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ZZdFactorizationMap || ZZdFactorizationMap -- join or concatenate maps vertically

Description

Given ZZ/d-graded factorization maps with the same source, this method constructs the associated map from the source to the direct sum of the targets.

First, we define some non-trivial maps of factorizations.

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 : D1 = randomTailMF(a^3+b^3, 2, 4, 2)

      4      4      4
o6 = S  <-- S  <-- S
                    
     0      1      0

o6 : ZZdFactorization
 
i7 : D2 = tailMF (m^2)

      3      3      3
o7 = S  <-- S  <-- S
                    
     0      1      0

o7 : ZZdFactorization
 
i8 : f = randomFactorizationMap(D1, C)

          4                                                  2
o8 = 0 : S  <---------------------------------------------- S  : 0
               | -13a2-43ab-15b2 -47a3+47a2b+19ab2-16b3 |
               | -28a2-47ab+38b2 7a3+15a2b-23ab2+39b3   |
               | 2a2+16ab+22b2   43a3-17a2b-11ab2+48b3  |
               | 45a2-34ab-48b2  36a3+35a2b+11ab2-38b3  |

          4                                                           2
     1 : S  <------------------------------------------------------- S  : 1
               {1} | 33a3+40a2b+11ab2+46b3 15a3-37a2b-13ab2-10b3 |
               {1} | -28a3+a2b-3ab2+22b3   30a3-18a2b+39ab2+27b3 |
               {2} | -47a2-23ab-7b2        -22a2+32ab-9b2        |
               {2} | 2a2+29ab-47b2         -32a2-20ab+24b2       |

o8 : ZZdFactorizationMap
 
i9 : g = randomFactorizationMap(D2, C, InternalDegree => 2)

          3                                        2
o9 = 0 : S  <------------------------------------ S  : 0
               {3} | -30a-48b -49a2-33ab-19b2 |
               {3} | -15a+39b 17a2-20ab+44b2  |
               {3} | 33b      -39a2+36ab+9b2  |

          3                               2
     1 : S  <--------------------------- S  : 1
               {5} | -39a+4b -11a-8b |
               {5} | 13a-26b 43a-8b  |
               {5} | 22a-49b 36a-3b  |

o9 : ZZdFactorizationMap
 
i10 : h = f||g

           7                                                      2
o10 = 0 : S  <-------------------------------------------------- S  : 0
                {0} | -13a2-43ab-15b2 -47a3+47a2b+19ab2-16b3 |
                {0} | -28a2-47ab+38b2 7a3+15a2b-23ab2+39b3   |
                {0} | 2a2+16ab+22b2   43a3-17a2b-11ab2+48b3  |
                {0} | 45a2-34ab-48b2  36a3+35a2b+11ab2-38b3  |
                {3} | -30a-48b        -49a2-33ab-19b2        |
                {3} | -15a+39b        17a2-20ab+44b2         |
                {3} | 33b             -39a2+36ab+9b2         |

           7                                                           2
      1 : S  <------------------------------------------------------- S  : 1
                {1} | 33a3+40a2b+11ab2+46b3 15a3-37a2b-13ab2-10b3 |
                {1} | -28a3+a2b-3ab2+22b3   30a3-18a2b+39ab2+27b3 |
                {2} | -47a2-23ab-7b2        -22a2+32ab-9b2        |
                {2} | 2a2+29ab-47b2         -32a2-20ab+24b2       |
                {5} | -39a+4b               -11a-8b               |
                {5} | 13a-26b               43a-8b                |
                {5} | 22a-49b               36a-3b                |

o10 : ZZdFactorizationMap
 
i11 : assert isWellDefined h
 
i12 : assert(target h === target f ++ target g)
 
i13 : assert(source h === source f)

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:4206:0.