SimplicialModuleMap ++ SimplicialModuleMap -- direct sum of simplicial module maps

Operator: ++
Usage:

h = f ++ g

h = directSum(f,g,...)

h = directSum(name1 => f, name2 => g, ...)
Inputs:
- f, a Map of Simplicial Modules,
- g, a Map of Simplicial Modules,
Outputs:
- h, a Map of Simplicial Modules, that is the direct sum of the input simplicial module maps

Description

The direct sum of two simplicial module maps is a simplicial module map from the direct sum of the sources to the direct sum of the targets.

First, we define some non-trivial maps of simplicial modules.

i1 : R = ZZ/101[a..d];

i2 : C1 = simplicialModule((freeResolution coker matrix{{a,b,c}})[1], Degeneracy => true)

      3      6      10
o2 = R  <-- R  <-- R  <-- ...
                    
     0      1      2

o2 : SimplicialModule

i3 : C2 = simplicialModule(freeResolution coker matrix{{a*b,a*c,b*c}}, Degeneracy => true)

      1      4      9
o3 = R  <-- R  <-- R <-- ...
                    
     0      1      2

o3 : SimplicialModule

i4 : D1 = simplicialModule((freeResolution coker matrix{{a,b,c}}),2,  Degeneracy => true)

      1      4      10
o4 = R  <-- R  <-- R  <-- ...
                    
     0      1      2

o4 : SimplicialModule

i5 : D2 = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}}[-1], 2, Degeneracy => true)

            1      5
o5 = 0 <-- R  <-- R <-- ...
                   
     0     1      2

o5 : SimplicialModule

i6 : f = randomSimplicialMap(D1, C1, Cycle => true)

          1                                                        3
o6 = 0 : R  <---------------------------------------------------- R  : 0
               | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d |

          4                                                                                                             6
     1 : R  <--------------------------------------------------------------------------------------------------------- R  : 1
               {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0               0               0                |
               {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          |
               {1} | 0               0              0             46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  |
               {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d |

          10                                                                                                                                                                              10
     2 : R   <-------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 2
                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0               0               0                0               0               0                0               |
                {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0               0                0               |
                {1} | 0               0              0             46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0               0                0               |
                {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0                0               |
                {1} | 0               0              0             0               0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               |
                {1} | 0               0              0             0               0               0                46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               |
                {1} | 0               0              0             0               0               0                -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               |
                {2} | 0               0              0             0               0               0                0               0               0                24a-36b-30c-29d |
                {2} | 0               0              0             0               0               0                0               0               0                19a+19b-10c-29d |
                {2} | 0               0              0             0               0               0                0               0               0                -8a-22b-29c-24d |

o6 : SimplicialModuleMap

i7 : g = randomSimplicialMap(D2, C2, Cycle => true)

                   1
o7 = 0 : 0 <----- R  : 0
              0

          1                                                                                                                    4
     1 : R  <---------------------------------------------------------------------------------------------------------------- R  : 1
               | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |

          5                                                                                                                                                                                                                                                                9
     2 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ R  : 2
               {0} | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                                  0                                 0                                 0                0               |
               {0} | 0 0                                  0                                 0                                 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                0               |
               {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 -7b+19c          -38a+5b-16c-22d |
               {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 -45a+34b+32c+47d 30a-34b-48c-47d |
               {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 -43a+8b-28c-47d  -39a-23b        |

o7 : SimplicialModuleMap

i8 : h = f ++ g

          1                                                          4
o8 = 0 : R  <------------------------------------------------------ R  : 0
               | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0 |

          5                                                                                                                                                                                                                      10
     1 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ R   : 1
               {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0               0               0                0 0                                  0                                 0                                 |
               {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0 0                                  0                                 0                                 |
               {1} | 0               0              0             46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0 0                                  0                                 0                                 |
               {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0 0                                  0                                 0                                 |
               {0} | 0               0              0             0               0               0                0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |

          15                                                                                                                                                                                                                                                                                                                                                                                                                               19
     2 : R   <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 2
                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {1} | 0               0              0             46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {1} | 0               0              0             0               0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {1} | 0               0              0             0               0               0                46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {1} | 0               0              0             0               0               0                -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {2} | 0               0              0             0               0               0                0               0               0                24a-36b-30c-29d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {2} | 0               0              0             0               0               0                0               0               0                19a+19b-10c-29d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {2} | 0               0              0             0               0               0                0               0               0                -8a-22b-29c-24d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               |
                {0} | 0               0              0             0               0               0                0               0               0                0               0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                                  0                                 0                                 0                0               |
                {0} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                0               |
                {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -7b+19c          -38a+5b-16c-22d |
                {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -45a+34b+32c+47d 30a-34b-48c-47d |
                {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -43a+8b-28c-47d  -39a-23b        |

o8 : SimplicialModuleMap

i9 : assert isWellDefined h

i10 : assert isSimplicialMorphism h

The direct sum of any sequence of simplicial module maps can be computed as follows.

i11 : directSum(f, g, f[2])

           4                                                                              5
o11 = 0 : R  <-------------------------------------------------------------------------- R  : 0
                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0 0               |
                {2} | 0               0              0             0 24a-36b-30c-29d |
                {2} | 0               0              0             0 19a+19b-10c-29d |
                {2} | 0               0              0             0 -8a-22b-29c-24d |

           8                                                                                                                                                                                                                                      11
      1 : R  <---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 1
                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0               0               0                0 0                                  0                                 0                                 0               |
                {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0 0                                  0                                 0                                 0               |
                {1} | 0               0              0             46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0 0                                  0                                 0                                 0               |
                {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0 0                                  0                                 0                                 0               |
                {0} | 0               0              0             0               0               0                0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0               |
                {2} | 0               0              0             0               0               0                0 0                                  0                                 0                                 24a-36b-30c-29d |
                {2} | 0               0              0             0               0               0                0 0                                  0                                 0                                 19a+19b-10c-29d |
                {2} | 0               0              0             0               0               0                0 0                                  0                                 0                                 -8a-22b-29c-24d |

           18                                                                                                                                                                                                                                                                                                                                                                                                                                               20
      2 : R   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 2
                 {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             0               0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             0               0               0                46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             0               0               0                -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0               0              0             0               0               0                0               0               0                24a-36b-30c-29d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0               0              0             0               0               0                0               0               0                19a+19b-10c-29d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0               0              0             0               0               0                0               0               0                -8a-22b-29c-24d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {0} | 0               0              0             0               0               0                0               0               0                0               0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                                  0                                 0                                 0                0               0               |
                 {0} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                0               0               |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -7b+19c          -38a+5b-16c-22d 0               |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -45a+34b+32c+47d 30a-34b-48c-47d 0               |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -43a+8b-28c-47d  -39a-23b        0               |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               24a-36b-30c-29d |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               19a+19b-10c-29d |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               -8a-22b-29c-24d |

o11 : SimplicialModuleMap

i12 : h2 = directSum(peanut => f, butter => g, jelly => f[2])

           4                                                                              5
o12 = 0 : R  <-------------------------------------------------------------------------- R  : 0
                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0 0               |
                {2} | 0               0              0             0 24a-36b-30c-29d |
                {2} | 0               0              0             0 19a+19b-10c-29d |
                {2} | 0               0              0             0 -8a-22b-29c-24d |

           8                                                                                                                                                                                                                                      11
      1 : R  <---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 1
                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0               0               0                0 0                                  0                                 0                                 0               |
                {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0 0                                  0                                 0                                 0               |
                {1} | 0               0              0             46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0 0                                  0                                 0                                 0               |
                {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0 0                                  0                                 0                                 0               |
                {0} | 0               0              0             0               0               0                0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0               |
                {2} | 0               0              0             0               0               0                0 0                                  0                                 0                                 24a-36b-30c-29d |
                {2} | 0               0              0             0               0               0                0 0                                  0                                 0                                 19a+19b-10c-29d |
                {2} | 0               0              0             0               0               0                0 0                                  0                                 0                                 -8a-22b-29c-24d |

           18                                                                                                                                                                                                                                                                                                                                                                                                                                               20
      2 : R   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 2
                 {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             0               0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             0               0               0                46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0               0              0             0               0               0                -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0               0              0             0               0               0                0               0               0                24a-36b-30c-29d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0               0              0             0               0               0                0               0               0                19a+19b-10c-29d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0               0              0             0               0               0                0               0               0                -8a-22b-29c-24d 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {0} | 0               0              0             0               0               0                0               0               0                0               0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                                  0                                 0                                 0                0               0               |
                 {0} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                0               0               |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -7b+19c          -38a+5b-16c-22d 0               |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -45a+34b+32c+47d 30a-34b-48c-47d 0               |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 -43a+8b-28c-47d  -39a-23b        0               |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               24a-36b-30c-29d |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               19a+19b-10c-29d |
                 {2} | 0               0              0             0               0               0                0               0               0                0               0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               -8a-22b-29c-24d |

o12 : SimplicialModuleMap

i13 : indices h2

o13 = {peanut, butter, jelly}

o13 : List

i14 : h2_[butter,jelly]

           4                                 2
o14 = 0 : R  <----------------------------- R  : 0
                {0} | 0 0               |
                {2} | 0 24a-36b-30c-29d |
                {2} | 0 19a+19b-10c-29d |
                {2} | 0 -8a-22b-29c-24d |

           8                                                                                                                                        5
      1 : R  <------------------------------------------------------------------------------------------------------------------------------------ R  : 1
                {0} | 0 0                                  0                                 0                                 0               |
                {1} | 0 0                                  0                                 0                                 0               |
                {1} | 0 0                                  0                                 0                                 0               |
                {1} | 0 0                                  0                                 0                                 0               |
                {0} | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0               |
                {2} | 0 0                                  0                                 0                                 24a-36b-30c-29d |
                {2} | 0 0                                  0                                 0                                 19a+19b-10c-29d |
                {2} | 0 0                                  0                                 0                                 -8a-22b-29c-24d |

           18                                                                                                                                                                                                                                                                                10
      2 : R   <---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 2
                 {0} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {1} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0               |
                 {0} | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                                  0                                 0                                 0                0               0               |
                 {0} | 0 0                                  0                                 0                                 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                0               0               |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 -7b+19c          -38a+5b-16c-22d 0               |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 -45a+34b+32c+47d 30a-34b-48c-47d 0               |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 -43a+8b-28c-47d  -39a-23b        0               |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               24a-36b-30c-29d |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               19a+19b-10c-29d |
                 {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                0               -8a-22b-29c-24d |

o14 : SimplicialModuleMap

i15 : assert(source oo == C2 ++ C1[2])

One can easily obtain the compositions with canonical injections and surjections.

i16 : h_[0]^[0] == f

o16 = true

i17 : h_[1]^[1] == g

o17 = true

i18 : h_[0]^[1] == 0

o18 = true

i19 : h_[1]^[0] == 0

o19 = true

i20 : h_[0] == h * (C1 ++ C2)_[0]

o20 = true

i21 : h_[1] == h * (C1 ++ C2)_[1]

o21 = true

i22 : h^[0] == (D1 ++ D2)^[0] * h

o22 = true

i23 : h^[1] == (D1 ++ D2)^[1] * h

o23 = true

Ways to use this method:

SimplicialModuleMap ++ SimplicialModuleMap -- direct sum of simplicial module maps

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:2729:0.

SimplicialModuleMap ++ SimplicialModuleMap -- direct sum of simplicial module maps

Description

See also

Ways to use this method: