FilteredComplex -- the type of all filtered complexes

Description

An ascending filtration of a bounded (homological, lower index, or degree -1) chain complex C : …→C_i →C_{i - 1} →… is an ordered family of chain subcomplexes FC : …⊆F_{n - 1} C ⊆F_n C ⊆…. Such a filtration is said to be bounded if F_s C = C for all sufficiently large s and F_t 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 →C^{i + 1} →… is an ordered family of subchain complexes FC : …⊆F^{n + 1} C ⊆Fⁿ C ⊆…. Such a filtration is said to be bounded if F^s C = 0 for all sufficiently large s and F^t 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.

Functions and methods returning a filtered chain complex :

ChainComplex ** FilteredComplex -- filtered tensor product of complexes
FilteredComplex ** ChainComplex, see ChainComplex ** FilteredComplex -- filtered tensor product of complexes
filteredComplex(ChainComplex) -- obtain a filtered complex from a chain complex
filteredComplex(Ideal,ChainComplex,ZZ) -- I-adic filtrations of chain complexes
filteredComplex(List) -- obtain a filtered complex from a list of chain complex maps or a nested list of simplicial complexes
filteredComplex(SpectralSequence) -- obtain the filtered complex associated to the spectral sequence
Hom(ChainComplex,FilteredComplex), see Hom(FilteredComplex,ChainComplex) -- the filtered Hom complex
Hom(FilteredComplex,ChainComplex) -- the filtered Hom complex

Methods that use a filtered chain complex :

associatedGradedHomologyObject(ZZ,ZZ,FilteredComplex), see associatedGradedHomologyObject -- compute the associated graded homology object
chainComplex(FilteredComplex) -- the ambient chain complex of a filtered complex
FilteredComplex ^ InfiniteNumber, see FilteredComplex ^ ZZ -- the filtered pieces
FilteredComplex ^ ZZ -- the filtered pieces
FilteredComplex _ InfiniteNumber, see FilteredComplex _ ZZ -- the filtered pieces
FilteredComplex _ ZZ -- the filtered pieces
filteredHomologyObject(ZZ,ZZ,FilteredComplex) -- compute the filtered homology object
inducedMap(FilteredComplex,ZZ) -- the i th inclusion map in a filtered complex
max(FilteredComplex) -- maximum spot where the given filtered complex has a module.
min(FilteredComplex) -- minimum spot where the given filtered complex has a module.
net(FilteredComplex)
spectralSequence(FilteredComplex) -- construct a spectral sequence from a filtered complex
spectralSequencePage(FilteredComplex,ZZ) -- construct a spectral sequence page from a filtered complex
spectralSequencePageMap(FilteredComplex,ZZ), see spectralSequencePageMap -- compute the maps on a spectral sequence page
spots(FilteredComplex) -- which spots does the given filtered complex has a module.
support(FilteredComplex)

For the programmer

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