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# Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map

- Function: Hom
- Usage:
    `h = Hom(f,g)`
- Inputs:
  - f, a map of complexes, or a map of ZZ/d-graded factorizations
  - g, a map of ZZ/d-graded factorizations, or a map of complexes
- Optional inputs:
  - DegreeLimit => ..., default value null,
  - MinimalGenerators => ..., default value true,
  - Strategy => ..., default value null,
- Outputs:
  - h, a map of ZZ/d-graded factorizations,

## Description

The maps $f : C \to D$ and $g : E \to F$ of ZZ/d-graded factorizations induces the map $h = Hom(f,g) : Hom(D,E) \to Hom(C,F)$ defined by $\phi \mapsto g \phi f$.

    i1 : S = ZZ/101[a,b];

    i2 : R = S/(a^3+b^3);

    i3 : m = ideal vars R

    o3 = ideal (a, b)

    o3 : Ideal of R

    i4 : C = tailMF m

          2      2      2
    o4 = S  <-- S  <-- S

         0      1      0

    o4 : ZZdFactorization

    i5 : D = tailMF (m^2)

          3      3      3
    o5 = S  <-- S  <-- S

         0      1      0

    o5 : ZZdFactorization

    i6 : K = koszulComplex vars S

          1      2      1
    o6 = S  <-- S  <-- S

         0      1      2

    o6 : Complex

    i7 : H = Hom(K, C)

          8      8      8
    o7 = S  <-- S  <-- S

         0      1      0

    o7 : ZZdFactorization

    i8 : potential H

            3    3
    o8 = - a  - b

    o8 : S

    i9 : H' = Hom(D, K)

          12      12      12
    o9 = S   <-- S   <-- S

         0       1       0

    o9 : ZZdFactorization

    i10 : potential H'

           3    3
    o10 = a  + b

    o10 : S

    i11 : isZZdComplex (H**H')

    o11 = true

## See also

- homomorphism(ZZdFactorizationMap) -- get the homomorphism from an element of Hom
- homomorphism'(ZZdFactorizationMap) -- get the element of Hom from a map of factorizations
- randomFactorizationMap(ZZdFactorization,ZZdFactorization) -- a random map of ZZ/d-graded factorizations
- Hom(ZZdFactorization,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations

## Ways to use this method:

- Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
- `Hom(ComplexMap,ZZdFactorizationMap,RingElement)`
- `Hom(ComplexMap,ZZdFactorizationMap,Symbol)`
- `Hom(ZZdFactorizationMap,ComplexMap)`
- `Hom(ZZdFactorizationMap,ComplexMap,RingElement)`
- `Hom(ZZdFactorizationMap,ComplexMap,Symbol)`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixFactorizations/MatrixFactorizationsDOC.m2:3377:0.