Making maps between ZZ/d-graded factorizations -- information about the basic constructors

Basic constructors

• map(ZZdFactorization,ZZdFactorization,HashTable) -- make a map of ZZ/d-graded factorizations
• 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)
• id _ ZZdFactorization -- the identity map of a ZZ/d-graded factorization
• differential of a ZZ/d-graded factorization -- get the maps between the terms in a ZZ/d-graded factorization
• ZZdFactorizationMap Array -- shift a ZZ/d-graded factorization or map of ZZ/d-graded factorizations
• isWellDefined(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is well-defined

Important computations creating new ZZ/d-graded factorization maps

• HH ZZdFactorizationMap -- induced map on homology or cohomology
• ZZdFactorization ** Matrix -- create the tensor product of a ZZ/d-graded factorization and a map of modules
• nullHomotopy(ZZdFactorizationMap) -- a map which is a candidate for being a null homotopy

Canonical maps between ZZ/d-graded factorizations

Some ZZ/d-graded factorizations come with canonical maps. To access the ZZ/d-graded factorization map, one uses inducedMap(ZZdFactorization,ZZdFactorization).

• kernel(ZZdFactorizationMap) -- make the kernel of a map of ZZ/d-graded factorizations
• cokernel(ZZdFactorizationMap) -- make the cokernel of a map of ZZ/d-graded factorizations
• image(ZZdFactorizationMap) -- make the image of a map of ZZ/d-graded factorizations
• coimage(ZZdFactorizationMap) -- make the coimage of a map of ZZ/d-graded factorizations
• cone(ZZdFactorizationMap) -- make the mapping cone of a morphism of ZZ/d-graded factorizations
• inducedMap(ZZdFactorization,ZZdFactorization) -- make the map of ZZ/d-graded factorizations induced at each term by the identity map

Random maps of ZZ/d-graded factorizations

The method randomFactorizationMap(ZZdFactorization,ZZdFactorization) allows one to construct random ZZ/d-graded factorization maps, random morphisms between factorizations, and random null homotopies between factorizaions.

• isCommutative(ZZdFactorizationMap) -- whether a ZZ/d-graded factorization map commutes with the differentials
• isFactorizationMorphism(ZZdFactorizationMap) -- whether a ZZdFactorizationMap is a morphism of factorizations
• isNullHomotopic(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is null-homotopic

Elementary operations on factorization maps

• arithmetic with complex maps
• ZZdFactorizationMap + ZZdFactorizationMap -- perform arithmetic operations on ZZ/d-graded factorization maps
• ZZdFactorizationMap | ZZdFactorizationMap -- join or concatenate maps horizontally
• ZZdFactorizationMap || ZZdFactorizationMap -- join or concatenate maps vertically
• ZZdFactorizationMap ++ ZZdFactorizationMap -- direct sum of ZZ/d-graded factorization maps
• ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
• Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
• dual(ZZdFactorizationMap) -- the dual of a map of ZZ/d-graded factorizations
• ZZdFactorizationMap _ Array -- the composition with the canonical inclusion or projection map
• ZZdFactorizationMap ^ Array -- the composition with the canonical inclusion or projection map
• RingMap ZZdFactorizationMap -- apply a ring map to a map of ZZ/d-graded factorizations
• RingMap ** ZZdFactorizationMap -- tensor a map of ZZ/d-graded factorizations along a ring map