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SimplicialModuleMap | SimplicialModuleMap -- join or concatenate maps horizontally

Description

Given simplicial module maps with the same target, this method constructs the associated map from the direct sum of the sources to the target.

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 : D = simplicialModule(freeResolution coker matrix{{a^2,b^2,c*d}}, Degeneracy => true)

      1      4      10      20
o4 = R  <-- R  <-- R   <-- R  <-- ...
                            
     0      1      2       3

o4 : SimplicialModule
 
i5 : f = randomSimplicialMap(D, C1)

          1                                                           3
o5 = 0 : R  <------------------------------------------------------- R  : 0
               | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d |

          4                                                                          6
     1 : R  <---------------------------------------------------------------------- R  : 1
               {0} | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d 0   0  0   |
               {2} | 0               0               0               -38 21 -47 |
               {2} | 0               0               0               -16 34 -39 |
               {2} | 0               0               0               39  19 -18 |

          10                                                                                       10
     2 : R   <----------------------------------------------------------------------------------- R   : 2
                {0} | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d 0   0  0   0   0  0   0 |
                {2} | 0               0               0               -38 21 -47 0   0  0   0 |
                {2} | 0               0               0               -16 34 -39 0   0  0   0 |
                {2} | 0               0               0               39  19 -18 0   0  0   0 |
                {2} | 0               0               0               0   0  0   -38 21 -47 0 |
                {2} | 0               0               0               0   0  0   -16 34 -39 0 |
                {2} | 0               0               0               0   0  0   39  19 -18 0 |
                {4} | 0               0               0               0   0  0   0   0  0   0 |
                {4} | 0               0               0               0   0  0   0   0  0   0 |
                {4} | 0               0               0               0   0  0   0   0  0   0 |

o5 : SimplicialModuleMap
 
i6 : g = randomSimplicialMap(D, C2)

          1               1
o6 = 0 : R  <----------- R  : 0
               | -13 |

          4                              4
     1 : R  <-------------------------- R  : 1
               {0} | -13 0   0   0  |
               {2} | 0   -43 -47 16 |
               {2} | 0   -15 38  22 |
               {2} | 0   -28 2   45 |

          10                                             9
     2 : R   <----------------------------------------- R  : 2
                {0} | -13 0   0   0  0   0   0  0 0 |
                {2} | 0   -43 -47 16 0   0   0  0 0 |
                {2} | 0   -15 38  22 0   0   0  0 0 |
                {2} | 0   -28 2   45 0   0   0  0 0 |
                {2} | 0   0   0   0  -43 -47 16 0 0 |
                {2} | 0   0   0   0  -15 38  22 0 0 |
                {2} | 0   0   0   0  -28 2   45 0 0 |
                {4} | 0   0   0   0  0   0   0  0 0 |
                {4} | 0   0   0   0  0   0   0  0 0 |
                {4} | 0   0   0   0  0   0   0  0 0 |

o6 : SimplicialModuleMap
 
i7 : h = f|g

          1                                                               4
o7 = 0 : R  <----------------------------------------------------------- R  : 0
               | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d -13 |

          4                                                                                         10
     1 : R  <------------------------------------------------------------------------------------- R   : 1
               {0} | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d 0   0  0   -13 0   0   0  |
               {2} | 0               0               0               -38 21 -47 0   -43 -47 16 |
               {2} | 0               0               0               -16 34 -39 0   -15 38  22 |
               {2} | 0               0               0               39  19 -18 0   -28 2   45 |

          10                                                                                                                     19
     2 : R   <----------------------------------------------------------------------------------------------------------------- R   : 2
                {0} | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d 0   0  0   0   0  0   0 -13 0   0   0  0   0   0  0 0 |
                {2} | 0               0               0               -38 21 -47 0   0  0   0 0   -43 -47 16 0   0   0  0 0 |
                {2} | 0               0               0               -16 34 -39 0   0  0   0 0   -15 38  22 0   0   0  0 0 |
                {2} | 0               0               0               39  19 -18 0   0  0   0 0   -28 2   45 0   0   0  0 0 |
                {2} | 0               0               0               0   0  0   -38 21 -47 0 0   0   0   0  -43 -47 16 0 0 |
                {2} | 0               0               0               0   0  0   -16 34 -39 0 0   0   0   0  -15 38  22 0 0 |
                {2} | 0               0               0               0   0  0   39  19 -18 0 0   0   0   0  -28 2   45 0 0 |
                {4} | 0               0               0               0   0  0   0   0  0   0 0   0   0   0  0   0   0  0 0 |
                {4} | 0               0               0               0   0  0   0   0  0   0 0   0   0   0  0   0   0  0 0 |
                {4} | 0               0               0               0   0  0   0   0  0   0 0   0   0   0  0   0   0  0 0 |

o7 : SimplicialModuleMap
 
i8 : assert isWellDefined h
 
i9 : assert(source h === source f ++ source g)
 
i10 : assert(target h === target f)

See also

Ways to use this method:

The source of this document is in SimplicialModules/SimplicialModuleDOC.m2:2628:0.