SimplicialModuleMap ** SimplicialModuleMap -- the map of simplicial modules between tensor simplicial modules

Operator: **
Usage:

h = f ** g

h = tensor(f, g)
Inputs:
- f, a Map of Simplicial Modules,
- g, a Map of Simplicial Modules,
Outputs:
- h, a Map of Simplicial Modules,

Description

The maps $f : C \to D$ and $g : E \to F$ of simplicial modules induces the map $h = f \otimes g : C \otimes E \to D \otimes F$ defined by $c \otimes e \mapsto f(c) \otimes g(e)$.

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 : D = simplicialModule((freeResolution coker matrix{{a^2,a*b,b^3}})[-1], Degeneracy => true)

            1      5      14
o3 = 0 <-- S  <-- S  <-- S  <-- ...
                          
     0     1      2      3

o3 : SimplicialModule

i4 : f = randomSimplicialMap(D,C)

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

          1                                                  4
     1 : S  <---------------------------------------------- S  : 1
               | 0 24a-36b-30c -29a+19b+19c -10a-29b-8c |

          5                                                                                                       10
     2 : S  <--------------------------------------------------------------------------------------------------- S   : 2
               {0} | 0 24a-36b-30c -29a+19b+19c -10a-29b-8c 0           0            0           0   0   0   |
               {0} | 0 0           0            0           24a-36b-30c -29a+19b+19c -10a-29b-8c 0   0   0   |
               {2} | 0 0           0            0           0           0            0           -22 -24 -16 |
               {2} | 0 0           0            0           0           0            0           -29 -38 39  |
               {3} | 0 0           0            0           0           0            0           0   0   0   |

          14                                                                                                                                                                       20
     3 : S   <------------------------------------------------------------------------------------------------------------------------------------------------------------------- S   : 3
                {0} | 0 24a-36b-30c -29a+19b+19c -10a-29b-8c 0           0            0           0           0            0           0   0   0   0   0   0   0   0   0   0  |
                {0} | 0 0           0            0           24a-36b-30c -29a+19b+19c -10a-29b-8c 0           0            0           0   0   0   0   0   0   0   0   0   0  |
                {0} | 0 0           0            0           0           0            0           24a-36b-30c -29a+19b+19c -10a-29b-8c 0   0   0   0   0   0   0   0   0   0  |
                {2} | 0 0           0            0           0           0            0           0           0            0           -22 -24 -16 0   0   0   0   0   0   0  |
                {2} | 0 0           0            0           0           0            0           0           0            0           -29 -38 39  0   0   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  |
                {2} | 0 0           0            0           0           0            0           0           0            0           0   0   0   -22 -24 -16 0   0   0   0  |
                {2} | 0 0           0            0           0           0            0           0           0            0           0   0   0   -29 -38 39  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  |
                {2} | 0 0           0            0           0           0            0           0           0            0           0   0   0   0   0   0   -22 -24 -16 0  |
                {2} | 0 0           0            0           0           0            0           0           0            0           0   0   0   0   0   0   -29 -38 39  0  |
                {3} | 0 0           0            0           0           0            0           0           0            0           0   0   0   0   0   0   0   0   0   0  |
                {3} | 0 0           0            0           0           0            0           0           0            0           0   0   0   0   0   0   0   0   0   21 |
                {4} | 0 0           0            0           0           0            0           0           0            0           0   0   0   0   0   0   0   0   0   0  |

o4 : SimplicialModuleMap

i5 : E = simplicialModule((dual C.complex)[-3], Degeneracy => true)

      1      4      10      20
o5 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o5 : SimplicialModule

i6 : F = simplicialModule((dual D.complex)[-3], 3, Degeneracy => true)

      2      5      9      14
o6 = S  <-- S  <-- S  <-- S  <-- ...
                           
     0      1      2      3

o6 : SimplicialModule

i7 : g = randomSimplicialMap(F,E)

          2                            1
o7 = 0 : S  <------------------------ S  : 0
               {-3} | 34          |
               {-4} | 19a-47b-39c |

          5                                                                 4
     1 : S  <------------------------------------------------------------- S  : 1
               {-3} | 34          0            0          0            |
               {-4} | 19a-47b-39c 0            0          0            |
               {-2} | 0           -18          -47        45           |
               {-2} | 0           -13          38         -34          |
               {-3} | 0           -43a-15b-28c 2a+16b+22c -48a-47b+47c |

          9                                                                                                            10
     2 : S  <-------------------------------------------------------------------------------------------------------- S   : 2
               {-3} | 34          0            0          0            0            0          0            0 0 0 |
               {-4} | 19a-47b-39c 0            0          0            0            0          0            0 0 0 |
               {-2} | 0           -18          -47        45           0            0          0            0 0 0 |
               {-2} | 0           -13          38         -34          0            0          0            0 0 0 |
               {-3} | 0           -43a-15b-28c 2a+16b+22c -48a-47b+47c 0            0          0            0 0 0 |
               {-2} | 0           0            0          0            -18          -47        45           0 0 0 |
               {-2} | 0           0            0          0            -13          38         -34          0 0 0 |
               {-3} | 0           0            0          0            -43a-15b-28c 2a+16b+22c -48a-47b+47c 0 0 0 |
               {0}  | 0           0            0          0            0            0          0            0 0 0 |

          14                                                                                                                                                               20
     3 : S   <----------------------------------------------------------------------------------------------------------------------------------------------------------- S   : 3
                {-3} | 34          0            0          0            0            0          0            0            0          0            0 0 0 0 0 0 0 0 0 0 |
                {-4} | 19a-47b-39c 0            0          0            0            0          0            0            0          0            0 0 0 0 0 0 0 0 0 0 |
                {-2} | 0           -18          -47        45           0            0          0            0            0          0            0 0 0 0 0 0 0 0 0 0 |
                {-2} | 0           -13          38         -34          0            0          0            0            0          0            0 0 0 0 0 0 0 0 0 0 |
                {-3} | 0           -43a-15b-28c 2a+16b+22c -48a-47b+47c 0            0          0            0            0          0            0 0 0 0 0 0 0 0 0 0 |
                {-2} | 0           0            0          0            -18          -47        45           0            0          0            0 0 0 0 0 0 0 0 0 0 |
                {-2} | 0           0            0          0            -13          38         -34          0            0          0            0 0 0 0 0 0 0 0 0 0 |
                {-3} | 0           0            0          0            -43a-15b-28c 2a+16b+22c -48a-47b+47c 0            0          0            0 0 0 0 0 0 0 0 0 0 |
                {-2} | 0           0            0          0            0            0          0            -18          -47        45           0 0 0 0 0 0 0 0 0 0 |
                {-2} | 0           0            0          0            0            0          0            -13          38         -34          0 0 0 0 0 0 0 0 0 0 |
                {-3} | 0           0            0          0            0            0          0            -43a-15b-28c 2a+16b+22c -48a-47b+47c 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 0 0 0 |

o7 : SimplicialModuleMap

i8 : h = f ** g;

i9 : assert isWellDefined h

i10 : assert(prune source h == C ** E)

i11 : assert(prune target h == D ** F)

If one argument is a SimplicialModule or Module, then the identity map of the corresponding complex is used.

i12 : fE = f ** E;

i13 : assert(fE == f ** id_E)

i14 : k = coker vars S

o14 = cokernel | a b c |

                             1
o14 : S-module, quotient of S

i15 : gk = g ** k;

This routine is functorial.

i16 : D' = simplicialModule((freeResolution coker matrix{{a^2,a*b,c^3}})[-1], 3, Degeneracy => true)

             1      5      15
o16 = 0 <-- S  <-- S  <-- S  <-- ...
                           
      0     1      2      3

o16 : SimplicialModule

i17 : f' = randomSimplicialMap(D', D)

o17 = 0 : 0 <----- 0 : 0
               0

           1              1
      1 : S  <---------- S  : 1
                | 19 |

           5                                          5
      2 : S  <-------------------------------------- S  : 2
                {0} | 19 0  0   0   0            |
                {0} | 0  19 0   0   0            |
                {2} | 0  0  -16 15  39a+43b-17c  |
                {2} | 0  0  7   -23 -11a+48b+36c |
                {3} | 0  0  0   0   35           |

           15                                                                                                       14
      3 : S   <--------------------------------------------------------------------------------------------------- S   : 3
                 {0} | 19 0  0  0   0   0            0   0   0            0   0   0            0  0            |
                 {0} | 0  19 0  0   0   0            0   0   0            0   0   0            0  0            |
                 {0} | 0  0  19 0   0   0            0   0   0            0   0   0            0  0            |
                 {2} | 0  0  0  -16 15  39a+43b-17c  0   0   0            0   0   0            0  0            |
                 {2} | 0  0  0  7   -23 -11a+48b+36c 0   0   0            0   0   0            0  0            |
                 {3} | 0  0  0  0   0   35           0   0   0            0   0   0            0  0            |
                 {2} | 0  0  0  0   0   0            -16 15  39a+43b-17c  0   0   0            0  0            |
                 {2} | 0  0  0  0   0   0            7   -23 -11a+48b+36c 0   0   0            0  0            |
                 {3} | 0  0  0  0   0   0            0   0   35           0   0   0            0  0            |
                 {2} | 0  0  0  0   0   0            0   0   0            -16 15  39a+43b-17c  0  0            |
                 {2} | 0  0  0  0   0   0            0   0   0            7   -23 -11a+48b+36c 0  0            |
                 {3} | 0  0  0  0   0   0            0   0   0            0   0   35           0  0            |
                 {3} | 0  0  0  0   0   0            0   0   0            0   0   0            11 -38a+33b+40c |
                 {5} | 0  0  0  0   0   0            0   0   0            0   0   0            0  0            |
                 {5} | 0  0  0  0   0   0            0   0   0            0   0   0            0  0            |

o17 : SimplicialModuleMap

i18 : (f' * f) ** g == (f' ** g) * (f ** id_E)

o18 = true

i19 : (f' * f) ** g == (f' ** id_F) * (f ** g)

o19 = true

i20 : F' = simplicialModule(dual (freeResolution coker matrix{{a^2,a*b,a*c,b^3}})[-3], Degeneracy => true)

       1      5      13      26
o20 = S  <-- S  <-- S   <-- S  <-- ...
                             
      0      1      2       3

o20 : SimplicialModule

i21 : g' = randomSimplicialMap(F', F)

           1                              2
o21 = 0 : S  <-------------------------- S  : 0
                {-4} | 11a+46b-28c 1 |

           5                                                                                                   5
      1 : S  <----------------------------------------------------------------------------------------------- S  : 1
                {-4} | 11a+46b-28c 1 0                              0                         0           |
                {-3} | 0           0 -3a+22b-47c                    27a-22b+32c               -19         |
                {-3} | 0           0 -23a-7b+2c                     -9a-32b-20c               17          |
                {-3} | 0           0 29a-47b+15c                    24a-30b-48c               -20         |
                {-4} | 0           0 -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-49bc-33c2 44a-39b+36c |

           13                                                                                                                                                                                                                             9
      2 : S   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 2
                 {-4} | 11a+46b-28c 1 0                              0                         0           0                              0                         0           0                                                    |
                 {-3} | 0           0 -3a+22b-47c                    27a-22b+32c               -19         0                              0                         0           0                                                    |
                 {-3} | 0           0 -23a-7b+2c                     -9a-32b-20c               17          0                              0                         0           0                                                    |
                 {-3} | 0           0 29a-47b+15c                    24a-30b-48c               -20         0                              0                         0           0                                                    |
                 {-4} | 0           0 -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-49bc-33c2 44a-39b+36c 0                              0                         0           0                                                    |
                 {-3} | 0           0 0                              0                         0           -3a+22b-47c                    27a-22b+32c               -19         0                                                    |
                 {-3} | 0           0 0                              0                         0           -23a-7b+2c                     -9a-32b-20c               17          0                                                    |
                 {-3} | 0           0 0                              0                         0           29a-47b+15c                    24a-30b-48c               -20         0                                                    |
                 {-4} | 0           0 0                              0                         0           -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-49bc-33c2 44a-39b+36c 0                                                    |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           9a2-39ab+13b2+4ac-26bc+22c2                          |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           -49a2-11ab+43b2-8ac-8bc+36c2                         |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           -3a2-22ab+41b2-30ac+16bc-28c2                        |
                 {-3} | 0           0 0                              0                         0           0                              0                         0           -6a3+35a2b-35ab2+3b3-9a2c+6abc-31b2c+40ac2+25bc2-2c3 |

