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SimplicialModuleMap _ ZZ -- access individual matrices in a simplicial module map

Description

A simplicial module map $f : C \to D$ of degree $d$ is a sequence of maps $f_i : C_i \to D_{i+d}$. This method allows one to access the individual $f_i$.

i1 : S = ZZ/101[a..c];
 
i2 : C = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}}, Degeneracy => true)

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

o2 : SimplicialModule
 
i3 : D = simplicialModule(freeResolution coker vars S, Degeneracy => true)

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

o3 : SimplicialModule
 
i4 : f = randomSimplicialMap(D, C)

          1              1
o4 = 0 : S  <---------- S  : 0
               | 24 |

          4                                                         4
     1 : S  <----------------------------------------------------- S  : 1
               {0} | 24 0            0            0            |
               {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c |
               {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c |
               {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  |

          10                                                                                                                                                                                           10
     2 : S   <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S   : 2
                {0} | 24 0            0            0            0            0            0            0                             0                             0                              |
                {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0            0            0            0                             0                             0                              |
                {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0            0            0            0                             0                             0                              |
                {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0            0            0            0                             0                             0                              |
                {1} | 0  0            0            0            -36a-30b-29c -29a-24b-38c -39a-18b-13c 0                             0                             0                              |
                {1} | 0  0            0            0            19a+19b-10c  -16a+39b+21c -43a-15b-28c 0                             0                             0                              |
                {1} | 0  0            0            0            -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0                             0                             0                              |
                {2} | 0  0            0            0            0            0            0            16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2 |
                {2} | 0  0            0            0            0            0            0            47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2   |
                {2} | 0  0            0            0            0            0            0            39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc       |

          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  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0            0            0            0            0            0            0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0            0            0            0            0            0            0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0            0            0            0            0            0            0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {1} | 0  0            0            0            -36a-30b-29c -29a-24b-38c -39a-18b-13c 0            0            0            0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {1} | 0  0            0            0            19a+19b-10c  -16a+39b+21c -43a-15b-28c 0            0            0            0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {1} | 0  0            0            0            -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0            0            0            0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {1} | 0  0            0            0            0            0            0            -36a-30b-29c -29a-24b-38c -39a-18b-13c 0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {1} | 0  0            0            0            0            0            0            19a+19b-10c  -16a+39b+21c -43a-15b-28c 0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {1} | 0  0            0            0            0            0            0            -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0                             0                             0                              0                             0                             0                              0                             0                             0                              0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2 0                             0                             0                              0                             0                             0                              0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2   0                             0                             0                              0                             0                             0                              0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc       0                             0                             0                              0                             0                             0                              0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            0                             0                             0                              16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2 0                             0                             0                              0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            0                             0                             0                              47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2   0                             0                             0                              0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            0                             0                             0                              39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc       0                             0                             0                              0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            0                             0                             0                              0                             0                             0                              16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2 0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            0                             0                             0                              0                             0                             0                              47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2   0                                                       |
                {2} | 0  0            0            0            0            0            0            0            0            0            0                             0                             0                              0                             0                             0                              39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc       0                                                       |
                {3} | 0  0            0            0            0            0            0            0            0            0            0                             0                             0                              0                             0                             0                              0                             0                             0                              33a3-49a2b-19ab2+44b3-33a2c+17abc-39b2c-20ac2+36bc2+9c3 |

o4 : SimplicialModuleMap
 
i5 : f_2

o5 = {0} | 24 0            0            0            0           
     {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0           
     {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0           
     {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0           
     {1} | 0  0            0            0            -36a-30b-29c
     {1} | 0  0            0            0            19a+19b-10c 
     {1} | 0  0            0            0            -29a-8b-22c 
     {2} | 0  0            0            0            0           
     {2} | 0  0            0            0            0           
     {2} | 0  0            0            0            0           
     ------------------------------------------------------------------------
     0            0            0                            
     0            0            0                            
     0            0            0                            
     0            0            0                            
     -29a-24b-38c -39a-18b-13c 0                            
     -16a+39b+21c -43a-15b-28c 0                            
     34a+19b-47c  -47a+38b+2c  0                            
     0            0            16a2+22ab-34b2+45ac-48bc-47c2
     0            0            47a2+19ab+7b2-16ac+15bc-23c2 
     0            0            39a2+43ab-11b2-17ac+48bc+36c2
     ------------------------------------------------------------------------
     0                             0                              |
     0                             0                              |
     0                             0                              |
     0                             0                              |
     0                             0                              |
     0                             0                              |
     0                             0                              |
     35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2 |
     46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2   |
     -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc       |

             10      10
o5 : Matrix S   <-- S
 
i6 : f_0

o6 = | 24 |

             1      1
o6 : Matrix S  <-- S

The face/degeneracy maps of a simplicial module are indexed by sequences $(a,b)$ of integers, where $b \leq a$.

i7 : dd^C_(2,0)

o7 = {0} | 1 a2 b2 c2 0 0 0 0   0   0   |
     {2} | 0 0  0  0  1 0 0 -b2 -c2 0   |
     {2} | 0 0  0  0  0 1 0 a2  0   -c2 |
     {2} | 0 0  0  0  0 0 1 0   a2  b2  |

             4      10
o7 : Matrix S  <-- S
 
i8 : dd^C_(2,1)

o8 = {0} | 1 0 0 0 0 0 0 0 0 0 |
     {2} | 0 1 0 0 1 0 0 0 0 0 |
     {2} | 0 0 1 0 0 1 0 0 0 0 |
     {2} | 0 0 0 1 0 0 1 0 0 0 |

             4      10
o8 : Matrix S  <-- S
 
i9 : dd^C_(2,2)

o9 = {0} | 1 0 0 0 0 0 0 0 0 0 |
     {2} | 0 1 0 0 0 0 0 0 0 0 |
     {2} | 0 0 1 0 0 0 0 0 0 0 |
     {2} | 0 0 0 1 0 0 0 0 0 0 |

             4      10
o9 : Matrix S  <-- S
 
i10 : ss^C_(0,0)

o10 = {0} | 1 |
      {2} | 0 |
      {2} | 0 |
      {2} | 0 |

              4      1
o10 : Matrix S  <-- S
 
i11 : ss^C_(1,0)

o11 = {0} | 1 0 0 0 |
      {2} | 0 0 0 0 |
      {2} | 0 0 0 0 |
      {2} | 0 0 0 0 |
      {2} | 0 1 0 0 |
      {2} | 0 0 1 0 |
      {2} | 0 0 0 1 |
      {4} | 0 0 0 0 |
      {4} | 0 0 0 0 |
      {4} | 0 0 0 0 |

              10      4
o11 : Matrix S   <-- S
 
i12 : ss^C_(1,1)

o12 = {0} | 1 0 0 0 |
      {2} | 0 1 0 0 |
      {2} | 0 0 1 0 |
      {2} | 0 0 0 1 |
      {2} | 0 0 0 0 |
      {2} | 0 0 0 0 |
      {2} | 0 0 0 0 |
      {4} | 0 0 0 0 |
      {4} | 0 0 0 0 |
      {4} | 0 0 0 0 |

              10      4
o12 : Matrix S   <-- S

Indices that are outside of the top degree or range of sequences return 0 by default.

i13 : topDegree f

o13 = 3
 
i14 : f_-1

o14 = 0

o14 : Matrix 0 <-- 0
 
i15 : f_3

o15 = {0} | 24 0            0            0            0           
      {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0           
      {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0           
      {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0           
      {1} | 0  0            0            0            -36a-30b-29c
      {1} | 0  0            0            0            19a+19b-10c 
      {1} | 0  0            0            0            -29a-8b-22c 
      {1} | 0  0            0            0            0           
      {1} | 0  0            0            0            0           
      {1} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {2} | 0  0            0            0            0           
      {3} | 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           
      -29a-24b-38c -39a-18b-13c 0            0            0           
      -16a+39b+21c -43a-15b-28c 0            0            0           
      34a+19b-47c  -47a+38b+2c  0            0            0           
      0            0            -36a-30b-29c -29a-24b-38c -39a-18b-13c
      0            0            19a+19b-10c  -16a+39b+21c -43a-15b-28c
      0            0            -29a-8b-22c  34a+19b-47c  -47a+38b+2c 
      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            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                            
      0                             0                            
      0                             0                            
      16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2
      47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2   
      39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2 
      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                              0                            
      -37a2-13ab+30b2-10ac-18bc+39c2 0                            
      27a2-22ab-9b2+32ac-32bc-20c2   0                            
      24a2-30ab-15b2-48ac+39bc       0                            
      0                              16a2+22ab-34b2+45ac-48bc-47c2
      0                              47a2+19ab+7b2-16ac+15bc-23c2 
      0                              39a2+43ab-11b2-17ac+48bc+36c2
      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                             0                             
      35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2
      46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2  
      -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc      
      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                             0                            
      0                             0                            
      0                             0                            
      0                             0                            
      16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2
      47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2   
      39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2 
      0                             0                            
      -----------------------------------------------------------------------
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      0                             
      -37a2-13ab+30b2-10ac-18bc+39c2
      27a2-22ab-9b2+32ac-32bc-20c2  
      24a2-30ab-15b2-48ac+39bc      
      0                             
      -----------------------------------------------------------------------
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      0                                                       |
      33a3-49a2b-19ab2+44b3-33a2c+17abc-39b2c-20ac2+36bc2+9c3 |

              20      20
o15 : Matrix S   <-- S
 
i16 : f_4

o16 = 0

o16 : Matrix 0 <-- 0
 
i17 : ss^D_(2,3)

o17 = 0

              20      10
o17 : Matrix S   <-- S
 
i18 : ss^D_(4,0)

o18 = 0

o18 : Matrix 0 <-- 0

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:1859:0.