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# inducedMap(ZZdFactorization,ZZdFactorization) -- make the map of ZZ/d-graded factorizations induced at each term by the identity map

- Function: inducedMap
- Usage:
    `f = inducedMap(D, C)`
- Inputs:
  - C, a ZZ/d graded factorization,
  - D, a ZZ/d graded factorization,
- Optional inputs:
  - Degree => an integer, default value null, specify the degree of the map of ZZ/d-graded factorizations, if not 0
  - Verify => a Boolean value, default value true, if true, check that the resulting maps are well-defined
- Outputs:
  - f, a map of ZZ/d-graded factorizations,

## Description

Let $d$ be the value of the optional argument Degree, or zero, if not given. For each $i$, the terms $D_{i+d}$ and $C_i$ must be subquotients of the same ambient free module. This method returns the factorization map induced by the identity on each of these free modules.

If Verify => true is given, then this method also checks that these identity maps induced well-defined maps. This can be a relatively expensive computation.

We illustrate this method by inducing some of the natural inclusions and surjections induced by taking kernels/cokernels of morphisms of factorizations.

    i1 : R = ZZ/101[a,b,c]

    o1 = R

    o1 : PolynomialRing

    i2 : C = koszulMF({a,b,c}, a^3+b^3+c^3)

          4      4      4
    o2 = R  <-- R  <-- R

         0      1      0

    o2 : ZZdFactorization

    i3 : f = randomFactorizationMap(C, C, Cycle => true, InternalDegree => 1)

              4                                                           4
    o3 = 0 : R  <------------------------------------------------------- R  : 0
                   | 24a-36b-30c 0           0           0           |
                   | -29c        24a-36b-30c 0           0           |
                   | -29a        0           24a-36b-30c 0           |
                   | 29b         0           0           24a-36b-30c |

              4                                                           4
         1 : R  <------------------------------------------------------- R  : 1
                   | 24a-36b-30c 0           0           0           |
                   | -29c        24a-36b-30c -29b        -29a        |
                   | 0           0           24a-36b-30c 0           |
                   | 0           0           0           24a-36b-30c |

    o3 : ZZdFactorizationMap

    i4 : isWellDefined coker f

    o4 = true

    i5 : (coker f).dd^2 == (a^3 + b^3 + c^3)*id_(coker f)

    o5 = true

    i6 : i1 = inducedMap(coker f, C)

                                                                                            4
    o6 = 0 : cokernel | 24a-36b-30c 0           0           0           | <--------------- R  : 0
                      | -29c        24a-36b-30c 0           0           |    | 1 0 0 0 |
                      | -29a        0           24a-36b-30c 0           |    | 0 1 0 0 |
                      | 29b         0           0           24a-36b-30c |    | 0 0 1 0 |
                                                                             | 0 0 0 1 |

                                                                                            4
         1 : cokernel | 24a-36b-30c 0           0           0           | <--------------- R  : 1
                      | -29c        24a-36b-30c -29b        -29a        |    | 1 0 0 0 |
                      | 0           0           24a-36b-30c 0           |    | 0 1 0 0 |
                      | 0           0           0           24a-36b-30c |    | 0 0 1 0 |
                                                                             | 0 0 0 1 |

    o6 : ZZdFactorizationMap

    i7 : i2 = inducedMap(C, image f)

              4
    o7 = 0 : R  <------------------------------------------------------- image | 24a-36b-30c 0           0           0           | : 0
                   | 24a-36b-30c 0           0           0           |         | -29c        24a-36b-30c 0           0           |
                   | -29c        24a-36b-30c 0           0           |         | -29a        0           24a-36b-30c 0           |
                   | -29a        0           24a-36b-30c 0           |         | 29b         0           0           24a-36b-30c |
                   | 29b         0           0           24a-36b-30c |

              4
         1 : R  <------------------------------------------------------- image | 24a-36b-30c 0           0           0           | : 1
                   | 24a-36b-30c 0           0           0           |         | -29c        24a-36b-30c -29b        -29a        |
                   | -29c        24a-36b-30c -29b        -29a        |         | 0           0           24a-36b-30c 0           |
                   | 0           0           24a-36b-30c 0           |         | 0           0           0           24a-36b-30c |
                   | 0           0           0           24a-36b-30c |

    o7 : ZZdFactorizationMap

    i8 : ker i1 == image i2

    o8 = true

## See also

- inducedMap(Module,Module) -- compute the map induced by the identity

## Ways to use this method:

- inducedMap(ZZdFactorization,ZZdFactorization) -- make the map of ZZ/d-graded factorizations induced at each term by the identity map

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixFactorizations/MatrixFactorizationsDOC.m2:4045:0.