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

Function: cokernel
Usage:

cokernel f

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

Description

If $f : C \to D$ is a map of simplicial modules of degree $d$, then the cokernel is the simplicial module $E$ whose $i$-th term is $cokernel(f_{i-d})$, 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 coker f

o5 = 0 <-- cokernel {1} | -b-43c-36d    -8c+45d    50c+43d     | <-- cokernel {1} | -b-43c-36d    -8c+45d    50c+43d     0             0          0           0               0          | <-- cokernel {1} | -b-43c-36d    -8c+45d    50c+43d     0             0          0           0             0          0           0               0          0               0          0               0          |<-- ...
                    {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d |              {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d 0             0          0           0               0          |              {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d 0             0          0           0             0          0           0               0          0               0          0               0          |
     0              {1} | -39c          b+40c      a+35c       |              {1} | -39c          b+40c      a+35c       0             0          0           0               0          |              {1} | -39c          b+40c      a+35c       0             0          0           0             0          0           0               0          0               0          0               0          |
                                                                              {1} | 0             0          0           -b-43c-36d    -8c+45d    50c+43d     0               0          |              {1} | 0             0          0           -b-43c-36d    -8c+45d    50c+43d     0             0          0           0               0          0               0          0               0          |
           1                                                                  {1} | 0             0          0           a+43b+36c+39d 8b-46c-40d 50b-43c-35d 0               0          |              {1} | 0             0          0           a+43b+36c+39d 8b-46c-40d 50b-43c-35d 0             0          0           0               0          0               0          0               0          |
                                                                              {1} | 0             0          0           -39c          b+40c      a+35c       0               0          |              {1} | 0             0          0           -39c          b+40c      a+35c       0             0          0           0               0          0               0          0               0          |
                                                                              {2} | 0             0          0           0             0          0           13a-10b-34c+26d 3b+25c-49d |              {1} | 0             0          0           0             0          0           -b-43c-36d    -8c+45d    50c+43d     0               0          0               0          0               0          |
                                                                              {2} | 0             0          0           0             0          0           a-26b+42c-12d   7b+5c-5d   |              {1} | 0             0          0           0             0          0           a+43b+36c+39d 8b-46c-40d 50b-43c-35d 0               0          0               0          0               0          |
                                                                              {2} | 0             0          0           0             0          0           34b+49c+13d     a-3b+9c-3d |              {1} | 0             0          0           0             0          0           -39c          b+40c      a+35c       0               0          0               0          0               0          |
                                                                                                                                                                                                        {2} | 0             0          0           0             0          0           0             0          0           13a-10b-34c+26d 3b+25c-49d 0               0          0               0          |
                                                                     2                                                                                                                                  {2} | 0             0          0           0             0          0           0             0          0           a-26b+42c-12d   7b+5c-5d   0               0          0               0          |
                                                                                                                                                                                                        {2} | 0             0          0           0             0          0           0             0          0           34b+49c+13d     a-3b+9c-3d 0               0          0               0          |
                                                                                                                                                                                                        {2} | 0             0          0           0             0          0           0             0          0           0               0          13a-10b-34c+26d 3b+25c-49d 0               0          |
                                                                                                                                                                                                        {2} | 0             0          0           0             0          0           0             0          0           0               0          a-26b+42c-12d   7b+5c-5d   0               0          |
                                                                                                                                                                                                        {2} | 0             0          0           0             0          0           0             0          0           0               0          34b+49c+13d     a-3b+9c-3d 0               0          |
                                                                                                                                                                                                        {2} | 0             0          0           0             0          0           0             0          0           0               0          0               0          13a-10b-34c+26d 3b+25c-49d |
                                                                                                                                                                                                        {2} | 0             0          0           0             0          0           0             0          0           0               0          0               0          a-26b+42c-12d   7b+5c-5d   |
                                                                                                                                                                                                        {2} | 0             0          0           0             0          0           0             0          0           0               0          0               0          34b+49c+13d     a-3b+9c-3d |
                                                                                                                                                                                                        {3} | 0             0          0           0             0          0           0             0          0           0               0          0               0          0               0          |
                                                                                                                                                                                                
                                                                                                                                                                                               3

o5 : SimplicialModule

i6 : prune normalize oo

                                                                                                                1
o6 = cokernel {1} | -b-43c-36d    -8c+45d    50c+43d     | <-- cokernel {2} | 13a-10b-34c+26d 3b+25c-49d | <-- S
              {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d |              {2} | a-26b+42c-12d   7b+5c-5d   |      
              {1} | -39c          b+40c      a+35c       |              {2} | 34b+49c+13d     a-3b+9c-3d |     3
                                                                
     1                                                         2

o6 : Complex

i7 : prune HH coker f

o7 = 0 <-- cokernel {1} | -d -c -b 0  -d -c | <-- cokernel {1} | -d -c -b 0  -d -c 0  0  0  0  0  0  | <-- cokernel {1} | -d -c -b 0  -d -c 0  0  0  0  0  0  0  0  0  0  0  0  |<-- ...
                    {1} | c  b  a  -d 0  0  |              {1} | c  b  a  -d 0  0  0  0  0  0  0  0  |              {1} | c  b  a  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
     0              {1} | 0  0  0  c  b  a  |              {1} | 0  0  0  c  b  a  0  0  0  0  0  0  |              {1} | 0  0  0  c  b  a  0  0  0  0  0  0  0  0  0  0  0  0  |
                                                           {1} | 0  0  0  0  0  0  -d -c -b 0  -d -c |              {1} | 0  0  0  0  0  0  -d -c -b 0  -d -c 0  0  0  0  0  0  |
           1                                               {1} | 0  0  0  0  0  0  c  b  a  -d 0  0  |              {1} | 0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  0  0  |
                                                           {1} | 0  0  0  0  0  0  0  0  0  c  b  a  |              {1} | 0  0  0  0  0  0  0  0  0  c  b  a  0  0  0  0  0  0  |
                                                                                                                    {1} | 0  0  0  0  0  0  0  0  0  0  0  0  -d -c -b 0  -d -c |
                                                  2                                                                 {1} | 0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  |
                                                                                                                    {1} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  |
                                                                                                            
                                                                                                           3

o7 : SimplicialModule

i8 : g1 = inducedMap(coker f, target f)

                                  1
o8 = 0 : cokernel | -22 | <----- S  : 0
                             0

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

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

                                                                                                                                                                                                                                                                                                         20
     3 : cokernel {0} | -22 0            0           0           0            0           0           0            0           0           0                0                0                0                0                0                | <--------------------------------------------------- S   : 3
                  {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0           0           0            0           0           0                0                0                0                0                0                |    {0} | 0 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 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 1 0 0 0 0 0 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 1 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 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 1 0 0 0 0 0 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 1 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 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 0 1 0 0 0 0 0 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                |    {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 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 1 0 0 0 0 0 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 1 0 0 0 0 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 1 0 0 0 0 0 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 0 1 0 0 0 0 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 1 0 0 0 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 0 1 0 0 0 0 |
                  {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 0 0 1 0 0 0 |
                  {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 0 0 0 1 0 0 |
                  {3} | 0   0            0           0           0            0           0           0            0           0           0                0                0                0                0                0                |    {2} | 0 0 0 0 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 0 0 0 0 1 |

o8 : SimplicialModuleMap

i9 : coker f == image g1

o9 = true

i10 : coker g1 == 0

o10 = true

Ways to use this method:

cokernel(SimplicialModuleMap) -- make the cokernel 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.m2:2895:0.

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

Description

See also

Ways to use this method: