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components(ZZdFactorization) -- list the components of a direct sum

Description

A ZZ/d-graded factorization which has been constructed as a direct sum stores its component factorizations.

i1 : S = ZZ/101[a,b,c];
 
i2 : f = a^3 + b^3 + c^3;
 
i3 : C1 = randomTailMF(f)

      6      6      6
o3 = S  <-- S  <-- S
                    
     0      1      0

o3 : ZZdFactorization
 
i4 : C2 = randomTailMF(f)

      12      12      12
o4 = S   <-- S   <-- S
                      
     0       1       0

o4 : ZZdFactorization
 
i5 : D = C1 ++ C2

      18      18      18
o5 = S   <-- S   <-- S
                      
     0       1       0

o5 : ZZdFactorization
 
i6 : L = components D

       6      6      6   12      12      12
o6 = {S  <-- S  <-- S , S   <-- S   <-- S  }
                                         
      0      1      0   0       1       0

o6 : List
 
i7 : L_0 === C1

o7 = true
 
i8 : L_1 === C2

o8 = true
 
i9 : E = (peanut => C1) ++ (butter => C2)

      18      18      18
o9 = S   <-- S   <-- S
                      
     0       1       0

o9 : ZZdFactorization
 
i10 : components E

        6      6      6   12      12      12
o10 = {S  <-- S  <-- S , S   <-- S   <-- S  }
                                          
       0      1      0   0       1       0

o10 : List

The names of the component complexes are called indices, and are used to access the relevant inclusion and projection maps.

i11 : indices D

o11 = {0, 1}

o11 : List
 
i12 : D^[0]

           6                                                    18
o12 = 0 : S  <------------------------------------------------ S   : 0
                {9}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {9}  | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {9}  | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {10} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {10} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {10} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |

           6                                                    18
      1 : S  <------------------------------------------------ S   : 1
                {11} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {11} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {11} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {11} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {11} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {11} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |

o12 : ZZdFactorizationMap
 
i13 : indices E

o13 = {peanut, butter}

o13 : List
 
i14 : E_[butter]

           18                                        12
o14 = 0 : S   <------------------------------------ S   : 0
                 {9}  | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {9}  | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {9}  | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {10} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {10} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {10} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {7}  | 1 0 0 0 0 0 0 0 0 0 0 0 |
                 {7}  | 0 1 0 0 0 0 0 0 0 0 0 0 |
                 {7}  | 0 0 1 0 0 0 0 0 0 0 0 0 |
                 {7}  | 0 0 0 1 0 0 0 0 0 0 0 0 |
                 {7}  | 0 0 0 0 1 0 0 0 0 0 0 0 |
                 {7}  | 0 0 0 0 0 1 0 0 0 0 0 0 |
                 {7}  | 0 0 0 0 0 0 1 0 0 0 0 0 |
                 {7}  | 0 0 0 0 0 0 0 1 0 0 0 0 |
                 {7}  | 0 0 0 0 0 0 0 0 1 0 0 0 |
                 {7}  | 0 0 0 0 0 0 0 0 0 1 0 0 |
                 {7}  | 0 0 0 0 0 0 0 0 0 0 1 0 |
                 {7}  | 0 0 0 0 0 0 0 0 0 0 0 1 |

           18                                        12
      1 : S   <------------------------------------ S   : 1
                 {11} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {11} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {11} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {11} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {11} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {11} | 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {8}  | 1 0 0 0 0 0 0 0 0 0 0 0 |
                 {8}  | 0 1 0 0 0 0 0 0 0 0 0 0 |
                 {8}  | 0 0 1 0 0 0 0 0 0 0 0 0 |
                 {8}  | 0 0 0 1 0 0 0 0 0 0 0 0 |
                 {8}  | 0 0 0 0 1 0 0 0 0 0 0 0 |
                 {8}  | 0 0 0 0 0 1 0 0 0 0 0 0 |
                 {9}  | 0 0 0 0 0 0 1 0 0 0 0 0 |
                 {9}  | 0 0 0 0 0 0 0 1 0 0 0 0 |
                 {9}  | 0 0 0 0 0 0 0 0 1 0 0 0 |
                 {9}  | 0 0 0 0 0 0 0 0 0 1 0 0 |
                 {9}  | 0 0 0 0 0 0 0 0 0 0 1 0 |
                 {9}  | 0 0 0 0 0 0 0 0 0 0 0 1 |

o14 : ZZdFactorizationMap

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:1298:0.