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# ComplexMap -- the class of all maps between chain complexes

## Description

A map of chain complexes $f \colon C \rightarrow D$ of degree $d$ is a sequence of maps $f_i \colon C_i \rightarrow D_{d+i}$. No relationship between the maps $f_i$ and and the differentials of either $C$ or $D$ is assumed.

The set of all maps from $C$ to $D$ form the complex $\Hom(C,D)$ where $\Hom(C,D)_d$ consists of the maps of degree $d$.

The usual algebraic operations are available: addition, subtraction, scalar multiplication, and composition. The identity map from a chain complex to itself can be produced with id. An attempt to add (subtract, or compare) a ring element to a chain complex will result in the ring element being multiplied by the appropriate identity map.

## See also

- Complex -- the class of all chain complexes

## Functions and methods returning a map of complexes:

- `ComplexMap + Number` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `ComplexMap + RingElement` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `ComplexMap - ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `ComplexMap - Number` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `ComplexMap - RingElement` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `Number * ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `Number + ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `Number - ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `RingElement * ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `RingElement + ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `RingElement - ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `augmentationMap(Complex)` -- see augmentationMap -- map from a free resolution to a module regarded as a complex
- basis(List,ComplexMap) (missing documentation)
- basis(ZZ,ComplexMap) (missing documentation)
- `canonicalMap(Complex,Complex)` -- see canonicalMap -- gets the natural map arising from various constructions
- `canonicalTruncation(ComplexMap,InfiniteNumber,InfiniteNumber)` -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- `canonicalTruncation(ComplexMap,InfiniteNumber,ZZ)` -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- `canonicalTruncation(ComplexMap,Nothing,ZZ)` -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- `canonicalTruncation(ComplexMap,Sequence)` -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- `canonicalTruncation(ComplexMap,ZZ,InfiniteNumber)` -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- `canonicalTruncation(ComplexMap,ZZ,Nothing)` -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- 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 ^ 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
- `ComplexMap Array` -- see 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
- ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- `ComplexMap ** Module` -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- `Module ** ComplexMap` -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- `tensor(ComplexMap,ComplexMap)` -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- ComplexMap ++ ComplexMap -- direct sum of complex maps
- ComplexMap ^ Array -- the composition with the canonical inclusion or projection map
- `ComplexMap _ Array` -- see ComplexMap ^ Array -- the composition with the canonical inclusion or projection map
- ComplexMap ^ ZZ -- the n-fold composition
- ComplexMap | ComplexMap -- join or concatenate maps horizontally
- ComplexMap || ComplexMap -- join or concatenate maps vertically
- `connectingExtMap(Matrix,Matrix,Module)` -- see connectingExtMap(Module,Matrix,Matrix) -- makes the connecting maps in Ext
- connectingExtMap(Module,Matrix,Matrix) -- makes the connecting maps in Ext
- connectingMap(ComplexMap,ComplexMap) -- construct the connecting homomorphism on homology
- `connectingTorMap(Matrix,Matrix,Module)` -- see connectingTorMap(Module,Matrix,Matrix) -- makes the connecting maps in Tor
- connectingTorMap(Module,Matrix,Matrix) -- makes the connecting maps in Tor
- `Symbol ^ Complex` -- see differential of a chain complex -- get the maps between the terms in a complex
- dual(ComplexMap) -- the dual of a map of complexes
- `transpose(ComplexMap)` -- see dual(ComplexMap) -- the dual of a map of complexes
- epicResolutionMap(Complex) (missing documentation)
- `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
- freeResolution(Matrix) -- compute the induced map between free resolutions
- HH ComplexMap -- induced map on homology or cohomology
- `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(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- `Hom(ComplexMap,Matrix)` -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- `Hom(ComplexMap,Module)` -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- `Hom(ComplexMap,Ring)` -- 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
- `Hom(Matrix,ComplexMap)` -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- `Hom(Module,ComplexMap)` -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- `Hom(Ring,ComplexMap)` -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- homomorphism'(ComplexMap) -- get the element of Hom from a map of complexes
- homomorphism(ComplexMap) -- get the homomorphism from an element of Hom
- homomorphism(ZZ,Matrix,Complex) -- get the homomorphism from an element of Hom
- inducedMap(Complex,Complex) -- make the map of complexes induced at each term by the identity map
- `homotopyMap(ComplexMap)` -- see liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- 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
- `minimalPresentation(ComplexMap)` -- see minimalPresentation(Complex) -- minimal presentation of all terms in a complex
- `prune(ComplexMap)` -- see minimalPresentation(Complex) -- minimal presentation of all terms in a complex
- `naiveTruncation(ComplexMap,InfiniteNumber,InfiniteNumber)` -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- `naiveTruncation(ComplexMap,InfiniteNumber,ZZ)` -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- `naiveTruncation(ComplexMap,Nothing,ZZ)` -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- `naiveTruncation(ComplexMap,Sequence)` -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- `naiveTruncation(ComplexMap,Sequence,Sequence)` -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- `naiveTruncation(ComplexMap,ZZ,InfiniteNumber)` -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- `naiveTruncation(ComplexMap,ZZ,Nothing)` -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- nullHomotopy(ComplexMap) -- a map which is a candidate for being a null homotopy
- part(List,ComplexMap) -- extract a graded component of a map of complexes
- `part(ZZ,ComplexMap)` -- see part(List,ComplexMap) -- extract a graded component of a map of complexes
- randomComplexMap(Complex,Complex) -- a random map of chain complexes
- `resolutionMap(Complex)` -- see resolutionMap -- map from a free resolution to the given complex
- `ComplexMap ** Ring` -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- `ComplexMap ** RingMap` -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- `Ring ** ComplexMap` -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- `tensor(ComplexMap,RingMap)` -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- `tensor(RingMap,ComplexMap)` -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- RingMap ComplexMap -- apply a ring map to a map of complexes
- tensorAssociativity(Complex,Complex,Complex) -- make the canonical isomorphism arising from associativity
- tensorCommutativity(Complex,Complex) -- make the canonical isomorphism arising from commutativity
- truncate(List,ComplexMap) -- truncation of a complex map at a specified degree or set of degrees
- `truncate(ZZ,ComplexMap)` -- see truncate(List,ComplexMap) -- truncation of a complex map at a specified degree or set of degrees
- yonedaMap(Matrix) -- creates a chain complex map representing an extension of modules

