---- $R$ is a ([[commutative ring|commutative]]) [[ring]]. > [!theorem] Theorem. ([[Artin-Rees Lemma]]) > > > Suppose: > - $R$ is [[Noetherian ring|Noetherian]]; > - $\mathfrak{a} \subset R$ is an [[ideal]] of $R$; > - $M$ is a [[submodule generated by a subset|finitely generated]] $R$-[[module]]; > - $(M_{\ell})_{\ell \geq 0}$ is a [[filtration|stable]] $\mathfrak{a}$-[[filtration|filtration]]; > - $N \subset M$ is a [[submodule]]. > > We get an induced $\mathfrak{a}$-filtration on $N$ as $(N \cap M_{\ell})_{\ell \geq 0}$. The **Artin-Rees Lemma** asserts it is a stable $\mathfrak{a}$-filtration. > [!proof]+ Proof. ([[Artin-Rees Lemma]]) > $R^{*}=\bigoplus_{n \geq 0}\mathfrak{a}^{n}$ denotes the [[Rees ring]] of $(R, \mathfrak{a})$. [[Rees ring|Note]] that $R^{*}$ is [[Noetherian ring|Noetherian]] because $R$ is. > First need to verify $(N \cap M_{\ell})_{\ell \geq 0}$ is an $\mathfrak{a}$-filtration at all. We have $\mathfrak{a}(N \cap M_{\ell}) \subset \underbrace{ N }_{ N \text{ a submod.} } \cap \underbrace{ \mathfrak{a}M_{\ell} }_{ (M_{\ell}) \text{ an } \mathfrak{a}\text{-filt.} } \subset \overbrace{ N \cap M_{\ell + 1} }^{ := N _{\ell} },$ so it indeed is. > > Now, [[Rees ring|the]] [[graded module|graded]] $R^{*}$-[[module]] $N^{*}=\bigoplus_{\ell \geq 0}N_{\ell}$ is an $M^{*}$-[[submodule]]: it is closed wrt the $R^{*}$-[[module|action]] because it is an $\mathfrak{a}$-filtration. We have assumed $M^{*}$ is $\mathfrak{a}$-stable, so by the [[the Rees characterization of stable filtrations over Noetherian rings]] it is finitely generated over $R^{*}$. By [[module is Noetherian (resp. Artinian) iff submodule and quotient is|finitely generated module over a Noetherian ring is Noetherian]], $M^{*}$ is thus a [[Noetherian module|Noetherian]] $R^{*}$-[[module]]. The [[submodule]] $N^{*}$ is therefore [[submodule generated by a subset|finitely generated]], and now from (the other direction of) [[the Rees characterization of stable filtrations over Noetherian rings]] we may conclude $N^{*}$ is a stable $\mathfrak{a}$-filtration. Summary: - Consider the Rees ring $R^{*}\text{ of }(R, \mathfrak{a})$; note that it is Noetherian because $R$ is Noetherian. - First briefly justify that $(N_{\ell})_{\ell \geq 0}=(N \cap M_{\ell})_{\ell \geq 0}$ is, indeed, an $\mathfrak{a}$-filtration - Justify that $N^{*}=\bigoplus_{\ell \geq 0} N_{\ell}$ is a submodule of $M^{*}$. If we can show $N^{*}$ is finitely-generated, then we are done by the [[the Rees characterization of stable filtrations over Noetherian rings]] - Do this by showing $M^{*}$ is Noetherian, which it is: it is a finitely-generated $R^{*}$-module [[the Rees characterization of stable filtrations over Noetherian rings|because it is]] $\mathfrak{a}$-stable, and a finitely generated module over a Noetherian ring is Noetherian ---- #### Secondly, note that each $M_{\ell}$ is a [[submodule generated by a subset|finitely generated]] $R$-[[module]], by [[module is Noetherian (resp. Artinian) iff submodule and quotient is|finitely generated module over a Noetherian ring is Noetherian]]. ----- #### 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 > ```