The Stacks project

Lemma 85.4.3. Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Then

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

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

Proof. Let $M_\lambda $, $\lambda \in \Lambda $ be a fundamental system of open submodules. Then $N \cap M_\lambda $ is a fundamental system of open submodules of $N$ and $M_\lambda + N/N$ is a fundamental system of open submodules of $M/N$. Thus we see that (1) follows from the exactness of the sequences

\[ 0 \to N/N \cap M_\lambda \to M/M_\lambda \to M/(M_\lambda + N) \to 0 \]

and the fact that taking limits commutes with limits. The second statement follows from this and the fact that $N \to N^\wedge $ has dense image and that the kernel of $M^\wedge \to (M/N)^\wedge $ is closed. $\square$

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