The Stacks project

[Theorem 8.1, Ma]

Lemma 87.4.5. Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Assume $M$ has a countable fundamental system of neighbourhoods of $0$. Then

  1. $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge \to 0$ is exact,

  2. $N^\wedge $ is the closure of the image of $N \to M^\wedge $,

  3. $M^\wedge \to (M/N)^\wedge $ is open.

Proof. We have $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge $ is exact and statement (2) by Lemma 87.4.3. This produces a canonical map $c : M^\wedge /N^\wedge \to (M/N)^\wedge $. The module $M^\wedge /N^\wedge $ is complete and $M^\wedge \to M^\wedge /N^\wedge $ is open by Lemma 87.4.4. By the universal property of completion we obtain a canonical map $b : (M/N)^\wedge \to M^\wedge /N^\wedge $. Then $b$ and $c$ are mutually inverse as they are on a dense subset. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 87.4: Topological rings and modules

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Your email address will not be published.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 0AS0. Beware of the difference between the letter 'O' and the digit '0'.