stacks/stacks-project

Thanks to Kestutis Cesnavicius
 @@ -801,6 +801,37 @@ \section{Topological rings and modules} Then $b$ and $c$ are mutually inverse as they are on a dense subset. \end{proof} \begin{lemma} \label{lemma-completion-adic-star} Let $R$ be a topological ring. Let $M$ be a topological $R$-module. Let $I \subset R$ be a finitely generated ideal. Assume $M$ has an open submodule whose topology is $I$-adic. Then $M^\wedge$ has an open submodule whose topology is $I$-adic and we have $M^\wedge/I^n M^\wedge = M/I^nM$ for all $n \geq 1$. \end{lemma} \begin{proof} Let $M' \subset M$ be an open submodule whose topology is $I$-adic. Then $\{I^nM'\}_{n \geq 1}$ is a fundamental system of open submodules of $M$. Thus $M^\wedge = \lim M/I^nM'$ contains $(M')^\wedge = \lim M'/I^nM'$ as an open submodule and the topology on $(M')^\wedge$ is $I$-adic by Algebra, Lemma \ref{algebra-lemma-hathat-finitely-generated}. Since $I$ is finitely generated, $I^n$ is finitely generated, say by $f_1, \ldots, f_r$. Observe that the surjection $(f_1, \ldots, f_r) : M^{\oplus r} \to I^n M$ is continuous and open by our description of the topology on $M$ above. By Lemma \ref{lemma-ses} applied to this surjection and to the short exact sequence $0 \to I^nM \to M \to M/I^nM \to 0$ we conclude that $$(f_1, \ldots, f_r) : (M^\wedge)^{\oplus r} \longrightarrow M^\wedge$$ surjects onto the kernel of the surjection $M^\wedge \to M/I^nM$. Since $f_1, \ldots, f_r$ generate $I^n$ we conclude. \end{proof} \begin{definition} \label{definition-toplogy-tensor-product} Let $R$ be a topological ring. Let $M$ and $N$ be linearly