----- > [!proposition] Proposition. ([[the Rees characterization of stable filtrations over Noetherian rings]]) > > - Let $R$ be a [[Noetherian ring|Noetherian]] [[ring]]. > - Let $M$ be a [[submodule generated by a subset|finitely generated]] $R$-[[module]]. > - Let $(M_{n})_{n \geq 0}$ be an $\mathfrak{a}$-[[filtration|filtration]] of $M$. > > Then the following are equivalent: > 1. The [[Rees ring|module]] $M^{*}$ is [[submodule generated by a subset|finitely generated]] over the [[Rees ring]] $R^{*}$ > 2. The $\mathfrak{a}$-[[filtration|filtration]] $(M_{n})_{n \geq 0}$ is [[filtration|stable]]. > [!proof]- Proof. ([[the Rees characterization of stable filtrations over Noetherian rings]]) > - [ ] in notes, not a ton of work ----- #### ---- #### 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 > ```