---- > [!definition] Definition. ([[associated graded module to an ideal]]) > Let $R$ be a ([[commutative ring|commutative]]) [[ring]], $\mathfrak{a} \subset R$ an [[ideal]]. Denote by $G_{\mathfrak{a}}(R)$ the [[associated graded ring to an ideal|associated graded ring]]. > > Let $(M_{n})_{n \geq 0}$ be an $\mathfrak{a}$-[[filtration|filtration]]. The **associated graded module** is a [[graded module|graded]] $G_{\mathfrak{a}}(R)$-[[module]] $G(M):= \bigoplus_{n \geq 0} \frac{M_{n}}{M_{n+1}}$ where the graded ring action is specified on homogeneous elements via $(\overbrace{ x + \mathfrak{a}^{n+1} }^{ \in \mathfrak{a}^{n} / \mathfrak{a}^{n+1} }) \cdot (\overbrace{ m + M_{\ell+1} }^{ \in M_{\ell} / M_{\ell+1} }):= xm + M^{n + \ell + 1}.$ > By construction, $\big( G_{\mathfrak{a}}(R) \big)_{n} \big( G(M) \big)_{\ell}= \frac{\mathfrak{a}^{n}}{\mathfrak{a}^{n+1}} \cdot \frac{M_{\ell}}{M_{\ell+1}} \subset \frac{M_{n+\ell}}{M_{n + \ell + 1}}=\big( G(M) \big)_{n+\ell},$ > so this is indeed a [[graded module|graded]] $G_{\mathfrak{a}}(R)$-[[module]]. ---- #### ---- #### 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 > ```