Lemma 87.5.3. Let $A$ be a linearly topologized ring. The map $A \to A^\wedge $ from $A$ to its completion is taut.
Proof. Let $I_\lambda $ be a fundamental system of open ideals of $A$. Recall that $A^\wedge = \mathop{\mathrm{lim}}\nolimits A/I_\lambda $ with the limit topology, which means that the kernels $J_\lambda = \mathop{\mathrm{Ker}}(A^\wedge \to A/I_\lambda )$ form a fundamental system of open ideals of $A^\wedge $. Since $J_\lambda $ is the closure of $I_\lambda A^\wedge $ (compare with Lemma 87.4.11) we conclude. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)