Filtered algebra

In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.

A filtered algebra over the field ${\displaystyle k}$ is an algebra ${\displaystyle (A,\cdot )}$ over ${\displaystyle k}$ which has an increasing sequence ${\displaystyle \{0\}\subset F_{0}\subset F_{1}\subset \cdots \subset F_{i}\subset \cdots \subset A}$ of subspaces of ${\displaystyle A}$ such that

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