ZZdFactorizationMap -- the class of all maps between ZZ/d-graded factorizations

Description

A map of ZZ/d-graded factorizations $f \colon C \rightarrow D$ 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 another factorization $\Hom(C,D)$ where $\Hom(C,D)_d$ consists of the maps of degree $d$; this is a factorization of the differential of the respective potentials.

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

Functions and methods returning a map of ZZ/d-graded factorizations:

Number + ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
Number - ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
RingElement + ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
RingElement - ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
ZZdFactorizationMap + Number -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
ZZdFactorizationMap + RingElement -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
ZZdFactorizationMap - Number -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
ZZdFactorizationMap - RingElement -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
ZZdFactorizationMap - ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
Complex ** ZZdFactorizationMap -- see ComplexMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations, obtained by first folding the relevant complex map
ComplexMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations, obtained by first folding the relevant complex map
ZZdFactorizationMap ** Complex -- see ComplexMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations, obtained by first folding the relevant complex map
ZZdFactorizationMap ** ComplexMap -- see ComplexMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations, obtained by first folding the relevant complex map
Symbol ^ ZZdFactorization -- see differential of a ZZ/d-graded factorization -- get the maps between the terms in a ZZ/d-graded factorization
dual(ZZdFactorizationMap) -- the dual of a map of ZZ/d-graded factorizations
dual(ZZdFactorizationMap,RingElement) -- the dual of a map of ZZ/d-graded factorizations
dual(ZZdFactorizationMap,Symbol) -- see dual(ZZdFactorizationMap,RingElement) -- the dual of a map of ZZ/d-graded factorizations
extend(ZZdFactorization,ZZdFactorization,Matrix) (missing documentation)
extend(ZZdFactorization,ZZdFactorization,Matrix,Sequence) (missing documentation)
HH ZZdFactorizationMap -- induced map on homology or cohomology
Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
Hom(ComplexMap,ZZdFactorizationMap,RingElement) -- see Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
Hom(ComplexMap,ZZdFactorizationMap,Symbol) -- see Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
Hom(ZZdFactorizationMap,ComplexMap) -- see Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
Hom(ZZdFactorizationMap,ComplexMap,RingElement) -- see Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
Hom(ZZdFactorizationMap,ComplexMap,Symbol) -- see Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
Hom(Ring,ZZdFactorization) -- see Hom(ZZdFactorization,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
Hom(Matrix,ZZdFactorization) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(Matrix,ZZdFactorizationMap) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(Module,ZZdFactorizationMap) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(Ring,ZZdFactorizationMap) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,Matrix) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,ZZdFactorizationMap) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Matrix) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Module) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Ring) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorization) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,Matrix,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,Matrix,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,ZZdFactorizationMap,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,ZZdFactorizationMap,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Matrix,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Matrix,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Module,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Module,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Ring,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,Ring,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorization,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorization,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorizationMap,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
homomorphism'(ZZdFactorizationMap) -- get the element of Hom from a map of factorizations
homomorphism(ZZ,Matrix,ZZdFactorization) -- get the homomorphism from an element of Hom
homomorphism(ZZdFactorizationMap) -- get the homomorphism from an element of Hom
inducedMap(ZZdFactorization,ZZdFactorization) -- make the map of ZZ/d-graded factorizations induced at each term by the identity map
inducedMap(ZZdFactorization,ZZdFactorization,ZZdFactorizationMap) (missing documentation)
injectiveCover(ZZdFactorization) -- see injectiveCover -- Constructs the injective hull of a d-fold matrix factorization
map(ZZdFactorization,ZZdFactorization,Function) (missing documentation)
map(ZZdFactorization,ZZdFactorization,HashTable) -- make a map of ZZ/d-graded factorizations
map(ZZdFactorization,ZZdFactorization,List) (missing documentation)
map(ZZdFactorization,ZZdFactorization,ZZ) -- make the zero map or identity between ZZ/d-graded factorizations
map(ZZdFactorization,ZZdFactorization,ZZdFactorizationMap) -- make a new map of ZZ/d-graded factorizations from an existing one (for inducing maps)
minimalPresentation(ZZdFactorizationMap) -- see minimalPresentation(ZZdFactorization) -- minimal presentation of all terms in a ZZ/d-graded factorization
prune(ZZdFactorizationMap) -- see minimalPresentation(ZZdFactorization) -- minimal presentation of all terms in a ZZ/d-graded factorization
nullHomotopy(ZZdFactorizationMap) -- a map which is a candidate for being a null homotopy
projectiveCover(ZZdFactorization) -- see projectiveCover -- Constructs the projective cover of a d-fold matrix factorization
randomFactorizationMap(ZZdFactorization,ZZdFactorization) -- see randomFactorizationMap -- a random map of ZZ/d-graded factorizations
Ring ** ZZdFactorizationMap -- see RingMap ** ZZdFactorizationMap -- tensor a map of ZZ/d-graded factorizations along a ring map
RingMap ** ZZdFactorizationMap -- tensor a map of ZZ/d-graded factorizations along a ring map
tensor(RingMap,ZZdFactorizationMap) -- see RingMap ** ZZdFactorizationMap -- tensor a map of ZZ/d-graded factorizations along a ring map
tensor(ZZdFactorizationMap,RingMap) -- see RingMap ** ZZdFactorizationMap -- tensor a map of ZZ/d-graded factorizations along a ring map
ZZdFactorizationMap ** Ring -- see RingMap ** ZZdFactorizationMap -- tensor a map of ZZ/d-graded factorizations along a ring map
ZZdFactorizationMap ** RingMap -- see RingMap ** ZZdFactorizationMap -- tensor a map of ZZ/d-graded factorizations along a ring map
RingMap ZZdFactorizationMap -- apply a ring map to a map of ZZ/d-graded factorizations
tensor(ZZdFactorizationMap,ZZdFactorizationMap,RingElement) (missing documentation)
tensor(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) (missing documentation)
tensorAssociativity(ZZdFactorization,ZZdFactorization,ZZdFactorization) -- make the canonical isomorphism arising from associativity
tensorCommutativity(ZZdFactorization,ZZdFactorization) -- make the canonical isomorphism arising from commutativity of the tensor product operation
Matrix ** ZZdFactorization -- see ZZdFactorization ** Matrix -- create the tensor product of a ZZ/d-graded factorization and a map of modules
ZZdFactorization ** Matrix -- create the tensor product of a ZZ/d-graded factorization and a map of modules
ZZdFactorization ^ Array -- see ZZdFactorization _ Array -- the canonical inclusion or projection map of a direct sum
ZZdFactorization _ Array -- the canonical inclusion or projection map of a direct sum
ZZdFactorizationMap Array -- see ZZdFactorization Array -- shift a ZZ/d-graded factorization or map of ZZ/d-graded factorizations
Module ** ZZdFactorizationMap -- see ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
tensor(ZZdFactorizationMap,ZZdFactorizationMap) -- see ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
ZZdFactorization ** ZZdFactorizationMap -- see ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
ZZdFactorizationMap ** Module -- see ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
ZZdFactorizationMap ** ZZdFactorization -- see ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
ZZdFactorizationMap ++ ZZdFactorizationMap -- direct sum of ZZ/d-graded factorization maps
ZZdFactorizationMap ^ Array -- the composition with the canonical inclusion or projection map
ZZdFactorizationMap _ Array -- see ZZdFactorizationMap ^ Array -- the composition with the canonical inclusion or projection map
ZZdFactorizationMap ^ ZZ -- the n-fold composition
ZZdFactorizationMap | ZZdFactorizationMap -- join or concatenate maps horizontally
ZZdFactorizationMap || ZZdFactorizationMap -- join or concatenate maps vertically

Methods that use a map of ZZ/d-graded factorizations:

adjoinRoot(ZZdFactorizationMap,RingElement) -- see adjoinRoot -- Adjoin a root of unity to the underlying object input
adjoinRoot(ZZdFactorizationMap,Symbol) -- see adjoinRoot -- Adjoin a root of unity to the underlying object input
- ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
Number * ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
RingElement * ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
ZZdFactorizationMap * Number -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
ZZdFactorizationMap * RingElement -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
ZZdFactorizationMap + ZZdFactorizationMap -- see arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
coimage(ZZdFactorizationMap) -- make the coimage of a map of ZZ/d-graded factorizations
cokernel(ZZdFactorizationMap) -- make the cokernel of a map of ZZ/d-graded factorizations
components(ZZdFactorizationMap) -- list the components of a direct sum
cone(ZZdFactorizationMap) -- make the mapping cone of a morphism of ZZ/d-graded factorizations
degree(ZZdFactorizationMap) -- get the degree of a map of ZZ/d-graded factorizations
expression(ZZdFactorizationMap) (missing documentation)
HH^ZZ ZZdFactorizationMap -- see HH ZZdFactorizationMap -- induced map on homology or cohomology
HH_ZZ ZZdFactorizationMap -- see HH ZZdFactorizationMap -- induced map on homology or cohomology
image(ZZdFactorizationMap) -- make the image of a map of ZZ/d-graded factorizations
isCommutative(ZZdFactorizationMap) -- whether a ZZ/d-graded factorization map commutes with the differentials
isFactorizationMorphism(ZZdFactorizationMap) -- whether a ZZdFactorizationMap is a morphism of factorizations
isHomogeneous(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is homogeneous
isNullHomotopic(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is null-homotopic
isNullHomotopyOf(ZZdFactorizationMap,ZZdFactorizationMap) -- whether the first map of ZZ/d-graded factorizations is a null homotopy for the second
isQuasiIsomorphism(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is a quasi-isomorphism
isShortExactSequence(ZZdFactorizationMap,ZZdFactorizationMap) -- whether a pair of ZZ/d-graded factorization maps forms a short exact sequence
isWellDefined(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is well-defined
kernel(ZZdFactorizationMap) -- make the kernel of a map of ZZ/d-graded factorizations
net(ZZdFactorizationMap) (missing documentation)
period(ZZdFactorizationMap) -- see period -- the period of a ZZ/d-graded factorization or map
ring(ZZdFactorizationMap) -- see ring(ZZdFactorization) -- access the ring of a ZZ/d-graded factorization or a factorization map
source(ZZdFactorizationMap) -- get the source of a map of ZZ/d-graded factorizations
suspension(ZZdFactorizationMap) -- see suspension -- Constructs the suspension of a d-fold factorization
target(ZZdFactorizationMap) -- get the target of a map of ZZ/d-graded factorizations
unfold(ZZdFactorizationMap,List) -- see unfold -- Converts a factorization into a complex
unfold(ZZdFactorizationMap,ZZ) -- see unfold -- Converts a factorization into a complex
ZZdFactorizationMap * ZZdFactorizationMap -- composition of homomorphisms of ZZ/d-graded factorizations
directSum(ZZdFactorizationMap) -- see ZZdFactorizationMap ++ ZZdFactorizationMap -- direct sum of ZZ/d-graded factorization maps
RingElement == ZZdFactorizationMap -- see ZZdFactorizationMap == ZZdFactorizationMap -- whether two ZZ/d-graded factorization maps are equal
ZZ == ZZdFactorizationMap -- see ZZdFactorizationMap == ZZdFactorizationMap -- whether two ZZ/d-graded factorization maps are equal
ZZdFactorizationMap == RingElement -- see ZZdFactorizationMap == ZZdFactorizationMap -- whether two ZZ/d-graded factorization maps are equal
ZZdFactorizationMap == ZZ -- see ZZdFactorizationMap == ZZdFactorizationMap -- whether two ZZ/d-graded factorization maps are equal
ZZdFactorizationMap == ZZdFactorizationMap -- whether two ZZ/d-graded factorization maps are equal
ZZdFactorizationMap _ ZZ -- access individual matrices in a ZZ/d-graded factorization map

For the programmer

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

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:2349:0.

ZZdFactorizationMap -- the class of all maps between ZZ/d-graded factorizations

Description

See also

Functions and methods returning a map of ZZ/d-graded factorizations:

Methods that use a map of ZZ/d-graded factorizations:

For the programmer