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# dual(ZZdFactorizationMap) -- the dual of a map of ZZ/d-graded factorizations

- Function: dual
- Usage:
    `h = dual f`
- Inputs:
  - f, a map of ZZ/d-graded factorizations,
- Outputs:
  - h, a map of ZZ/d-graded factorizations,

## Description

The map $f : C \to D$ of ZZ/d-graded factorizations over the ring $S$ induces the map $h = Hom(f, S^1) : Hom(D, S^1) \to Hom(C,S^1)$ defined by $\phi \mapsto \phi f$.

    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 : f = randomFactorizationMap(C,D)

              2                                         3
    o6 = 0 : S  <------------------------------------- S  : 0
                   {2} | 24a-36b -29a+19b -10a-29b |
                   {3} | -30     19       -8       |

              2                                          3
         1 : S  <-------------------------------------- S  : 1
                   {4} | -22a-29b -16a+39b 19a-47b  |
                   {4} | -24a-38b 21a+34b  -39a-18b |

    o6 : ZZdFactorizationMap

    i7 : h = dual f

              3                                  2
    o7 = 0 : S  <------------------------------ S  : 0
                   {-3} | -22a-29b -24a-38b |
                   {-3} | -16a+39b 21a+34b  |
                   {-3} | 19a-47b  -39a-18b |

              3                             2
         1 : S  <------------------------- S  : 1
                   {-5} | 24a-36b  -30 |
                   {-5} | -29a+19b 19  |
                   {-5} | -10a-29b -8  |

    o7 : ZZdFactorizationMap

    i8 : assert isWellDefined h

This routine is functorial.

    i9 : D' = tailMF (trim m^3)

          3      3      3
    o9 = S  <-- S  <-- S

         0      1      0

    o9 : ZZdFactorization

    i10 : f' = randomFactorizationMap(D, D')

               3                                          3
    o10 = 0 : S  <-------------------------------------- S  : 0
                    {3} | -13a-43b 2a+16b   -47a+47b |
                    {3} | -15a-28b 22a+45b  19a-16b  |
                    {3} | -47a+38b -34a-48b 7a+15b   |

               3                                        3
          1 : S  <------------------------------------ S  : 1
                    {5} | -23a+39b 36a+35b 11a+46b |
                    {5} | 43a-17b  11a-38b -28a+b  |
                    {5} | -11a+48b 33a+40b -3a+22b |

    o10 : ZZdFactorizationMap

    i11 : dual(f * f') == dual f' * dual f

    o11 = true

## See also

- Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
- randomFactorizationMap(ZZdFactorization,ZZdFactorization) -- a random map of ZZ/d-graded factorizations
- dual(Matrix) -- dual of a map

## Ways to use this method:

- dual(ZZdFactorizationMap) -- the dual of a map of ZZ/d-graded factorizations

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixFactorizations/MatrixFactorizationsDOC.m2:3664:0.