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# 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

## See also

- Making chain complexes -- information about the basic constructors
- Basic invariants and properties -- information about accessing basic features
- Working with Ext -- information about functorial properties
- Working with Tor -- information about functorial properties
- Towards computing in the derived category

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Complexes/ChainComplexMapDoc.m2:95:0.