Lemma 87.4.3 (0ARZ)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.4: Topological rings and modules
Lemma 87.4.3 (cite)

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Lemma 87.4.3. Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$ -module. Let $N \subset M$ be a submodule. Then

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