Spectral_Sequences

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title: Spectral Sequences permalink: wiki/Spectral_Sequences/ layout: wiki

General Goal

Implement the data types and methods for creating and manipulating the spectral sequences arising from a filtered chain complex.

Specific Goals

Implemented the following data types:

1. FilteredComplex
2. SpectralSequence
3. SpectralSequenceSheet

Implemented the following methods:

1. truncate(ZZ, ChainComplex) := ChainComplex
2. Hom(ChainComplex, ChainComplex) := ChainComplex
3. FilteredComplex ^ ZZ := ChainComplex
4. filteredComplex List := FilteredComplex (given a list of chain complex maps)
5. filteredComplex ChainComplex := FilteredComplex (needs an option to cover various cases?)
6. Hom(FilteredComplex, ChainComplex) := FilteredComplex
7. Hom(ChainComplex, FilteredComplex) := FilteredComplex
8. FilteredComplex ** ChainComplex := FilteredComplex
9. ChainComplex ** FilteredComplex := FilteredComplex
10. spectralSequence FilteredComplex := SpectralSequence
11. SpectralSequence _ ZZ := SpectralSequenceSheet
12. SpectralSequence _ * := SpectralSequenceSheet
13. support SpectralSequenceSheet := something like BettiTally ? (which modules are nonzero)
14. minimalPresentation SpectralSequenceSheet := SpectralSequenceSheet

Required changes:

1. need chainComplex(SpectralSequenceSheet, List) := ChainComplex -- What do you mean here? What is the purpose of this function? - Thanh
2. need an isWellDefined command that checks to make sure the ChainComplexMaps are well-defined -- Implemented - Thanh
3. need SpectralSequenceSheet . dd := ChainComplexMap
4. need to make the output/net look better
5. want both cohomological and homological spectral sequences
6. convert SpectralSequenceSheets into a type of ChainComplex
7. can we get the associated long exact sequences in special cases
8. when Hom(Matrix, Module) is implemented for non - free modules --- I implemented already - Thanh
9. need to add tests

Examples wanted:

1. change of ring spectral sequences for Tor -- Have it already
2. change of ring spectral sequences for Ext -- can we without injectives?
3. spectral sequence relating local and global Ext
4. simplicial complex filtered by its skeleton
5. Serre spectral sequence for the Hopf filtration

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