Complex -- the class of all chain complexes

Description

A complex is a sequence of objects $C_i$, connected by maps $dd^C_i : C_i \rightarrow C_{i-1}$ such that the composition of any two consecutive maps is zero.

TODO: more needs to be added here explaining how to used complexes. and links to "landing pages".

Functions and methods returning a complex :

canonicalTruncation(Complex,InfiniteNumber,InfiniteNumber) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
canonicalTruncation(Complex,InfiniteNumber,ZZ) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
canonicalTruncation(Complex,Nothing,ZZ) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
canonicalTruncation(Complex,Sequence) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
canonicalTruncation(Complex,ZZ,InfiniteNumber) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
canonicalTruncation(Complex,ZZ,Nothing) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
coimage(ComplexMap) -- make the coimage of a map of complexes
cokernel(ComplexMap) -- make the cokernel of a map of complexes
complex(List) -- see complex -- make a chain complex
Complex ** Complex -- tensor product of complexes
Complex ** Module -- see Complex ** Complex -- tensor product of complexes
Module ** Complex -- see Complex ** Complex -- tensor product of complexes
tensor(Complex,Complex) -- see Complex ** Complex -- tensor product of complexes
complex(ChainComplex) -- translate between data types for chain complexes
complex(Complex) -- make a complex by reindexing the terms of the complex
complex(ComplexMap) -- make a complex by specifying the differential
complex(HashTable) -- make a chain complex
complex(Ideal) -- see complex(Module) -- make a chain complex of length zero
complex(Module) -- make a chain complex of length zero
complex(Ring) -- see complex(Module) -- make a chain complex of length zero
cone(ComplexMap) -- make the mapping cone of a morphism of chain complexes
cylinder(ComplexMap) -- see cylinder -- make the mapping cylinder of a morphism of chain complexes
Complex ++ Complex -- see directSum(Complex) -- direct sum of complexes
dual(Complex) -- make the dual of a complex
freeResolution(Ideal) -- see freeResolution -- compute a free resolution of a module or ideal
freeResolution(Module) -- see freeResolution -- compute a free resolution of a module or ideal
freeResolution(MonomialIdeal) -- see freeResolution -- compute a free resolution of a module or ideal
gradedModule(Complex) -- a new complex in which the differential is zero
HH Complex -- homology of a complex
Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
Hom(Complex,Module) -- see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
Hom(Complex,Ring) -- see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
Hom(Module,Complex) -- see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
Hom(Ring,Complex) -- see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
image(ComplexMap) -- make the image of a map of complexes
kernel(ComplexMap) -- make the kernel of a map of complexes
koszulComplex(List) -- see koszulComplex(Matrix) -- makes the Koszul complex
koszulComplex(Matrix) -- makes the Koszul complex
longExactSequence(ComplexMap,ComplexMap) -- make the long exact sequence in homology
minimalPresentation(Complex) -- minimal presentation of all terms in a complex
prune(Complex) -- see minimalPresentation(Complex) -- minimal presentation of all terms in a complex
naiveTruncation(Complex,InfiniteNumber,InfiniteNumber) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
naiveTruncation(Complex,InfiniteNumber,ZZ) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
naiveTruncation(Complex,Nothing,ZZ) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
naiveTruncation(Complex,Sequence) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
naiveTruncation(Complex,ZZ,InfiniteNumber) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
naiveTruncation(Complex,ZZ,Nothing) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
part(List,Complex) -- extract a graded component of a complex
part(ZZ,Complex) -- see part(List,Complex) -- extract a graded component of a complex
ring(ComplexMap) -- see ring(Complex) -- access the ring of a complex or a complex map
Complex ** Ring -- see RingMap ** Complex -- tensor a complex along a ring map
Complex ** RingMap -- see RingMap ** Complex -- tensor a complex along a ring map
Ring ** Complex -- see RingMap ** Complex -- tensor a complex along a ring map
RingMap ** Complex -- tensor a complex along a ring map
tensor(Complex,RingMap) -- see RingMap ** Complex -- tensor a complex along a ring map
tensor(RingMap,Complex) -- see RingMap ** Complex -- tensor a complex along a ring map
RingMap Complex -- apply a ring map
source(ComplexMap) -- get the source of a map of chain complexes
target(ComplexMap) -- get the target of a map of chain complexes
truncate(List,Complex) -- truncation of a complex at a specified degree or set of degrees
truncate(ZZ,Complex) -- see truncate(List,Complex) -- truncation of a complex at a specified degree or set of degrees
yonedaExtension(Matrix) -- creates a chain complex representing an extension of modules

