# Filtered module

A module $M$ endowed with an increasing or decreasing filtration, that is, an increasing or decreasing family of submodules $( M _ {n} ) _ {n \in \mathbf Z }$. A filtration is called exhaustive if $M = \cup _ {n \in \mathbf Z } M _ {n}$, and separable if $\cap _ {n \in \mathbf Z } M _ {n} = 0$. If $N$ is a submodule of a filtered module $M$, then filtrations are defined on $N$ and $M/N$ in a natural way. If $M = \sum _ {n \in \mathbf Z } M _ {(} n)$ is a graded module, then the submodules $M _ {n} = \sum _ {i \geq n } M _ {(} i)$ define an exhaustive separable decreasing filtration on $M$. Conversely, with every filtered module $M$ endowed, for example, with a decreasing filtration, there is associated the graded module
$$\mathop{\rm gr} M = \ \oplus _ {n \in \mathbf Z } \mathop{\rm gr} _ {n} M,$$
where $\mathop{\rm gr} _ {n} M = M _ {n} /M _ {n + 1 }$. A filtration $( M _ {n} ) _ {n \in \mathbf Z }$ constitutes a fundamental system of neighbourhoods of zero. Its topology is separable if and only if the filtration is separable, and discrete if and only if $M _ {n} = 0$ for some $n$.