Suppose that $R$ is a topological ring with a complete system of neighbourhoods of zero given by left ideals. Let $\widehat{R}$ be its completion. Are the following four categories equivalent (perhaps under some reasonable conditions):

(1) the pro-category associated to the category of discrete topological left $R$-modules,

(2) the category of left $\widehat{R}$-modules,

(3) the category of topological left $\widehat{R}$-modules,

(4) the full subcategory of the category of topological left $R$-modules containing the discrete modules and their limits?

I apologize in advance if my question is rather basic, but I'm not very knowledgeable about pro-categories and topological rings.

  • 1
    $\begingroup$ Considering the case when $R=\widehat R=k$ is a field with the discrete topology should already be pretty illuminating. $\endgroup$ – Leonid Positselski Dec 24 '17 at 0:02

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