.93.7 and 15.11.2. $\square$

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