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SpectralSequences : Index
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A spectral sequence which fails to degenerate quickly
associatedGradedHomologyObject
-- compute the associated graded homology object
associatedGradedHomologyObject(ZZ,ZZ,FilteredComplex)
-- compute the associated graded homology object
Balancing Tor
basis(List,SpectralSequencePage)
-- generators of a particular degree
basis(ZZ,SpectralSequencePage)
-- generators of a particular degree
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
Computing the Serre Spectral Sequence associated to a Hopf Fibration
connectingMorphism
-- use spectral sequences to compute connecting morphisms
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
Edge homomorphisms
edgeComplex
-- the edge homomorphisms
edgeComplex(SpectralSequence)
-- the edge homomorphisms
Example 1
-- Easy example of a filtered simplicial complex
Example 2
-- Easy example of a filtered simplicial complex
Example 3
-- Easy example of a filtered simplicial complex
Examples of change of rings Spectral Sequences
Examples of filtered complexes and spectral sequences
-- How to use this package
expression(SpectralSequence)
Filtered complexes and simplicial complexes
filtered complexes and spectral sequences from simplicial complexes
FilteredComplex
-- the type of all filtered complexes
filteredComplex
-- make a filtered complex
FilteredComplex ** ChainComplex
-- filtered tensor product of complexes
FilteredComplex ^ InfiniteNumber
-- the filtered pieces
FilteredComplex ^ ZZ
-- the filtered pieces
FilteredComplex _ InfiniteNumber
-- the filtered pieces
FilteredComplex _ ZZ
-- the filtered pieces
filteredComplex(...,ReducedHomology=>...)
filteredComplex(...,Shift=>...)
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
-- compute the filtered homology object
filteredHomologyObject(ZZ,ZZ,FilteredComplex)
-- compute the filtered homology object
Filtrations and homomorphism complexes
Filtrations and tensor product complexes
hilbertPolynomial(SpectralSequencePage)
-- the Hilbert polynomial of a spectral sequence page
Hom(ChainComplex,FilteredComplex)
-- the filtered Hom complex
Hom(FilteredComplex,ChainComplex)
-- the filtered Hom complex
homologyIsomorphism
-- compute the homology isomorphism
homologyIsomorphism(SpectralSequence,ZZ,ZZ,ZZ)
-- the homology isomorphism
How to make filtered complexes from chain complex maps
I-adic filtrations of chain complexes and their spectral sequences
Identifying anti-podal points of the two sphere
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
minimalPresentation(SpectralSequencePage)
-- a minimal presentation of a spectral sequence page
net(FilteredComplex)
net(Page)
net(PageMap)
net(SpectralSequence)
net(SpectralSequencePage)
netPage
-- display a small portion of a given Spectral Sequence page
netPage(Page,List,List)
-- display a small portion of a given Spectral Sequence page
new Page
Page
-- the type of all pages
page
Page _ List
page(...,Prune=>...)
page(List,List,Page)
page(SpectralSequencePage)
PageMap
-- the type of all page maps
pageMap
PageMap _ List
prune(SpectralSequence)
-- a minimal presentation of a spectral sequence
prune(SpectralSequencePage)
-- a minimal presentation of a spectral sequence page
pruningMaps
-- compute the pruning maps on a spectral sequence page
pruningMaps(SpectralSequencePage)
-- compute the pruning maps on a spectral sequence page
ReducedHomology
-- name for an optional argument
ring(Page)
Seeing Cancellations
Shift
-- name for an optional argument
sourcePruningMap
Spectral sequences and connecting morphisms
Spectral sequences and hypercohomology calculations
Spectral sequences and non-Koszul syzygies
SpectralSequence
-- the type of all spectral sequences
spectralSequence
-- construct a spectral sequence
SpectralSequence ^ InfiniteNumber
-- the infinity page of a spectral sequence
SpectralSequence ^ ZZ
-- the kth page of a spectral sequence
SpectralSequence _ InfiniteNumber
-- the infinity page of a spectral sequence
SpectralSequence _ ZZ
-- the kth page of a spectral sequence
spectralSequence(...,Prune=>...)
spectralSequence(FilteredComplex)
-- construct a spectral sequence from a filtered complex
SpectralSequencePage
-- the type of all spectral sequence pages
spectralSequencePage
-- construct a spectral sequence page 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(...,Prune=>...)
spectralSequencePage(FilteredComplex,ZZ)
-- construct a spectral sequence page from a filtered complex
SpectralSequencePageMap
-- the type of all spectral sequence page maps
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
spectralSequencePageMap(...,Prune=>...)
spectralSequencePageMap(FilteredComplex,ZZ)
-- compute the maps on a spectral sequence page
SpectralSequences
-- a package for working with filtered complexes and spectral sequences
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(Page)
-- which spots does the given page 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)
targetPruningMap
tensor(RingMap,ChainComplex)
-- tensor product of a chain complex by a ring map
The fibration of the Klein Bottle over the sphere with fibers the sphere
The trivial fibration over the sphere with fibers the sphere
truncate(ChainComplex,ZZ)
-- compute the hard truncation of a chain complex