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SpectralSequences :: FilteredComplex

FilteredComplex -- the type of all filtered complexes

Description

An ascending filtration of a bounded (homological, lower index, or degree -1) chain complex C : …→Ci →Ci - 1 →… is an ordered family of chain subcomplexes FC : …⊆Fn - 1 C ⊆Fn C ⊆…. Such a filtration is said to be bounded if Fs C = C for all sufficiently large s and Ft C = 0 for all sufficiently small t.

Alternatively, a descending filtration of a bounded (cohomological, or upper index, or degree 1) chain complex C : …→Ci →Ci + 1 →… is an ordered family of subchain complexes FC : …⊆Fn + 1 C ⊆Fn C ⊆…. Such a filtration is said to be bounded if Fs C = 0 for all sufficiently large s and Ft C = C for all sufficiently small t.

The type FilteredComplex is a data type for working with bounded filtrations of bounded chain complexes.

Caveat

By assumption all filtered complexes arise from bounded filtrations of bounded chain complexes. Filtrations on degree -1 chain complexes are ascending. Filtrations on degree 1 chain complexes are descending.

See also

Functions and methods returning a filtered chain complex :

Methods that use a filtered chain complex :

For the programmer

The object FilteredComplex is a type, with ancestor classes HashTable < Thing.