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. where, for all x ∈ F n {displaystyle xin F_{n}} and y ∈ F m {displaystyle yin F_{m}} . (More precisely, the multiplication map G ( A ) × G ( A ) → G ( A ) {displaystyle {mathcal {G}}(A) imes {mathcal {G}}(A) o {mathcal {G}}(A)} is combined from the maps 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 k {displaystyle k} is an algebra ( A , ⋅ ) {displaystyle (A,cdot )} over k {displaystyle k} which has an increasing sequence { 0 } ⊆ F 0 ⊆ F 1 ⊆ ⋯ ⊆ F i ⊆ ⋯ ⊆ A {displaystyle {0}subseteq F_{0}subseteq F_{1}subseteq cdots subseteq F_{i}subseteq cdots subseteq A} of subspaces of A {displaystyle A} such that