           26                                                                                                                                                                                                                                                                                                                                                                                                            14
      3 : S   <-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S   : 3
                 {-4} | 11a+46b-28c 1 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    0                                                    0                                                    |
                 {-3} | 0           0 -3a+22b-47c                    27a-22b+32c               -19         0                              0                         0           0                              0                         0           0                                                    0                                                    0                                                    |
                 {-3} | 0           0 -23a-7b+2c                     -9a-32b-20c               17          0                              0                         0           0                              0                         0           0                                                    0                                                    0                                                    |
                 {-3} | 0           0 29a-47b+15c                    24a-30b-48c               -20         0                              0                         0           0                              0                         0           0                                                    0                                                    0                                                    |
                 {-4} | 0           0 -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-49bc-33c2 44a-39b+36c 0                              0                         0           0                              0                         0           0                                                    0                                                    0                                                    |
                 {-3} | 0           0 0                              0                         0           -3a+22b-47c                    27a-22b+32c               -19         0                              0                         0           0                                                    0                                                    0                                                    |
                 {-3} | 0           0 0                              0                         0           -23a-7b+2c                     -9a-32b-20c               17          0                              0                         0           0                                                    0                                                    0                                                    |
                 {-3} | 0           0 0                              0                         0           29a-47b+15c                    24a-30b-48c               -20         0                              0                         0           0                                                    0                                                    0                                                    |
                 {-4} | 0           0 0                              0                         0           -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-49bc-33c2 44a-39b+36c 0                              0                         0           0                                                    0                                                    0                                                    |
                 {-3} | 0           0 0                              0                         0           0                              0                         0           -3a+22b-47c                    27a-22b+32c               -19         0                                                    0                                                    0                                                    |
                 {-3} | 0           0 0                              0                         0           0                              0                         0           -23a-7b+2c                     -9a-32b-20c               17          0                                                    0                                                    0                                                    |
                 {-3} | 0           0 0                              0                         0           0                              0                         0           29a-47b+15c                    24a-30b-48c               -20         0                                                    0                                                    0                                                    |
                 {-4} | 0           0 0                              0                         0           0                              0                         0           -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-49bc-33c2 44a-39b+36c 0                                                    0                                                    0                                                    |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           9a2-39ab+13b2+4ac-26bc+22c2                          0                                                    0                                                    |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           -49a2-11ab+43b2-8ac-8bc+36c2                         0                                                    0                                                    |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           -3a2-22ab+41b2-30ac+16bc-28c2                        0                                                    0                                                    |
                 {-3} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           -6a3+35a2b-35ab2+3b3-9a2c+6abc-31b2c+40ac2+25bc2-2c3 0                                                    0                                                    |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    9a2-39ab+13b2+4ac-26bc+22c2                          0                                                    |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    -49a2-11ab+43b2-8ac-8bc+36c2                         0                                                    |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    -3a2-22ab+41b2-30ac+16bc-28c2                        0                                                    |
                 {-3} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    -6a3+35a2b-35ab2+3b3-9a2c+6abc-31b2c+40ac2+25bc2-2c3 0                                                    |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    0                                                    9a2-39ab+13b2+4ac-26bc+22c2                          |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    0                                                    -49a2-11ab+43b2-8ac-8bc+36c2                         |
                 {-2} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    0                                                    -3a2-22ab+41b2-30ac+16bc-28c2                        |
                 {-3} | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    0                                                    -6a3+35a2b-35ab2+3b3-9a2c+6abc-31b2c+40ac2+25bc2-2c3 |
                 {0}  | 0           0 0                              0                         0           0                              0                         0           0                              0                         0           0                                                    0                                                    0                                                    |

o21 : SimplicialModuleMap

i22 : f ** (g' * g) == (f ** g') * (id_C ** g)

o22 = true

i23 : f ** (g' * g) == (id_D ** g') * (f ** g)

o23 = true

Ways to use this method:

Complex ** SimplicialModuleMap
ComplexMap ** SimplicialModuleMap
Module ** SimplicialModuleMap
SimplicialModule ** SimplicialModuleMap
SimplicialModuleMap ** Complex
SimplicialModuleMap ** ComplexMap
SimplicialModuleMap ** Module
SimplicialModuleMap ** SimplicialModule
SimplicialModuleMap ** SimplicialModuleMap -- the map of simplicial modules between tensor simplicial modules

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:2291:0.

SimplicialModuleMap ** SimplicialModuleMap -- the map of simplicial modules between tensor simplicial modules

Description

See also

Ways to use this method: