coimage(SimplicialModuleMap) -- make the coimage of a map of simplicial modules

Function: coimage
Usage:

coimage f
Inputs:
- f, a Map of Simplicial Modules,
Outputs:
- a Simplicial Module,

Description

The coimage of a simplicial module map $f : C \to D$ is the simplicial module $E$ whose $i$-th term is $coimage(f_i)$, and whose face/degeneracy map is induced from the face/degeneracy map on the source.

In the following example, we first construct a random simplicial morphism $f : C \to D$.

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

i2 : C = simplicialModule(freeResolution ideal(b^2-a*c, b*c-a*d, c^2-b*d), 3, Degeneracy => true)

      1      4      9      16
o2 = S  <-- S  <-- S  <-- S  <-- ...
                           
     0      1      2      3

o2 : SimplicialModule

i3 : D = simplicialModule(freeResolution ideal(a,b,c), Degeneracy => true)

      1      4      10      20
o3 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o3 : SimplicialModule

i4 : f = randomSimplicialMap(D, C, Cycle => true, InternalDegree => 0)

          1               1
o4 = 0 : S  <----------- S  : 0
               | -22 |

          4                                                        4
     1 : S  <---------------------------------------------------- S  : 1
               {0} | -22 0            0           0           |
               {1} | 0   36b+3c       30b-19c+22d -29b-10c    |
               {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d |
               {1} | 0   19a-29b      19a-8b      10a-29b-22c |

          10                                                                                                                               9
     2 : S   <--------------------------------------------------------------------------------------------------------------------------- S  : 2
                {0} | -22 0            0           0           0            0           0           0                0                |
                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0           0           0                0                |
                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0           0           0                0                |
                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0           0           0                0                |
                {1} | 0   0            0           0           36b+3c       30b-19c+22d -29b-10c    0                0                |
                {1} | 0   0            0           0           -36a-22b+29c -30a-14c    29a+29c+22d 0                0                |
                {1} | 0   0            0           0           19a-29b      19a-8b      10a-29b-22c 0                0                |
                {2} | 0   0            0           0           0            0           0           -29a-30b+31c-22d 29b+6c-36d       |
                {2} | 0   0            0           0           0            0           0           -10a+24b+3c      34b-19c+19d      |
                {2} | 0   0            0           0           0            0           0           24a-8b+29c       -24a-29b-14c-29d |

          20                                                                                                                                                                                                                                        16
     3 : S   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ S   : 3
                {0} | -22 0            0           0           0            0           0           0            0           0           0                0                0                0                0                0                |
                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0           0           0            0           0           0                0                0                0                0                0                |
                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0           0           0            0           0           0                0                0                0                0                0                |
                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0           0           0            0           0           0                0                0                0                0                0                |
                {1} | 0   0            0           0           36b+3c       30b-19c+22d -29b-10c    0            0           0           0                0                0                0                0                0                |
                {1} | 0   0            0           0           -36a-22b+29c -30a-14c    29a+29c+22d 0            0           0           0                0                0                0                0                0                |
                {1} | 0   0            0           0           19a-29b      19a-8b      10a-29b-22c 0            0           0           0                0                0                0                0                0                |
                {1} | 0   0            0           0           0            0           0           36b+3c       30b-19c+22d -29b-10c    0                0                0                0                0                0                |
                {1} | 0   0            0           0           0            0           0           -36a-22b+29c -30a-14c    29a+29c+22d 0                0                0                0                0                0                |
                {1} | 0   0            0           0           0            0           0           19a-29b      19a-8b      10a-29b-22c 0                0                0                0                0                0                |
                {2} | 0   0            0           0           0            0           0           0            0           0           -29a-30b+31c-22d 29b+6c-36d       0                0                0                0                |
                {2} | 0   0            0           0           0            0           0           0            0           0           -10a+24b+3c      34b-19c+19d      0                0                0                0                |
                {2} | 0   0            0           0           0            0           0           0            0           0           24a-8b+29c       -24a-29b-14c-29d 0                0                0                0                |
                {2} | 0   0            0           0           0            0           0           0            0           0           0                0                -29a-30b+31c-22d 29b+6c-36d       0                0                |
                {2} | 0   0            0           0           0            0           0           0            0           0           0                0                -10a+24b+3c      34b-19c+19d      0                0                |
                {2} | 0   0            0           0           0            0           0           0            0           0           0                0                24a-8b+29c       -24a-29b-14c-29d 0                0                |
                {2} | 0   0            0           0           0            0           0           0            0           0           0                0                0                0                -29a-30b+31c-22d 29b+6c-36d       |
                {2} | 0   0            0           0           0            0           0           0            0           0           0                0                0                0                -10a+24b+3c      34b-19c+19d      |
                {2} | 0   0            0           0           0            0           0           0            0           0           0                0                0                0                24a-8b+29c       -24a-29b-14c-29d |
                {3} | 0   0            0           0           0            0           0           0            0           0           0                0                0                0                0                0                |

o4 : SimplicialModuleMap

i5 : g1 = inducedMap(coimage f, source f)

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

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

          9                                 9
     2 : S  <----------------------------- S  : 2
               {0} | 1 0 0 0 0 0 0 0 0 |
               {2} | 0 1 0 0 0 0 0 0 0 |
               {2} | 0 0 1 0 0 0 0 0 0 |
               {2} | 0 0 0 1 0 0 0 0 0 |
               {2} | 0 0 0 0 1 0 0 0 0 |
               {2} | 0 0 0 0 0 1 0 0 0 |
               {2} | 0 0 0 0 0 0 1 0 0 |
               {3} | 0 0 0 0 0 0 0 1 0 |
               {3} | 0 0 0 0 0 0 0 0 1 |

          16                                               16
     3 : S   <------------------------------------------- S   : 3
                {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

o5 : SimplicialModuleMap

i6 : coimage g1 == coimage f

o6 = true

i7 : coker g1 == 0

o7 = true

Caveat

The coimage is more computationally intensive than image(SimplicialModuleMap) because, unlike image, it computes kernels of maps of modules.

Ways to use this method:

coimage(SimplicialModuleMap) -- make the coimage of a map of simplicial modules

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:2815:0.

coimage(SimplicialModuleMap) -- make the coimage of a map of simplicial modules

Description

Caveat

See also

Ways to use this method: