# Direct limit of completions of finitely generated submodules

Let $A$ be a noetherian, local, integral domain with maximal ideal $\mathfrak m$. Moreover let $M$ be an $A$-module; I'd like to know if there exists an explicit expression of the module: $$\varinjlim_{N\subseteq M} \widehat N$$ where $N$ varies among the finitely generated $A$-submodules of $M$ and $\widehat N$ is the $\mathfrak m$-adic completion of $N$.

In particular what happens if $M$ is already finitely generated?

I'm asking this question because it is well known that $$M\cong \varinjlim_{N\subseteq M} N$$ so I'm interested in the behavior of the completions with respect to the direct limit.

Many thanks in advance

For finitely generated modules the completion functor is isomorphic to $-\otimes_{A} \hat{A}$ and direct limits commute with tensor products, so you get:
$\varinjlim\limits_{N\subset M} \hat{N}\cong \varinjlim\limits_{N\subset M} (N\otimes_{A}\hat{A}) \cong (\varinjlim\limits_{N\subset M} N)\otimes_{A} \hat{A}\cong M\otimes_{A}\hat{A}\cong \hat{M}$
• Can you give a reference for the sentence: ''For finitely generated modules the completion functor is isomorphic to $−\otimes_A\widehat A$ '' – ginevracoal Mar 15 '16 at 10:00