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$\newcommand{\colim}{\operatorname{colim}}$ Let $k$ be a field (of characteristic 0, say) and $M$ be a module over $R=k[[h]]$. Recall that the $h$-adic completion of $M$ is $$ \hat M:=\lim M/h^nM, $$ ...

Over a ring $R$ we have a notion of finitely presented module, namely: Definition 1 A module $F$ is finitely presented if there are $m$, $n$ positive integers such that $R^m\to R^n\to F\to 0$ is ...