Ascending filtration ($F_{-1}=0\subseteq F_0\subseteq F_1\subseteq \cdots$)

An ascending filtration is an increasing chain of additive subgroups, or vector subspaces when a scalar field is fixed. It is exhaustive if their union is the whole object. For a filtered ring, multiplicative compatibility means $F_{i}R{F}_{j}R\subseteq F_{i+j}R$. For an ideal $I$, intersecting each level with $I$ gives the induced filtration. Its associated graded module has components $(I\cap F_{n}R)/(I\cap F_{n-1}R)$.