---- Let $R$ be a ([[commutative ring|commutative]]) [[ring]], $\mathfrak{a}$ an [[ideal]] of $R$. > [!definition] Definition. ([[adic completion]]) > The **$\mathfrak{a}$-adic completion of $R$** is the [[categorical limit|inverse limit]][^1] $\hat{R}= \lim\limits_{{\longleftarrow}} R / \mathfrak{a}^{i}= \left\{ (r_{i} + \mathfrak{a}^{i})_{i \in \mathbb{N}} \in \prod_{i \in \mathbb{N}}R / \mathfrak{a}^{i} : r_{i} \equiv r_{j} \text{ mod }\mathfrak{a}^{i} \ \forall i \leq j \right\}.$ > > More generally, the **$\mathfrak{a}$-adic completion of an $R$-module $M$** is the [[categorical limit|inverse limit]] $\hat{M}=\lim\limits_{{\longleftarrow}} M / \mathfrak{a}^{i}M$. It is naturally an $\hat{R}$-[[module]] (act componentwise). ^definition > [!basicproperties] > - [[adic completions are well-behaved like localization is for Noetherian rings]] ^properties ---- #### [^1]: Here, the [[filtered poset|directed set]] is $(\mathbb{N}, \leq)$, and the [[categorical limit|inverse system]] is $R / \mathfrak{a}^{i+1} \xrightarrow{r + \mathfrak{a}^{i+1} \mapsto r + \mathfrak{a}^{i} } R / \mathfrak{a}^{i} \to \dots$. The map corresponding to $i \leq j$ is obtained by composition: $h_{ij}: R / \mathfrak{a}^{j} \to R / \mathfrak{a}^{i}$ maps $r_{j}+\mathfrak{a}^{j} \mapsto r_{j}+\mathfrak{a}^{i}$ for $r_{j}+\mathfrak{a}^{j} \in R / \mathfrak{a}^{j}$. Thus, $r_{i}+\mathfrak{a}^{i} \in R / \mathfrak{a}^{i}$ equals $h_{ij}(r_{j} + \mathfrak{a}^{j})$ if and only if $r_{i}+\mathfrak{a}^{i}=r_{j}+\mathfrak{a}^{i}$, i.e., $r_{i} \equiv r_{j} \text{ mod }\mathfrak{a}^{i}$. ---- #### 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 > ```