The Stacks project

Example 87.19.11. Let $B$ be a weakly admissible topological ring. Let $B \to A$ be a ring map (no topology). Then we can consider

where the limit is over all weak ideals of definition $J$ of $B$. Then $A^\wedge$ (endowed with the limit topology) is a complete linearly topologized ring. The (open) kernel $I$ of the surjection $A^\wedge \to A/JA$ is the closure of $JA^\wedge$, see Lemma 87.4.2. By Lemma 87.4.10 we see that $I$ consists of topologically nilpotent elements. Thus $I$ is a weak ideal of definition of $A^\wedge$ and we conclude $A^\wedge$ is a weakly admissible topological ring. Thus $\varphi : B \to A^\wedge$ is taut map of weakly admissible topological rings and

is a special case of the phenomenon studied in Lemma 87.19.8.