FilteredComplex -- the type of all filtered complexes

Description

An ascending filtration of a bounded (homological, lower index, or degree $-1$) chain complex $C : \dots \rightarrow C_i \rightarrow C_{i - 1} \rightarrow \dots$ is an ordered family of chain subcomplexes $FC : \dots \subseteq F_{n - 1} C \subseteq F_n C \subseteq \dots $. 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 : \dots \rightarrow C^i \rightarrow C^{i + 1} \rightarrow \dots $ is an ordered family of subchain complexes $FC : \dots \subseteq F^{n + 1} C \subseteq F^n C \subseteq \dots$. 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.