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OldChainComplexes : Table of Contents
OldChainComplexes
-- legacy implementation of chain complexes
free resolutions of modules
computing resolutions
extracting information from chain complexes
making chain complexes by hand
manipulating chain complexes
maps between chain complexes
GradedModule
-- the class of all graded modules
gradedModule(List)
-- make a graded module
complete(GradedModule)
GradedModule Array
-- degree shift
max(GradedModule)
min(GradedModule)
GradedModule ** GradedModule
ChainComplex ** GradedModule
-- tensor product
GradedModule ** ChainComplex
-- tensor product
GradedModule ** Module
length(GradedModule)
-- length of a graded module
betti(GradedModule)
-- Betti diagram showing the of degrees in a graded module or chain complex
betti(...,Minimize=>...)
-- minimal betti numbers of a non-minimal free resolution
Ring ^ BettiTally
-- construct a chain complex with prescribed Betti table
GradedModuleMap
-- the class of all maps between graded modules
gradedModuleMap
-- make a map of graded modules
GradedModuleMap | GradedModuleMap
GradedModuleMap || GradedModuleMap
source(GradedModuleMap)
-- find the source of a map of graded modules
target(GradedModuleMap)
-- find the target of a map of graded modules
ChainComplex
-- the class of all chain complexes
chainComplex
-- make a chain complex
chainComplex(Ring)
-- make an empty chain complex over a ring
chainComplex(List)
-- make a chain complex
chainComplex(Matrix)
-- make a small chain complex
chainComplex(GradedModule)
-- make a chain complex from a graded module
dd
-- differential in a chain complex
status(ChainComplex)
-- status of a resolution computation
complete(ChainComplex)
-- complete the internal parts
length(ChainComplex)
-- length of a chain complex or graded module
ChainComplex ++ ChainComplex
-- direct sum
components(ChainComplex)
ChainComplex Array
-- degree shift
dual(ChainComplex)
-- dual
sum(ChainComplex)
-- direct sum of the components of a chain complex
new ChainComplex
-- make a new chain complex from scratch
ChainComplex ** ChainComplex
-- tensor product
ChainComplex ** Ring
HH^ZZ ChainComplex
-- cohomology of a chain complex
HH ChainComplex
-- homology of a chain complex
HH_ZZ ChainComplex
-- homology of a chain complex
poincare(ChainComplex)
-- assemble degrees of a chain complex into a polynomial
poincareN(ChainComplex)
-- assemble degrees into polynomial
regularity(ChainComplex)
-- compute the Castelnuovo-Mumford regularity
ChainComplex ^ ZZ
-- access member, cohomological degree
ChainComplex _ ZZ
-- component
ChainComplex _ ZZ = Thing
-- install component of chain complex
ChainComplex ^ Array
-- projection onto summand
ChainComplex _ Array
-- inclusion from summand
ChainComplexMap
-- the class of all maps between chain complexes
cone(ChainComplexMap)
ChainComplexMap Array
-- degree shift
sum(ChainComplexMap)
-- direct sum of the components of a chain map
ChainComplexMap ** ChainComplex
-- tensor product
ChainComplex ** ChainComplexMap
-- tensor product
ChainComplexMap ** ChainComplexMap
-- tensor product
dual(ChainComplexMap)
-- dual of a chain complex
HH^ZZ ChainComplexMap
-- cohomology of a chain complex map
HH_ZZ ChainComplexMap
-- homology of a chain complex map
HH ChainComplexMap
-- homology of a chain complex map
kernel(ChainComplexMap)
-- kernel of a chain complex map
source(ChainComplexMap)
-- find the source of a map of chain complexes
target(ChainComplexMap)
-- find the target of a map of chain complexes
transpose(ChainComplexMap)
-- transpose a map of chain complexes
ChainComplexMap ^ ZZ
-- iterated composition
ChainComplexMap _ ZZ
-- component map
ChainComplexMap _ Array
-- component of map corresponding to summand of source
ChainComplexMap _ ZZ = Thing
-- install component of chain complex map
extend(ChainComplex,ChainComplex,Matrix)
Hom(Module,ChainComplex)
-- the Hom functor
map(ChainComplex,ChainComplex,Function)
-- make a map of chain complexes
nullhomotopy
-- make a null homotopy
Resolution
-- the class of all resolution computations
resolution
-- projective resolution
resolution(Ideal)
-- compute a projective resolution of (the quotient ring corresponding to) an ideal
resolution(Matrix)
-- compute the comparison map between resolutions of the source and target of a module map represented by a matrix
resolution(Module)
-- compute a free resolution of a module
resolution(...,DegreeLimit=>...)
-- compute only up to this degree
resolution(...,SyzygyLimit=>...)
-- stop when this number of syzygies is reached
resolution(...,PairLimit=>...)
-- stop when this number of pairs has been handled
resolution(...,StopBeforeComputation=>...)
-- whether to stop the computation immediately
resolution(...,LengthLimit=>...)
-- stop when the resolution reaches this length
resolution(...,HardDegreeLimit=>...)
resolution(...,Strategy=>...)
resolution(...,SortStrategy=>...)
FastNonminimal
-- compute a non-minimal graded free resolution
syzygyScheme
-- construct a syzygy scheme