Methods that use a complex :

augmentationMap(Complex) -- see augmentationMap -- map from a free resolution to a module regarded as a complex
betti(Complex) -- display of degrees in a complex
canonicalMap(Complex,Complex) -- see canonicalMap -- gets the natural map arising from various constructions
chainComplex(Complex) -- translate between data types for chain complexes
Complex ** Matrix -- create the tensor product of a complex and a map of modules
Matrix ** Complex -- see Complex ** Matrix -- create the tensor product of a complex and a map of modules
Complex == Complex -- whether two complexes are equal
Complex == ZZ -- see Complex == Complex -- whether two complexes are equal
ZZ == Complex -- see Complex == Complex -- whether two complexes are equal
Complex ^ Array -- see Complex _ Array -- the canonical inclusion or projection map of a direct sum
Complex _ Array -- the canonical inclusion or projection map of a direct sum
Complex ^ ZZ -- see Complex _ ZZ -- access individual object in a complex
Complex _ ZZ -- access individual object in a complex
Complex Array -- shift a complex or complex map
Complex ** ComplexMap -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
ComplexMap ** Complex -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
components(Complex) -- list the components of a direct sum
concentration(Complex) -- see concentration -- indices on which a complex may be non-zero
max(Complex) -- see concentration -- indices on which a complex may be non-zero
min(Complex) -- see concentration -- indices on which a complex may be non-zero
Symbol ^ Complex -- see differential of a chain complex -- get the maps between the terms in a complex
directSum(Complex) -- direct sum of complexes
extend(Complex,Complex,Matrix) -- see extend(Complex,Complex,Matrix,Sequence) -- extend a map of modules to a map of chain complexes
extend(Complex,Complex,Matrix,Sequence) -- extend a map of modules to a map of chain complexes
HH^ZZ Complex -- see HH_ZZ Complex -- homology or cohomology module of a complex
HH_ZZ Complex -- homology or cohomology module of a complex
Hom(Complex,ComplexMap) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
Hom(Complex,Matrix) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
Hom(ComplexMap,Complex) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
Hom(Matrix,Complex) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
homomorphism(ZZ,Matrix,Complex) -- get the homomorphism from an element of Hom
horseshoeResolution(Complex) -- make the horseshoe resolution
inducedMap(Complex,Complex) -- make the map of complexes induced at each term by the identity map
isExact(Complex) -- see isExact -- whether a complex is exact
isExact(Complex,InfiniteNumber,InfiniteNumber) -- see isExact -- whether a complex is exact
isExact(Complex,InfiniteNumber,Number) -- see isExact -- whether a complex is exact
isExact(Complex,Number,InfiniteNumber) -- see isExact -- whether a complex is exact
isExact(Complex,Number,Number) -- see isExact -- whether a complex is exact
isFree(Complex) -- whether a complex consists of free modules
isHomogeneous(Complex) -- whether a complex is homogeneous
isShortExactSequence(Complex) -- whether a chain complex is a short exact sequence
isWellDefined(Complex) -- whether a complex is well-defined
length(Complex) -- length of a complex
map(Complex,Complex,ComplexMap) -- make a new map of chain complexes from an existing one
map(Complex,Complex,Function) -- make a map of chain complexes
map(Complex,Complex,HashTable) -- make a map of chain complexes
map(Complex,Complex,List) -- make a map of chain complexes
map(Complex,Complex,ZZ) -- make the zero map or identity between chain complexes
minimize(Complex) -- a quasi-isomorphic complex whose terms have minimal rank
poincare(Complex) -- assemble degrees of a chain complex into a polynomial
poincareN(Complex) -- assemble degrees of a chain complex into a polynomial
randomComplexMap(Complex,Complex) -- a random map of chain complexes
regularity(Complex) -- compute the Castelnuovo-Mumford regularity
resolution(Complex) -- minimal free resolution of a complex
resolutionMap(Complex) -- see resolutionMap -- map from a free resolution to the given complex
ring(Complex) -- access the ring of a complex or a complex map
sum(Complex) -- make the direct sum of all terms
tensorAssociativity(Complex,Complex,Complex) -- make the canonical isomorphism arising from associativity
tensorCommutativity(Complex,Complex) -- make the canonical isomorphism arising from commutativity
yonedaExtension'(Complex) -- identifies the element of Ext corresponding to an extension

For the programmer

The object Complex is a type, with ancestor classes MutableHashTable < HashTable < Thing.