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

Function: image
Usage:

E = image f
Inputs:
- f, a Map of Simplicial Modules,
Outputs:
- E, a Simplicial Module,

Description

If $f : C \to D$ is a map of simplicial modules of degree $d$, then the image is the simplicial module $E$ whose $i$-th term is $image(f_{i-d})$, and whose face/degeneracy map is induced from the face/degeneracy map on the target.

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 : prune image f

      1      4      9      16
o5 = S  <-- S  <-- S  <-- S  <-- ...
                           
     0      1      2      3

o5 : SimplicialModule

i6 : prune normalize oo

      1      3      2
o6 = S  <-- S  <-- S
                    
     0      1      2

o6 : Complex

i7 : i = inducedMap(forgetComplex target f, image f)

          1
o7 = 0 : S  <----------- image | -22 | : 0
               | -22 |

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

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

          20
     3 : S   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ image {0} | -22 0            0           0           0            0           0           0            0           0           0                0                0                0                0                0                | : 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   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   -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   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           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           -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           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           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           -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                |
                {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           -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           -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           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                -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                -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                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                -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                -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 |
                {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                |
                {3} | 0   0            0           0           0            0           0           0            0           0           0                0                0                0                0                0                |

o7 : SimplicialModuleMap

i8 : isSimplicialMorphism i

o8 = true

i9 : normalize i

          1
o9 = 0 : S  <----------- image | -22 | : 0
               | -22 |

     1 : image {0} | 0 0 0 | <------------------------------------------------ image {0} | 0             0          0           | : 1
               {1} | 1 0 0 |    {1} | -b-43c-36d    -8c+45d    50c+43d     |         {1} | -b-43c-36d    -8c+45d    50c+43d     |
               {1} | 0 1 0 |    {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d |         {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d |
               {1} | 0 0 1 |    {1} | -39c          b+40c      a+35c       |         {1} | -39c          b+40c      a+35c       |

     2 : image {0} | 0 0 0 | <-------------------------------------- image {0} | 0               0          | : 2
               {1} | 0 0 0 |    {2} | 13a-10b-34c+26d 3b+25c-49d |         {1} | 0               0          |
               {1} | 0 0 0 |    {2} | a-26b+42c-12d   7b+5c-5d   |         {1} | 0               0          |
               {1} | 0 0 0 |    {2} | 34b+49c+13d     a-3b+9c-3d |         {1} | 0               0          |
               {1} | 0 0 0 |                                               {1} | 0               0          |
               {1} | 0 0 0 |                                               {1} | 0               0          |
               {1} | 0 0 0 |                                               {1} | 0               0          |
               {2} | 1 0 0 |                                               {2} | 13a-10b-34c+26d 3b+25c-49d |
               {2} | 0 1 0 |                                               {2} | a-26b+42c-12d   7b+5c-5d   |
               {2} | 0 0 1 |                                               {2} | 34b+49c+13d     a-3b+9c-3d |

o9 : ComplexMap

There is a canonical map of simplicial modules from the image to the target.

i10 : g1 = inducedMap(target f, image f)

           1
o10 = 0 : S  <----------- image | -22 | : 0
                | -22 |

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

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

           20
      3 : S   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ image {0} | -22 0            0           0           0            0           0           0            0           0           0                0                0                0                0                0                | : 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   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   -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   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           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           -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           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           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           -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                |
                 {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           -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           -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           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                -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                -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                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                -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                -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 |
                 {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                |
                 {3} | 0   0            0           0           0            0           0           0            0           0           0                0                0                0                0                0                |

o10 : SimplicialModuleMap

i11 : ker g1 == 0

o11 = true

i12 : image g1 == image f

o12 = true

Ways to use this method:

image(SimplicialModuleMap) -- make the image 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.m2:2775:0.

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

Description

See also

Ways to use this method: