modulesfiltration

A filtration on a module M over a ring R is a sequence of submodules (F_i) indexed by a directed set (commonly the integers) such that 0 = F_0 ≤ F_1 ≤ F_2 ≤ ... ≤ F_n = M in the finite case, or an increasing/decreasing chain indexed by ordinals in the general case. Each F_i is a submodule of M, and the filtration is said to be exhaustive if it eventually reaches M and separated if the intersection of all F_i is zero. When the index set is the integers with F_i = M for i large enough, the filtration is called exhaustive; when the submodules eventually stabilize, it is called finite.

The successive quotients F_i / F_{i-1} are called the factors of the filtration. The associated graded module gr(M) is defined as the direct sum of these factors, gr(M) = ⊕_i (F_i / F_{i-1}). Filtrations can be finite or infinite; a finite filtration has a well-defined length equal to the number of nontrivial factors.

Filtrations are used to study the structure of modules by breaking them into simpler pieces. Common examples include the radical filtration (rad^i M), the socle filtration, and any composition series when it exists; the length of such filtrations yields invariants like the Loewy length or the composition length. In Noetherian or Artinian settings, finite length results provide a robust framework for analysis. Filtrations also interact with homological methods, giving rise to spectral sequences and connections between filtered and graded objects.

See also: filtration, graded module, composition series, radical series, Loewy length, spectral sequence, highest weight category.