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

Description

If $f : C \to D$ is a map of ZZ/d-graded factorizations of degree $d$, then the cokernel is the factorization $E$ whose $i-th$ term is $cokernel(f_{i-d})$, and whose differential is induced from the differential on the target.

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 : use S;
 
i5 : C = tailMF m

      2      2      2
o5 = S  <-- S  <-- S
                    
     0      1      0

o5 : ZZdFactorization
 
i6 : D = tailMF (m^2)

      3      3      3
o6 = S  <-- S  <-- S
                    
     0      1      0

o6 : ZZdFactorization
 
i7 : f = randomFactorizationMap(D, C, Cycle => true, InternalDegree => 2)

          3                                      2
o7 = 0 : S  <---------------------------------- S  : 0
               {3} | 20a+29b 24a2-17ab+9b2  |
               {3} | 10a-19b 30a2-48ab-7b2  |
               {3} | 32a-36b 19a2-11ab-21b2 |

          3                                2
     1 : S  <---------------------------- S  : 1
               {5} | 20a+21b 24a-36b  |
               {5} | -10a+9b -30a-29b |
               {5} | 32a-7b  19a+19b  |

o7 : ZZdFactorizationMap
 
i8 : g = inducedMap(coker f, D)

                                                                     3
o8 = 0 : cokernel {3} | 20a+29b 24a2-17ab+9b2  | <----------------- S  : 0
                  {3} | 10a-19b 30a2-48ab-7b2  |    {3} | 1 0 0 |
                  {3} | 32a-36b 19a2-11ab-21b2 |    {3} | 0 1 0 |
                                                    {3} | 0 0 1 |

                                                               3
     1 : cokernel {5} | 20a+21b 24a-36b  | <----------------- S  : 1
                  {5} | -10a+9b -30a-29b |    {5} | 1 0 0 |
                  {5} | 32a-7b  19a+19b  |    {5} | 0 1 0 |
                                              {5} | 0 0 1 |

o8 : ZZdFactorizationMap
 
i9 : coker g == 0

o9 = true
 
i10 : prune coker f

o10 = cokernel {3} | -12a+23b 27a2+17ab+2b2 | <-- cokernel {5} | -27a+15b 49a-42b | <-- cokernel {3} | -12a+23b 27a2+17ab+2b2 |
               {3} | -6a-29b  a2+29ab+13b2  |              {5} | a-5b     -36b    |              {3} | -6a-29b  a2+29ab+13b2  |
               {3} | a-39b    -36b2         |              {5} | -31b     a-32b   |              {3} | a-39b    -36b2         |
                                                                                         
      0                                           1                                     0

o10 : ZZdFactorization
 
i11 : (coker f).dd^2 == a^3+b^3

o11 = true

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:4441:0.