---- > [!definition] > Let $I$ be a [[poset|totally ordered set]] of 'indices'. Let $S$ be some object in a [[category]] $\mathsf{C}$. A family of [[subobject|subobjects]] $(S_{i})_{i \in I}$ indexed by $I$ is called a **filtration** of $S$ provided $i \leq j$ in $I$ implies $S_{i} \subset S_{j}$. ^definition > [!basicexample] > - Let $W=(W_{n})_{n \in \mathbb{N}}$ be a discrete-time [[stochastic process]] carried by [[probability space|triple]] $(\Omega, \mathcal{F}, \mathbb{P})$. The **natural filtration** $\{ \mathcal{W}_{n} \}_{n \in \mathbb{N}}$ is defined by taking $\mathcal{W_{n}}$ to be the smallest filtration of [[σ-algebra|σ-algebras]] wrt which $W$ is [[adapted stochastic process|adapted]] ([[σ-algebra generated by a set collection|namely]], $\mathcal{W}_{n}:=\sigma(W_{0}, \dots , W_{n})$). > - [[(abstract) simplicial complex]] - CAT notes > [!definition] Definition. ([[filtration|Of an R-module]]) > Let $R$ be a ([[commutative ring|commutative]]) [[ring]] and $M$ be an $R$-[[module]]. A **filtration** of $M$ is descending sequence of [[submodule|submodules]] $(M_{n})_{n \geq 1}$, $M_{n} \supset M_{n+1}$, where $M=M_{0}$. > For $\mathfrak{a}$ an [[ideal]] of $R$, we call $(M_{n})_{n \geq 0}$ an **$\mathfrak{a}$-filtration** if additionally $\mathfrak{a}M_{n} \subset M_{n+1}$ for all $n$. > An $\mathfrak{a}$-filtration $(M_{n})_{n \geq 0}$ is **stable** (or **$\mathfrak{a}$-stable**) if there exists $n_{0} \geq0$ such that $\mathfrak{a}M_{n}=M_{n+1}$ for all $n > n_{0}$. The canonical example of a stable $\mathfrak{a}$-filtration is $(\mathfrak{a}^{n}M)_{n \geq 0}$. [[up to equivalence, there is only one stable filtration for an ideal|In a sense, it is the only example.]] > > > > [!basicproperties] Basic Properties. > > - If $R$ is [[Noetherian ring|Noetherian]], then we can characterize $\mathfrak{a}$-stability of $(M_{n})_{n \geq 0}$ in terms of the [[Rees ring]] $R^{*}$. See [[the Rees characterization of stable filtrations over Noetherian rings]]. > ^properties ----- #### ---- #### 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 > ```