Filtrationsgrad

Filtrationsgrad is a term used in the mathematics of filtrations to denote the length or degree of a filtration on an object such as a vector space, module, or group. Given a filtration on an object A, typically written as a chain of subobjects 0 = F_0 ⊆ F_1 ⊆ ... ⊆ F_n = A (a finite filtration), the filtrationsgrad is the index n of the last step, i.e., the length of the filtration. In the associated graded construction, Gr_i = F_i / F_{i-1} for i = 1, ..., n, and the highest index with Gr_i ≠ 0 is the filtrationsgrad.

If the filtration is infinite, the filtrationsgrad may be infinite, indicating an unbounded depth. In practice, one often works with finite filtrations and denotes their length by n; the number of nontrivial graded pieces equals n when the inclusions are strict.

Examples: A finite-dimensional vector space V with a chain 0 ⊂ V_1 ⊂ ... ⊂ V_k = V has filtrationsgrad k. In a spectral-sequence context or in mixed Hodge theory, filtrationsgrad serves as a measure of the complexity of the filtration.

Properties: The filtrationsgrad depends on the chosen filtration and is not intrinsic to the object alone; refining a filtration by inserting more steps may increase the filtrationsgrad, while coarsening lowers it. It is related to, but distinct from, the dimension of the associated graded object.

See also: Filtration, associated graded, length of a filtration, spectral sequence, graded object.