SimplicialModuleMap == SimplicialModuleMap -- whether two simplicial module maps are equal

Operator: ==
Usage:

f == g

f == 0

f == 1
Inputs:
- f, a Map of Simplicial Modules, or 0, or 1.
- g, a Map of Simplicial Modules, or 0, or 1.
Outputs:
- a Boolean value, that is true when f and g are equal

Description

Two simplicial module maps are equal if they have the same source, the same target, and $f_i = g_i$ for all $i$.

i1 : S = ZZ/101[a..c]

o1 = S

o1 : PolynomialRing

i2 : C = simplicialModule(freeResolution coker vars S, Degeneracy => true)

      1      4      10      20
o2 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o2 : SimplicialModule

i3 : f = id_C

          1             1
o3 = 0 : S  <--------- S  : 0
               | 1 |

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

          10                                   10
     2 : S   <------------------------------- S   : 2
                {0} | 1 0 0 0 0 0 0 0 0 0 |
                {1} | 0 1 0 0 0 0 0 0 0 0 |
                {1} | 0 0 1 0 0 0 0 0 0 0 |
                {1} | 0 0 0 1 0 0 0 0 0 0 |
                {1} | 0 0 0 0 1 0 0 0 0 0 |
                {1} | 0 0 0 0 0 1 0 0 0 0 |
                {1} | 0 0 0 0 0 0 1 0 0 0 |
                {2} | 0 0 0 0 0 0 0 1 0 0 |
                {2} | 0 0 0 0 0 0 0 0 1 0 |
                {2} | 0 0 0 0 0 0 0 0 0 1 |

          20                                                       20
     3 : S   <--------------------------------------------------- S   : 3
                {0} | 1 0 0 0 0 0 0 0 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 0 1 0 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 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 0 1 0 0 0 0 0 0 0 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 1 0 0 0 0 0 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 1 0 0 0 0 0 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 0 0 1 0 0 0 0 0 |
                {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 0 0 1 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 |

o3 : SimplicialModuleMap

i4 : assert(f == 1)

A simplicial module map is equal to zero if all the maps are zero. This could require computation to determine if something that is superficially not zero is in fact zero.

i5 : assert(0 * id_C == 0)

i6 : g = randomSimplicialMap(C, C)

          1              1
o6 = 0 : S  <---------- S  : 0
               | 24 |

          4                              4
     1 : S  <-------------------------- S  : 1
               {0} | 24 0   0   0   |
               {1} | 0  -36 19  -29 |
               {1} | 0  -30 19  -8  |
               {1} | 0  -29 -10 -22 |

          10                                                      10
     2 : S   <-------------------------------------------------- S   : 2
                {0} | 24 0   0   0   0   0   0   0   0   0   |
                {1} | 0  -36 19  -29 0   0   0   0   0   0   |
                {1} | 0  -30 19  -8  0   0   0   0   0   0   |
                {1} | 0  -29 -10 -22 0   0   0   0   0   0   |
                {1} | 0  0   0   0   -36 19  -29 0   0   0   |
                {1} | 0  0   0   0   -30 19  -8  0   0   0   |
                {1} | 0  0   0   0   -29 -10 -22 0   0   0   |
                {2} | 0  0   0   0   0   0   0   -29 -16 34  |
                {2} | 0  0   0   0   0   0   0   -24 39  19  |
                {2} | 0  0   0   0   0   0   0   -38 21  -47 |

          20                                                                                              20
     3 : S   <------------------------------------------------------------------------------------------ S   : 3
                {0} | 24 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  -36 19  -29 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  -30 19  -8  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  -29 -10 -22 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  0   0   0   -36 19  -29 0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  0   0   0   -30 19  -8  0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  0   0   0   -29 -10 -22 0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  0   0   0   0   0   0   -36 19  -29 0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  0   0   0   0   0   0   -30 19  -8  0   0   0   0   0   0   0   0   0   0   |
                {1} | 0  0   0   0   0   0   0   -29 -10 -22 0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   -29 -16 34  0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   -24 39  19  0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   -38 21  -47 0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -29 -16 34  0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -24 39  19  0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -38 21  -47 0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -29 -16 34  0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -24 39  19  0   |
                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -38 21  -47 0   |
                {3} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -39 |

o6 : SimplicialModuleMap

i7 : h = inducedMap(coker g, target g)

                                 1
o7 = 0 : cokernel | 24 | <----- S  : 0
                            0

                                                 4
     1 : cokernel {0} | 24 0   0   0   | <----- S  : 1
                  {1} | 0  -36 19  -29 |    0
                  {1} | 0  -30 19  -8  |
                  {1} | 0  -29 -10 -22 |

                                                                         10
     2 : cokernel {0} | 24 0   0   0   0   0   0   0   0   0   | <----- S   : 2
                  {1} | 0  -36 19  -29 0   0   0   0   0   0   |    0
                  {1} | 0  -30 19  -8  0   0   0   0   0   0   |
                  {1} | 0  -29 -10 -22 0   0   0   0   0   0   |
                  {1} | 0  0   0   0   -36 19  -29 0   0   0   |
                  {1} | 0  0   0   0   -30 19  -8  0   0   0   |
                  {1} | 0  0   0   0   -29 -10 -22 0   0   0   |
                  {2} | 0  0   0   0   0   0   0   -29 -16 34  |
                  {2} | 0  0   0   0   0   0   0   -24 39  19  |
                  {2} | 0  0   0   0   0   0   0   -38 21  -47 |

                                                                                                                 20
     3 : cokernel {0} | 24 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   | <----- S   : 3
                  {1} | 0  -36 19  -29 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |    0
                  {1} | 0  -30 19  -8  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                  {1} | 0  -29 -10 -22 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                  {1} | 0  0   0   0   -36 19  -29 0   0   0   0   0   0   0   0   0   0   0   0   0   |
                  {1} | 0  0   0   0   -30 19  -8  0   0   0   0   0   0   0   0   0   0   0   0   0   |
                  {1} | 0  0   0   0   -29 -10 -22 0   0   0   0   0   0   0   0   0   0   0   0   0   |
                  {1} | 0  0   0   0   0   0   0   -36 19  -29 0   0   0   0   0   0   0   0   0   0   |
                  {1} | 0  0   0   0   0   0   0   -30 19  -8  0   0   0   0   0   0   0   0   0   0   |
                  {1} | 0  0   0   0   0   0   0   -29 -10 -22 0   0   0   0   0   0   0   0   0   0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   -29 -16 34  0   0   0   0   0   0   0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   -24 39  19  0   0   0   0   0   0   0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   -38 21  -47 0   0   0   0   0   0   0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -29 -16 34  0   0   0   0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -24 39  19  0   0   0   0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -38 21  -47 0   0   0   0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -29 -16 34  0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -24 39  19  0   |
                  {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -38 21  -47 0   |
                  {3} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -39 |

o7 : SimplicialModuleMap

i8 : assert(h == 0)

Testing whether a map is equal to 1 is a shorthand for determining if the simplicial module map is the identity. Although the matrices may appear to be the identity, the map is not the identity when the source and target are not equal.

i9 : g = randomSimplicialMap(C, C, InternalDegree=>1, Cycle=>true)

          1                       1
o9 = 0 : S  <------------------- S  : 0
               | -33a-3b+13c |

          4                                                             4
     1 : S  <--------------------------------------------------------- S  : 1
               {0} | -33a-3b+13c 0            0        0           |
               {1} | 0           -33a+25b-41c -37b+34c -45b+33c    |
               {1} | 0           -28a+16c     4a-3b+4c 45a-37c     |
               {1} | 0           -47a-16b     -34a+9b  35a+34b+13c |

          10                                                                                                                                  10
     2 : S   <------------------------------------------------------------------------------------------------------------------------------ S   : 2
                {0} | -33a-3b+13c 0            0        0           0            0        0           0          0           0           |
                {1} | 0           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0          0           0           |
                {1} | 0           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0          0           0           |
                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0          0           0           |
                {1} | 0           0            0        0           -33a+25b-41c -37b+34c -45b+33c    0          0           0           |
                {1} | 0           0            0        0           -28a+16c     4a-3b+4c 45a-37c     0          0           0           |
                {1} | 0           0            0        0           -47a-16b     -34a+9b  35a+34b+13c 0          0           0           |
                {2} | 0           0            0        0           0            0        0           4a+25b-43c 45a+38c     45b-22c     |
                {2} | 0           0            0        0           0            0        0           -34a+2b    35a-13b-41c -48b+34c    |
                {2} | 0           0            0        0           0            0        0           -47a+16b   47a+16c     -18a+34b+4c |

          20                                                                                                                                                                                                                                                       20
     3 : S   <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S   : 3
                {0} | -33a-3b+13c 0            0        0           0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           0            0        0           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           0            0        0           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           0            0        0           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           0            0        0           0            0        0           -33a+25b-41c -37b+34c -45b+33c    0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           0            0        0           0            0        0           -28a+16c     4a-3b+4c 45a-37c     0          0           0           0          0           0           0          0           0           0            |
                {1} | 0           0            0        0           0            0        0           -47a-16b     -34a+9b  35a+34b+13c 0          0           0           0          0           0           0          0           0           0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           4a+25b-43c 45a+38c     45b-22c     0          0           0           0          0           0           0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           -34a+2b    35a-13b-41c -48b+34c    0          0           0           0          0           0           0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           -47a+16b   47a+16c     -18a+34b+4c 0          0           0           0          0           0           0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           0          0           0           4a+25b-43c 45a+38c     45b-22c     0          0           0           0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           0          0           0           -34a+2b    35a-13b-41c -48b+34c    0          0           0           0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           0          0           0           -47a+16b   47a+16c     -18a+34b+4c 0          0           0           0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           0          0           0           0          0           0           4a+25b-43c 45a+38c     45b-22c     0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           0          0           0           0          0           0           -34a+2b    35a-13b-41c -48b+34c    0            |
                {2} | 0           0            0        0           0            0        0           0            0        0           0          0           0           0          0           0           -47a+16b   47a+16c     -18a+34b+4c 0            |
                {3} | 0           0            0        0           0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           -18a-13b-43c |

o9 : SimplicialModuleMap

i10 : h = inducedMap(coker g, target g)

                                               1
o10 = 0 : cokernel | -33a-3b+13c | <--------- S  : 0
                                      | 1 |

                                                                                               4
      1 : cokernel {0} | -33a-3b+13c 0            0        0           | <------------------- S  : 1
                   {1} | 0           -33a+25b-41c -37b+34c -45b+33c    |    {0} | 1 0 0 0 |
                   {1} | 0           -28a+16c     4a-3b+4c 45a-37c     |    {1} | 0 1 0 0 |
                   {1} | 0           -47a-16b     -34a+9b  35a+34b+13c |    {1} | 0 0 1 0 |
                                                                            {1} | 0 0 0 1 |

                                                                                                                                                                                10
      2 : cokernel {0} | -33a-3b+13c 0            0        0           0            0        0           0          0           0           | <------------------------------- S   : 2
                   {1} | 0           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0          0           0           |    {0} | 1 0 0 0 0 0 0 0 0 0 |
                   {1} | 0           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0          0           0           |    {1} | 0 1 0 0 0 0 0 0 0 0 |
                   {1} | 0           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0          0           0           |    {1} | 0 0 1 0 0 0 0 0 0 0 |
                   {1} | 0           0            0        0           -33a+25b-41c -37b+34c -45b+33c    0          0           0           |    {1} | 0 0 0 1 0 0 0 0 0 0 |
                   {1} | 0           0            0        0           -28a+16c     4a-3b+4c 45a-37c     0          0           0           |    {1} | 0 0 0 0 1 0 0 0 0 0 |
                   {1} | 0           0            0        0           -47a-16b     -34a+9b  35a+34b+13c 0          0           0           |    {1} | 0 0 0 0 0 1 0 0 0 0 |
                   {2} | 0           0            0        0           0            0        0           4a+25b-43c 45a+38c     45b-22c     |    {1} | 0 0 0 0 0 0 1 0 0 0 |
                   {2} | 0           0            0        0           0            0        0           -34a+2b    35a-13b-41c -48b+34c    |    {2} | 0 0 0 0 0 0 0 1 0 0 |
                   {2} | 0           0            0        0           0            0        0           -47a+16b   47a+16c     -18a+34b+4c |    {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} | -33a-3b+13c 0            0        0           0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           0            | <--------------------------------------------------- S   : 3
                   {1} | 0           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           0            |    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                   {1} | 0           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0            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           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0            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           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0          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           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0          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           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0          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           -33a+25b-41c -37b+34c -45b+33c    0          0           0           0          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           -28a+16c     4a-3b+4c 45a-37c     0          0           0           0          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           -47a-16b     -34a+9b  35a+34b+13c 0          0           0           0          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           4a+25b-43c 45a+38c     45b-22c     0          0           0           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           -34a+2b    35a-13b-41c -48b+34c    0          0           0           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           -47a+16b   47a+16c     -18a+34b+4c 0          0           0           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           0           4a+25b-43c 45a+38c     45b-22c     0          0           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           0           -34a+2b    35a-13b-41c -48b+34c    0          0           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           0           -47a+16b   47a+16c     -18a+34b+4c 0          0           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          0           0           4a+25b-43c 45a+38c     45b-22c     0            |    {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          0           0           -34a+2b    35a-13b-41c -48b+34c    0            |    {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          0           0           -47a+16b   47a+16c     -18a+34b+4c 0            |    {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           0          0           0           -18a-13b-43c |    {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 |

o10 : SimplicialModuleMap

i11 : assert(h != 1)

Testing for equality is not the same as testing for isomorphism. In particular, different presentations of a simplicial module need not be equal.

i12 : D = prune image g

       1      4      10      20
o12 = S  <-- S  <-- S   <-- S  <-- ...
                             
      0      1      2       3

o12 : SimplicialModule

i13 : p = D.cache.pruningMap

                                                1
o13 = 0 : image | -33a-3b+13c | <------------- S  : 0
                                   {1} | 1 |

                                                                                            4
      1 : image {0} | -33a-3b+13c 0            0        0           | <------------------- S  : 1
                {1} | 0           -33a+25b-41c -37b+34c -45b+33c    |    {0} | 1 0 0 0 |
                {1} | 0           -28a+16c     4a-3b+4c 45a-37c     |    {1} | 0 1 0 0 |
                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c |    {1} | 0 0 1 0 |
                                                                         {1} | 0 0 0 1 |

                                                                                                                                                                             10
      2 : image {0} | -33a-3b+13c 0            0        0           0            0        0           0          0           0           | <------------------------------- S   : 2
                {1} | 0           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0          0           0           |    {0} | 1 0 0 0 0 0 0 0 0 0 |
                {1} | 0           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0          0           0           |    {1} | 0 1 0 0 0 0 0 0 0 0 |
                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0          0           0           |    {1} | 0 0 1 0 0 0 0 0 0 0 |
                {1} | 0           0            0        0           -33a+25b-41c -37b+34c -45b+33c    0          0           0           |    {1} | 0 0 0 1 0 0 0 0 0 0 |
                {1} | 0           0            0        0           -28a+16c     4a-3b+4c 45a-37c     0          0           0           |    {1} | 0 0 0 0 1 0 0 0 0 0 |
                {1} | 0           0            0        0           -47a-16b     -34a+9b  35a+34b+13c 0          0           0           |    {1} | 0 0 0 0 0 1 0 0 0 0 |
                {2} | 0           0            0        0           0            0        0           4a+25b-43c 45a+38c     45b-22c     |    {1} | 0 0 0 0 0 0 1 0 0 0 |
                {2} | 0           0            0        0           0            0        0           -34a+2b    35a-13b-41c -48b+34c    |    {2} | 0 0 0 0 0 0 0 1 0 0 |
                {2} | 0           0            0        0           0            0        0           -47a+16b   47a+16c     -18a+34b+4c |    {2} | 0 0 0 0 0 0 0 0 1 0 |
                                                                                                                                              {2} | 0 0 0 0 0 0 0 0 0 1 |

                                                                                                                                                                                                                                                                                                                      20
      3 : image {0} | -33a-3b+13c 0            0        0           0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           0            | <--------------------------------------------------- S   : 3
                {1} | 0           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0            0        0           0          0           0           0          0           0           0          0           0           0            |    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {1} | 0           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0            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           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0            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           -33a+25b-41c -37b+34c -45b+33c    0            0        0           0          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           -28a+16c     4a-3b+4c 45a-37c     0            0        0           0          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           -47a-16b     -34a+9b  35a+34b+13c 0            0        0           0          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           -33a+25b-41c -37b+34c -45b+33c    0          0           0           0          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           -28a+16c     4a-3b+4c 45a-37c     0          0           0           0          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           -47a-16b     -34a+9b  35a+34b+13c 0          0           0           0          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           4a+25b-43c 45a+38c     45b-22c     0          0           0           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           -34a+2b    35a-13b-41c -48b+34c    0          0           0           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           -47a+16b   47a+16c     -18a+34b+4c 0          0           0           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           0           4a+25b-43c 45a+38c     45b-22c     0          0           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           0           -34a+2b    35a-13b-41c -48b+34c    0          0           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           0           -47a+16b   47a+16c     -18a+34b+4c 0          0           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          0           0           4a+25b-43c 45a+38c     45b-22c     0            |    {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          0           0           -34a+2b    35a-13b-41c -48b+34c    0            |    {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          0           0           -47a+16b   47a+16c     -18a+34b+4c 0            |    {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           0          0           0           -18a-13b-43c |    {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 |

o13 : SimplicialModuleMap

i14 : p == 1

o14 = false

i15 : assert(coker p == 0 and ker p == 0)

i16 : assert(prune p == 1)

Ways to use this method:

SimplicialModuleMap == SimplicialModuleMap -- whether two simplicial module maps are equal
SimplicialModuleMap == ZZ
ZZ == SimplicialModuleMap

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:2088:0.

SimplicialModuleMap == SimplicialModuleMap -- whether two simplicial module maps are equal

Description

See also

Ways to use this method: