Making maps between chain complexes -- information about the basic constructors

Basic constructors

• map(Complex,Complex,HashTable) -- make a map of chain complexes
• map(Complex,Complex,ZZ) -- make the zero map or identity between chain complexes
• map(Complex,Complex,Function) -- make a map of chain complexes
• map(Complex,Complex,List) -- make a map of chain complexes
• map(Complex,Complex,ComplexMap) -- make a new map of chain complexes from an existing one
• id _ Complex -- the identity map of a chain complex
• differential of a chain complex -- get the maps between the terms in a complex
• ComplexMap Array -- shift a complex or complex map
• isWellDefined(ComplexMap) -- whether a map of chain complexes is well-defined

Important computations creating new complex maps

• resolutionMap(Complex) -- map from a free resolution to the given complex
• HH ComplexMap -- induced map on homology or cohomology
• augmentationMap(Complex) -- map from a free resolution to a module regarded as a complex
• Complex ** Matrix -- create the tensor product of a complex and a map of modules
• freeResolution(Matrix) -- compute the induced map between free resolutions
• extend(Complex,Complex,Matrix) -- extend a map of modules to a map of chain complexes
• nullHomotopy(ComplexMap) -- a map which is a candidate for being a null homotopy

Canonical maps between complexes

Some complexes come with canonical maps. To access the complex map, one uses canonicalMap(Complex,Complex). The following operations have associated canonical maps.

• kernel(ComplexMap) -- make the kernel of a map of complexes
• cokernel(ComplexMap) -- make the cokernel of a map of complexes
• image(ComplexMap) -- make the image of a map of complexes
• coimage(ComplexMap) -- make the coimage of a map of complexes
• cone(ComplexMap) -- make the mapping cone of a morphism of chain complexes
• cylinder(ComplexMap) -- make the mapping cylinder of a morphism of chain complexes
• inducedMap(Complex,Complex) -- make the map of complexes induced at each term by the identity map

Random maps of chain complexes

The method randomComplexMap(Complex,Complex) allows one to construct random complex maps, random morphisms between complexes, and random null homotopies between complexes.

• isCommutative(ComplexMap) -- whether a complex map commutes with the differentials
• isComplexMorphism(ComplexMap) -- whether a complex map is a morphism of complexes
• isNullHomotopic(ComplexMap) -- whether a map of complexes is null-homotopic

Elementary operations on complex maps

• arithmetic with complex maps -- perform arithmetic operations on complex maps
• ComplexMap + ComplexMap -- perform arithmetic operations on complex maps
• ComplexMap | ComplexMap -- join or concatenate maps horizontally
• ComplexMap || ComplexMap -- join or concatenate maps vertically
• ComplexMap ++ ComplexMap -- direct sum of complex maps
• ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
• Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
• dual(ComplexMap) -- the dual of a map of complexes
• ComplexMap _ Array -- the composition with the canonical inclusion or projection map
• ComplexMap ^ Array -- the composition with the canonical inclusion or projection map
• naiveTruncation(ComplexMap,Sequence) -- drops all terms in the source of a complex outside a given interval
• canonicalTruncation(ComplexMap,Sequence) -- reducing the number of non-zero terms of a complex
• part(List,ComplexMap) -- extract a graded component of a map of complexes
• truncate(List,ComplexMap) -- truncation of a complex map at a specified degree or set of degrees
• RingMap ComplexMap -- apply a ring map to a map of complexes
• RingMap ** ComplexMap -- tensor a map of complexes along a ring map