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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

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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} $

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  • $\begingroup$ Can you give a reference for the sentence: ''For finitely generated modules the completion functor is isomorphic to $−\otimes_A\widehat A$ '' $\endgroup$ – ginevracoal Mar 15 '16 at 10:00
  • $\begingroup$ https://en.wikipedia.org/wiki/Completion_(algebra). You'll also find the statement in every introductory algebra book that treats completions. $\endgroup$ – Rieux Mar 15 '16 at 18:00

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