SimplicialModuleMap ^ Array -- the composition with the canonical inclusion or projection map

Operator: ^
Usage:

i = f_[name]

p = f^[name]
Inputs:
- f, a Map of Simplicial Modules,
- name, an array,
Outputs:
- a Map of Simplicial Modules, i is the composition of f with the canonical inclusion and p is the composition of the canonical projection with f

Description

The direct sum is an n-ary operator with projection and inclusion maps from each component satisfying appropriate identities.

One can access these maps as follows. 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], 3, Degeneracy => true)

      3      6      10      15
o2 = R  <-- R  <-- R   <-- R  <-- ...
                            
     0      1      2       3

o2 : SimplicialModule

i3 : C2 = simplicialModule(freeResolution coker matrix{{a*b,a*c,b*c}}, 3, Degeneracy => true)

      1      4      9      16
o3 = R  <-- R  <-- R  <-- R  <-- ...
                           
     0      1      2      3

o3 : SimplicialModule

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

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

o4 : SimplicialModule

i5 : D2 = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}}[-1], 3, Degeneracy => true)

            1      5      15
o5 = 0 <-- R  <-- R  <-- R  <-- ...
                          
     0     1      2      3

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 |

          20                                                                                                                                                                                                                                                               15
     3 : R   <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 3
                {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               |
                {1} | 0               0              0             -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          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               |
                {1} | 0               0              0             -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 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               |
                {1} | 0               0              0             0               0               0                46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  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               |
                {1} | 0               0              0             0               0               0                0               0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0               0               |
                {1} | 0               0              0             0               0               0                0               0               0                46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0               0               |
                {1} | 0               0              0             0               0               0                0               0               0                -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0               |
                {2} | 0               0              0             0               0               0                0               0               0                0               0               0                24a-36b-30c-29d 0               0               |
                {2} | 0               0              0             0               0               0                0               0               0                0               0               0                19a+19b-10c-29d 0               0               |
                {2} | 0               0              0             0               0               0                0               0               0                0               0               0                -8a-22b-29c-24d 0               0               |
                {2} | 0               0              0             0               0               0                0               0               0                0               0               0                0               24a-36b-30c-29d 0               |
                {2} | 0               0              0             0               0               0                0               0               0                0               0               0                0               19a+19b-10c-29d 0               |
                {2} | 0               0              0             0               0               0                0               0               0                0               0               0                0               -8a-22b-29c-24d 0               |
                {2} | 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               19a+19b-10c-29d |
                {2} | 0               0              0             0               0               0                0               0               0                0               0               0                0               0               -8a-22b-29c-24d |
                {3} | 0               0              0             0               0               0                0               0               0                0               0               0                0               0               0               |

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        |

          15                                                                                                                                                                                                                                                                                                                                                                                                                                         16
     3 : R   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 3
                {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                                 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                                 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               |
                {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                                  0                                 0                                 -7b+19c          -38a+5b-16c-22d 0                0               0                0               |
                {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                                  0                                 0                                 -45a+34b+32c+47d 30a-34b-48c-47d 0                0               0                0               |
                {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                                  0                                 0                                 -43a+8b-28c-47d  -39a-23b        0                0               0                0               |
                {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                                  0                                 0                                 0                0               -7b+19c          -38a+5b-16c-22d 0                0               |
                {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                                  0                                 0                                 0                0               -45a+34b+32c+47d 30a-34b-48c-47d 0                0               |
                {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                                  0                                 0                                 0                0               -43a+8b-28c-47d  -39a-23b        0                0               |
                {2} | 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               -45a+34b+32c+47d 30a-34b-48c-47d |
                {2} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0                0               -43a+8b-28c-47d  -39a-23b        |
                {4} | 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               |
                {4} | 0 0                                  0                                 0                                 0                                  0                                 0                                 0                                  0                                 0                                 0                0               0                0               0                0               |

o7 : SimplicialModuleMap

i8 : h = f ++ g;

The four basic maps are the inclusion from each summand of the source and the projection to each summand of the target.

i9 : h_[0] == h * (C1 ++ C2)_[0]

o9 = true

i10 : h_[1] == h * (C1 ++ C2)_[1]

o10 = true

i11 : h^[0] == (D1 ++ D2)^[0] * h

o11 = true

i12 : h^[1] == (D1 ++ D2)^[1] * h

o12 = true

These can be combined to obtain the blocks of the map of simplicial modules.

i13 : h_[0]^[0] == f

o13 = true

i14 : h_[1]^[1] == g

o14 = true

i15 : h_[0]^[1] == 0

o15 = true

i16 : h_[1]^[0] == 0

o16 = true

The default names for the components are the non-negative integers. However, one can choose any name.

i17 : h = (chicken => f) ++ (nuggets => g);

i18 : indices h

o18 = {chicken, nuggets}

o18 : List

i19 : h_[chicken]^[chicken] == f

o19 = true

i20 : h_[nuggets]^[nuggets] == g

o20 = true

Ways to use this method:

SimplicialModuleMap ^ Array -- the composition with the canonical inclusion or projection map

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:2962:0.

SimplicialModuleMap ^ Array -- the composition with the canonical inclusion or projection map

Description

See also

Ways to use this method: