----- > [!proposition] Proposition. ([[adic completions are well-behaved like localization is for Noetherian rings]]) > > Let $R$ be a [[Noetherian ring|Noetherian]] [[ring]], and let $\hat{R}$ be the $\mathfrak{a}$-[[adic completion]] of $R$, $\mathfrak{a}$ an [[ideal]] of $R$. Then: > 1. $\hat{R}$ is also [[Noetherian ring|Noetherian]]; > 2. $\hat{R} \otimes_{R} \_$ is an [[exact functor]]; > 3. If $M$ is a [[submodule generated by a subset|finitely generated]] $R$-[[module]], then the map $\hat{R} \otimes_{R} M \to \hat{M}$, $x \otimes m \mapsto xm$, is an $\hat{R}$-[[linear map|linear]] [[isomorphism]]. > [!proposition] Corollary. > If $R$ is a [[Noetherian ring]] then the [[power series]] [[ring]] $R[ [T_{1},\dots,T_{n}]]$ is too. ^proposition > [!proof]- Proof. ([[adic completions are well-behaved like localization is for Noetherian rings]]) > ~ ----- #### ---- #### 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 > ```