SpectralSequences -- a package for working with filtered complexes and spectral sequences

Description

Spectral sequences, although notoriously technical, are very useful in applications---especially when they degenerate quickly. By contrast, little is known about their general structure when they fail to degenerate quickly. Even in cases when the terms in the spectral sequences are well understood, the maps remain mysterious. One of the motivations behind this package is to shed light on spectral sequences through examples. Its purpose is to allow for effective calculations of particular kinds of spectral sequences. As one general situation, which illustrates some capabilities of this package, let k be a computable field, S a k-algebra of finite type, C a bounded chain complex of finitely generated S-modules, and FC a bounded ascending filtration of C. This package is capable of computing, under these assumptions, the spectral sequence---especially the differentials on each page---determined by FC.

Constructors used in this package

Other examples which illustrate this package

More easy topological examples

Authors

Version

This documentation describes version 1.0 of SpectralSequences.

Source code

The source code from which this documentation is derived is in the file SpectralSequences.m2.

Exports

Types
- FilteredComplex -- the type of all filtered complexes
- Page -- the type of all pages
- PageMap -- the type of all page maps
- SpectralSequence -- the type of all spectral sequences
- SpectralSequencePage -- the type of all spectral sequence pages
- SpectralSequencePageMap -- the type of all spectral sequence page maps
Functions and commands
- associatedGradedHomologyObject -- compute the associated graded homology object
- connectingMorphism -- use spectral sequences to compute connecting morphisms
- edgeComplex -- the edge homomorphisms
- filteredComplex -- make a filtered complex
- filteredHomologyObject -- compute the filtered homology object
- homologyIsomorphism -- compute the homology isomorphism
- netPage -- display a small portion of a given Spectral Sequence page
- page
- pruningMaps -- compute the pruning maps on a spectral sequence page
- spectralSequence -- construct a spectral sequence
- spectralSequencePage -- construct a spectral sequence page from a filtered complex
- spectralSequencePageMap -- compute the maps on a spectral sequence page
- spots -- which spots does the given page has a module.
Methods
- associatedGradedHomologyObject(ZZ,ZZ,FilteredComplex) -- see associatedGradedHomologyObject -- compute the associated graded homology object
- basis(List,SpectralSequencePage) -- generators of a particular degree
- basis(ZZ,SpectralSequencePage) -- see basis(List,SpectralSequencePage) -- generators of a particular degree
- ChainComplex ** FilteredComplex -- filtered tensor product of complexes
- FilteredComplex ** ChainComplex -- see ChainComplex ** FilteredComplex -- filtered tensor product of complexes
- chainComplex(FilteredComplex) -- the ambient chain complex of a filtered complex
- chainComplex(SpectralSequence) -- the underlying chain complex of a Spectral Sequence
- connectingMorphism(ChainComplexMap,ZZ) -- use spectral sequences to compute connecting morphisms
- degree(Page)
- describe(Page) -- real description
- describe(PageMap) -- real description
- describe(SpectralSequence) -- real description
- describe(SpectralSequencePage) -- real description
- describe(SpectralSequencePageMap) -- real description
- edgeComplex(SpectralSequence) -- see edgeComplex -- the edge homomorphisms
- expression(SpectralSequence)
- 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
- 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
- filteredHomologyObject(ZZ,ZZ,FilteredComplex) -- compute the filtered homology object
- hilbertPolynomial(SpectralSequencePage) -- the Hilbert polynomial of a spectral sequence page
- Hom(ChainComplex,FilteredComplex) -- see Hom(FilteredComplex,ChainComplex) -- the filtered Hom complex
- Hom(FilteredComplex,ChainComplex) -- the filtered Hom complex
- homologyIsomorphism(SpectralSequence,ZZ,ZZ,ZZ) -- the homology isomorphism
- 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.
- minimalPresentation(SpectralSequence) -- a minimal presentation of a spectral sequence
- prune(SpectralSequence) -- see minimalPresentation(SpectralSequence) -- a minimal presentation of a spectral sequence
- minimalPresentation(SpectralSequencePage) -- a minimal presentation of a spectral sequence page
- prune(SpectralSequencePage) -- see minimalPresentation(SpectralSequencePage) -- a minimal presentation of a spectral sequence page
- net(FilteredComplex)
- net(Page)
- net(PageMap)
- net(SpectralSequence)
- net(SpectralSequencePage)
- netPage(Page,List,List) -- see netPage -- display a small portion of a given Spectral Sequence page
- new Page
- page(List,List,Page) -- see page
- Page _ List
- page(SpectralSequencePage)
- PageMap _ List
- pruningMaps(SpectralSequencePage) -- compute the pruning maps on a spectral sequence page
- ring(Page)
- SpectralSequence ^ InfiniteNumber -- the infinity page of a spectral sequence
- SpectralSequence _ InfiniteNumber -- see SpectralSequence ^ InfiniteNumber -- the infinity page of a spectral sequence
- SpectralSequence ^ ZZ -- the kth page of a spectral sequence
- SpectralSequence _ ZZ -- the kth page of a spectral sequence
- spectralSequence(FilteredComplex) -- construct a spectral sequence from a filtered complex
- SpectralSequencePage ^ List -- the module in the i,j position on the page
- SpectralSequencePage _ List -- the module in the i,j position on the page
- 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
- SpectralSequencePageMap ^ List -- the p,q th map on of a spectral sequence page
- SpectralSequencePageMap _ List -- The p,q th map on of a spectral sequence page
- spots(Page) -- see spots -- which spots does the given page has a module.
- spots(ChainComplex) -- which spots does the given chain complex has a module.
- spots(FilteredComplex) -- which spots does the given filtered complex has a module.
- spots(PageMap)
- support(ChainComplex) -- nonzero parts of a chain complex
- support(FilteredComplex)
- support(Page) -- which non-zero modules appear in the given page.
- support(PageMap)
- support(SpectralSequencePage)
- tensor(RingMap,ChainComplex) -- tensor product of a chain complex by a ring map
- truncate(ChainComplex,ZZ) -- compute the hard truncation of a chain complex
Symbols
- pageMap
- ReducedHomology -- name for an optional argument
- Shift -- name for an optional argument
- sourcePruningMap
- targetPruningMap

For the programmer

The object SpectralSequences is a package.