## Methods that use a map of complexes:

- `- ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `ComplexMap * Number` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `ComplexMap * RingElement` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- `ComplexMap + ComplexMap` -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- coimage(ComplexMap) -- make the coimage of a map of complexes
- cokernel(ComplexMap) -- make the cokernel of a map of complexes
- complete(ComplexMap) (missing documentation)
- complex(ComplexMap) -- make a complex by specifying the differential
- ComplexMap * ComplexMap -- composition of homomorphisms of complexes
- `directSum(ComplexMap)` -- see ComplexMap ++ ComplexMap -- direct sum of complex maps
- ComplexMap == ComplexMap -- whether two complex maps are equal
- `ComplexMap == ZZ` -- see ComplexMap == ComplexMap -- whether two complex maps are equal
- `ZZ == ComplexMap` -- see ComplexMap == ComplexMap -- whether two complex maps are equal
- ComplexMap _ ZZ -- access individual matrices in a complex map
- components(ComplexMap) -- list the components of a direct sum
- concentration(ComplexMap) -- indices on which a complex map may be non-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
- degree(ComplexMap) -- get the degree of a map of chain complexes
- `HH^ZZ ComplexMap` -- see HH ComplexMap -- induced map on homology or cohomology
- `HH_ZZ ComplexMap` -- see HH ComplexMap -- induced map on homology or cohomology
- image(ComplexMap) -- make the image of a map of complexes
- isCommutative(ComplexMap) -- whether a complex map commutes with the differentials
- isComplexMorphism(ComplexMap) -- whether a complex map is a morphism of complexes
- isHomogeneous(ComplexMap) -- whether a map of complexes is homogeneous
- isNullHomotopic(ComplexMap) -- whether a map of complexes is null-homotopic
- isNullHomotopyOf(ComplexMap,ComplexMap) -- whether the first map of chain complexes is a null homotopy for the second
- isQuasiIsomorphism(ComplexMap) -- whether a map of complexes is a quasi-isomorphism
- isShortExactSequence(ComplexMap,ComplexMap) -- whether a pair of complex maps forms a short exact sequence
- isWellDefined(ComplexMap) -- whether a map of chain complexes is well-defined
- kernel(ComplexMap) -- make the kernel of a map of complexes
- `ComplexMap // ComplexMap` -- see liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- `quotient(ComplexMap,ComplexMap)` -- see liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- longExactSequence(ComplexMap,ComplexMap) -- make the long exact sequence in homology
- mathML(ComplexMap) (missing documentation)
- nullhomotopy(ComplexMap) (missing documentation)
- `ring(ComplexMap)` -- see ring(Complex) -- access the ring of a complex or a complex map
- source(ComplexMap) -- get the source of a map of chain complexes
- `sum(ComplexMap)` -- see sum(Complex) -- make the direct sum of all terms
- target(ComplexMap) -- get the target of a map of chain complexes
- yonedaMap'(ComplexMap) -- identifies the element of Ext corresponding to a map of free resolutions

## For the programmer

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Complexes/ChainComplexMapDoc.m2:146:0.