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

- Function: kernel
- Usage:
    `kernel f`
    `ker f`
- Inputs:
  - f, a map of ZZ/d-graded factorizations,
- Optional inputs:
  - DegreeLimit (missing documentation) => ..., default value {},
  - Strategy (missing documentation) => ..., default value {},
  - SubringLimit => ..., default value infinity,
- Outputs:
  - a ZZ/d graded factorization,

## Description

The kernel of a ZZ/d-graded factorization map $f : C \to D$ is the factorization $E$ whose $i-th$ term is $kernel(f_i)$, and whose differential is induced from the differential on the source.

In the following example, we first construct a random factorization morphism $f : C \to D$.

    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, Cycle => true, InternalDegree => 2)

              2                                                                                  3
    o6 = 0 : S  <------------------------------------------------------------------------------ S  : 0
                   {2} | -34a3+15a2b-31ab2-38b3 -29a3+35a2b+6ab2+38b3 -22a3-5a2b+41ab2-30b3 |
                   {3} | -38a2-16ab+39b2        -10a2-29ab-8b2        21a2+34ab+19b2        |

              2                                                                                  3
         1 : S  <------------------------------------------------------------------------------ S  : 1
                   {4} | -34a3-14a2b-12ab2-19b3 29a3+44a2b-35ab2+39b3 -22a3-39a2b+48ab2-8b3 |
                   {4} | -38a3+8a2b+3ab2-30b3   10a3+21a2b-40ab2+38b3 21a3+18a2b+32ab2-38b3 |

    o6 : ZZdFactorizationMap

    i7 : g = inducedMap(D, ker f)

              3
    o7 = 0 : S  <------------------------------------------------ image {3} | -29a5-27a4b+a3b2+2a2b3+10ab4+50b5    | : 0
                   {3} | -29a5-27a4b+a3b2+2a2b3+10ab4+50b5    |         {3} | -19a5-38a4b+30a3b2+14a2b3+29ab4+21b5 |
                   {3} | -19a5-38a4b+30a3b2+14a2b3+29ab4+21b5 |         {3} | a5+37a4b+a3b2+29a2b3-24ab4-30b5      |
                   {3} | a5+37a4b+a3b2+29a2b3-24ab4-30b5      |

              3
         1 : S  <----------------------------------------------- image {5} | -29a5+a4b-37a3b2+38a2b3+44ab4+30b5  | : 1
                   {5} | -29a5+a4b-37a3b2+38a2b3+44ab4+30b5  |         {5} | 19a5-10a4b-47a3b2+34a2b3+40ab4+50b5 |
                   {5} | 19a5-10a4b-47a3b2+34a2b3+40ab4+50b5 |         {5} | a5+17a4b+47a3b2+12a2b3-22ab4+21b5   |
                   {5} | a5+17a4b+47a3b2+12a2b3-22ab4+21b5   |

    o7 : ZZdFactorizationMap

    i8 : ker g == 0

    o8 = true

    i9 : prune ker f --since this is a 1x1 factorization of an irreducible,

          1      1      1
    o9 = S  <-- S  <-- S

         0      1      0

    o9 : ZZdFactorization

    i10 : oo.dd       --it must be the trivial factorization

               1                    1
    o10 = 1 : S  <---------------- S  : 0
                    {10} | a+b |

               1                        1
          0 : S  <-------------------- S  : 1
                    {8} | a2-ab+b2 |

    o10 : ZZdFactorizationMap

## See also

- Making ZZdFactorizations -- Information about the basic constructors
- Making maps between factorizations -- information about the basic constructors
- kernel -- kernel of a map
- 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
- cokernel(ZZdFactorizationMap) -- make the cokernel of a map of ZZ/d-graded factorizations

## Ways to use this method:

- kernel(ZZdFactorizationMap) -- make the kernel 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.m2:4399:0.