Associated Graded Space

The associated graded space of a vector space with an increasing filtration is the direct sum of its successive quotient vector spaces,

Its degree- component records the elements introduced at filtration level , modulo those already present at level . In particular, if is finite-dimensional, that component has dimension .

For a decreasing filtration , the corresponding convention is

The associated graded module is the corresponding construction for a module, including a filtration induced by powers of an ideal. For Vassiliev invariants, the increasing filtration is by order at most (Kneissler 1997).