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

Synopsis

Function: components
Usage:

components f
Inputs:
- f, a map of complexes,
Outputs:
- a list, the component maps of a direct sum of maps of complexes

Description

A map of complexes stores its component maps.

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

i2 : C = freeResolution coker vars S

      1      3      3      1
o2 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o2 : Complex

i3 : g1 = id_C

          1             1
o3 = 0 : S  <--------- S  : 0
               | 1 |

          3                     3
     1 : S  <----------------- S  : 1
               {1} | 1 0 0 |
               {1} | 0 1 0 |
               {1} | 0 0 1 |

          3                     3
     2 : S  <----------------- S  : 2
               {2} | 1 0 0 |
               {2} | 0 1 0 |
               {2} | 0 0 1 |

          1                 1
     3 : S  <------------- S  : 3
               {3} | 1 |

o3 : ComplexMap

i4 : g2 = randomComplexMap(C[1], C[2], Boundary => true)

                    1
o4 = -2 : 0 <----- S  : -2
               0

           1                                               3
     -1 : S  <------------------------------------------- S  : -1
                | -41a+30b+29c -19a+5b+10c 29a+8b+46c |

          3                                                     3
     0 : S  <------------------------------------------------- S  : 0
               {1} | 19a+7b-24c  -29a-16b-26c 5b           |
               {1} | 48a+30b-38c 8a-50c       -34a-8b+35c  |
               {1} | 14a-34b     40a-21b+29c  -19a+25b+10c |

          3                            1
     1 : S  <------------------------ S  : 1
               {2} | 34a+16b-29c  |
               {2} | 19a-39b-24c  |
               {2} | -47a-21b-38c |

o4 : ComplexMap

i5 : f = g1 ++ g2

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

           1                                               3
     -1 : S  <------------------------------------------- S  : -1
                | -41a+30b+29c -19a+5b+10c 29a+8b+46c |

          4                                                       4
     0 : S  <--------------------------------------------------- S  : 0
               {0} | 1 0           0            0            |
               {1} | 0 19a+7b-24c  -29a-16b-26c 5b           |
               {1} | 0 48a+30b-38c 8a-50c       -34a-8b+35c  |
               {1} | 0 14a-34b     40a-21b+29c  -19a+25b+10c |

          6                                  4
     1 : S  <------------------------------ S  : 1
               {1} | 1 0 0 0            |
               {1} | 0 1 0 0            |
               {1} | 0 0 1 0            |
               {2} | 0 0 0 34a+16b-29c  |
               {2} | 0 0 0 19a-39b-24c  |
               {2} | 0 0 0 -47a-21b-38c |

          4                     3
     2 : S  <----------------- S  : 2
               {2} | 1 0 0 |
               {2} | 0 1 0 |
               {2} | 0 0 1 |
               {3} | 0 0 0 |

          1                 1
     3 : S  <------------- S  : 3
               {3} | 1 |

o5 : ComplexMap

i6 : assert isWellDefined f

i7 : L = components f

           1             1             
o7 = {0 : S  <--------- S  : 0        ,
                | 1 |                  

           3                     3     
      1 : S  <----------------- S  : 1 
                {1} | 1 0 0 |          
                {1} | 0 1 0 |
                {1} | 0 0 1 |          
                                       
           3                     3     
      2 : S  <----------------- S  : 2 
                {2} | 1 0 0 |          
                {2} | 0 1 0 |
                {2} | 0 0 1 |          
                                       
           1                 1         
      3 : S  <------------- S  : 3     
                {3} | 1 |              
     ------------------------------------------------------------------------
                    1
     -2 : 0 <----- S  : -2                                           }
               0

           1                                               3
     -1 : S  <------------------------------------------- S  : -1
                | -41a+30b+29c -19a+5b+10c 29a+8b+46c |

          3                                                     3
     0 : S  <------------------------------------------------- S  : 0
               {1} | 19a+7b-24c  -29a-16b-26c 5b           |
               {1} | 48a+30b-38c 8a-50c       -34a-8b+35c  |
               {1} | 14a-34b     40a-21b+29c  -19a+25b+10c |

          3                            1
     1 : S  <------------------------ S  : 1
               {2} | 34a+16b-29c  |
               {2} | 19a-39b-24c  |
               {2} | -47a-21b-38c |

o7 : List

i8 : L_0 === g1

o8 = true

i9 : L_1 === g2

o9 = true

i10 : indices f

o10 = {0, 1}

o10 : List

i11 : f' = (greg => g1) ++ (mike => g2)

                     1
o11 = -2 : 0 <----- S  : -2
                0

            1                                               3
      -1 : S  <------------------------------------------- S  : -1
                 | -41a+30b+29c -19a+5b+10c 29a+8b+46c |

           4                                                       4
      0 : S  <--------------------------------------------------- S  : 0
                {0} | 1 0           0            0            |
                {1} | 0 19a+7b-24c  -29a-16b-26c 5b           |
                {1} | 0 48a+30b-38c 8a-50c       -34a-8b+35c  |
                {1} | 0 14a-34b     40a-21b+29c  -19a+25b+10c |

           6                                  4
      1 : S  <------------------------------ S  : 1
                {1} | 1 0 0 0            |
                {1} | 0 1 0 0            |
                {1} | 0 0 1 0            |
                {2} | 0 0 0 34a+16b-29c  |
                {2} | 0 0 0 19a-39b-24c  |
                {2} | 0 0 0 -47a-21b-38c |

           4                     3
      2 : S  <----------------- S  : 2
                {2} | 1 0 0 |
                {2} | 0 1 0 |
                {2} | 0 0 1 |
                {3} | 0 0 0 |

           1                 1
      3 : S  <------------- S  : 3
                {3} | 1 |

o11 : ComplexMap

i12 : components f'

            1             1             
o12 = {0 : S  <--------- S  : 0        ,
                 | 1 |                  

            3                     3     
       1 : S  <----------------- S  : 1 
                 {1} | 1 0 0 |          
                 {1} | 0 1 0 |
                 {1} | 0 0 1 |          
                                        
            3                     3     
       2 : S  <----------------- S  : 2 
                 {2} | 1 0 0 |          
                 {2} | 0 1 0 |
                 {2} | 0 0 1 |          
                                        
            1                 1         
       3 : S  <------------- S  : 3     
                 {3} | 1 |              
      -----------------------------------------------------------------------
                     1
      -2 : 0 <----- S  : -2                                           }
                0

            1                                               3
      -1 : S  <------------------------------------------- S  : -1
                 | -41a+30b+29c -19a+5b+10c 29a+8b+46c |

           3                                                     3
      0 : S  <------------------------------------------------- S  : 0
                {1} | 19a+7b-24c  -29a-16b-26c 5b           |
                {1} | 48a+30b-38c 8a-50c       -34a-8b+35c  |
                {1} | 14a-34b     40a-21b+29c  -19a+25b+10c |

           3                            1
      1 : S  <------------------------ S  : 1
                {2} | 34a+16b-29c  |
                {2} | 19a-39b-24c  |
                {2} | -47a-21b-38c |

o12 : List

i13 : indices f'

o13 = {greg, mike}

o13 : List

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

i14 : f'_[mike]

            1                                               3
o14 = -1 : S  <------------------------------------------- S  : -1
                 | -41a+30b+29c -19a+5b+10c 29a+8b+46c |

           4                                                     3
      0 : S  <------------------------------------------------- S  : 0
                {0} | 0           0            0            |
                {1} | 19a+7b-24c  -29a-16b-26c 5b           |
                {1} | 48a+30b-38c 8a-50c       -34a-8b+35c  |
                {1} | 14a-34b     40a-21b+29c  -19a+25b+10c |

           6                            1
      1 : S  <------------------------ S  : 1
                {1} | 0            |
                {1} | 0            |
                {1} | 0            |
                {2} | 34a+16b-29c  |
                {2} | 19a-39b-24c  |
                {2} | -47a-21b-38c |

o14 : ComplexMap

i15 : f'^[greg]

           1                   4
o15 = 0 : S  <--------------- S  : 0
                | 1 0 0 0 |

           3                       4
      1 : S  <------------------- S  : 1
                {1} | 1 0 0 0 |
                {1} | 0 1 0 0 |
                {1} | 0 0 1 0 |

           3                     3
      2 : S  <----------------- S  : 2
                {2} | 1 0 0 |
                {2} | 0 1 0 |
                {2} | 0 0 1 |

           1                 1
      3 : S  <------------- S  : 3
                {3} | 1 |

o15 : ComplexMap

i16 : f^[0]

           1                   4
o16 = 0 : S  <--------------- S  : 0
                | 1 0 0 0 |

           3                       4
      1 : S  <------------------- S  : 1
                {1} | 1 0 0 0 |
                {1} | 0 1 0 0 |
                {1} | 0 0 1 0 |

           3                     3
      2 : S  <----------------- S  : 2
                {2} | 1 0 0 |
                {2} | 0 1 0 |
                {2} | 0 0 1 |

           1                 1
      3 : S  <------------- S  : 3
                {3} | 1 |

o16 : ComplexMap

i17 : f_[0]

           4                 1
o17 = 0 : S  <------------- S  : 0
                {0} | 1 |
                {1} | 0 |
                {1} | 0 |
                {1} | 0 |

           6                     3
      1 : S  <----------------- S  : 1
                {1} | 1 0 0 |
                {1} | 0 1 0 |
                {1} | 0 0 1 |
                {2} | 0 0 0 |
                {2} | 0 0 0 |
                {2} | 0 0 0 |

           4                     3
      2 : S  <----------------- S  : 2
                {2} | 1 0 0 |
                {2} | 0 1 0 |
                {2} | 0 0 1 |
                {3} | 0 0 0 |

           1                 1
      3 : S  <------------- S  : 3
                {3} | 1 |

o17 : ComplexMap

Ways to use this method: