---- $R$ is a ([[commutative ring|commutative]]) [[ring]]. > [!definition] Definition. ([[completion of a module]]) > > Let $M$ be an $R$-[[module]] and $(M_{n})_{n \geq 1}$ a [[filtration|filtration]] of $M$. The **completion of $M$ with respect to the filtration $(M_{n})_{n \geq 1}$** is the [[categorical limit|inverse limit]] $\hat{M}=\lim\limits_{{\longleftarrow}} \frac{M}{M_{n}},$ > where the maps of the inverse system come from[^1] $\begin{align} > \frac{M}{M_{n+1}} &\to \frac{M}{M_{n}} \\ > m + M_{n+1} & \mapsto m + M_{n}. > \end{align}$ ---- #### [^1]: That such a map [[characterization of quotienting a group|is]] [[well-defined]] follows from $M_{n+1} \subset M_{n}$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```