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.

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
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
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(ChainComplexMap) -- translate between data types for chain complex maps
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
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
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
chainComplex(ComplexMap) -- translate between data types for chain complexes
coimage(ComplexMap) -- make the coimage of a map of complexes
cokernel(ComplexMap) -- make the cokernel of a map of complexes
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
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.

ComplexMap -- the class of all maps between chain complexes

Description

See also

Functions and methods returning a map of complexes :

Methods that use a map of complexes :

For the programmer