The Stacks project

Example 86.19.11. Let $B$ be a weakly admissible topological ring. Let $B \to A$ be a ring map (no topology). Then we can consider

\[ A^\wedge = \mathop{\mathrm{lim}}\nolimits A/JA \]

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 86.4.2. By Lemma 86.4.10 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

\[ \text{Spf}(A^\wedge ) \longrightarrow \text{Spf}(B) \]

is a special case of the phenomenon studied in Lemma 86.19.8.

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