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

Synopsis

Operator: ^
Usage:

i = f_[name]

p = f^[name]
Inputs:
- f, a map of complexes,
- name, an array,
Outputs:
- a map of complexes, 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 chain complexes.

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

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

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

o2 : Complex

i3 : C2 = freeResolution coker matrix{{a*b,a*c,b*c}}

      1      3      2
o3 = R  <-- R  <-- R
                    
     0      1      2

o3 : Complex

i4 : D1 = (freeResolution coker matrix{{a,b,c}})

      1      3      3      1
o4 = R  <-- R  <-- R  <-- R
                           
     0      1      2      3

o4 : Complex

i5 : D2 = freeResolution coker matrix{{a^2, b^2, c^2}}[-1]

      1      3      3      1
o5 = R  <-- R  <-- R  <-- R
                           
     1      2      3      4

o5 : Complex

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

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

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

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

          3                               1
     2 : R  <--------------------------- R  : 2
               {2} | 24a-36b-30c-29d |
               {2} | 19a+19b-10c-29d |
               {2} | -8a-22b-29c-24d |

o6 : ComplexMap

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

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

          1                                                                                                                  3
     1 : R  <-------------------------------------------------------------------------------------------------------------- R  : 1
               | 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |

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

o7 : ComplexMap

i8 : h = f ++ g

                    1
o8 = -1 : 0 <----- R  : -1
               0

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

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

          6                                                                3
     2 : R  <------------------------------------------------------------ R  : 2
               {2} | 24a-36b-30c-29d 0                0               |
               {2} | 19a+19b-10c-29d 0                0               |
               {2} | -8a-22b-29c-24d 0                0               |
               {2} | 0               -7b+19c          -38a+5b-16c-22d |
               {2} | 0               -45a+34b+32c+47d 30a-34b-48c-47d |
               {2} | 0               -43a+8b-28c-47d  -39a-23b        |

o8 : ComplexMap

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 chain complexes.

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

i17 : assert(h == map(D1 ++ D2, C1 ++ C2, {{f,0},{0,g}}))

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

i18 : h = (mike => f) ++ (greg => g)

                     1
o18 = -1 : 0 <----- R  : -1
                0

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

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

           6                                                                3
      2 : R  <------------------------------------------------------------ R  : 2
                {2} | 24a-36b-30c-29d 0                0               |
                {2} | 19a+19b-10c-29d 0                0               |
                {2} | -8a-22b-29c-24d 0                0               |
                {2} | 0               -7b+19c          -38a+5b-16c-22d |
                {2} | 0               -45a+34b+32c+47d 30a-34b-48c-47d |
                {2} | 0               -43a+8b-28c-47d  -39a-23b        |

o18 : ComplexMap

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

o19 = true

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

o20 = true

Ways to use this method: