The Stacks project

Example 85.15.11. 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 85.4.2. By Lemma 85.4.10 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 85.15.10.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AN6. Beware of the difference between the letter 'O' and the digit '0'.