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tensorCommutativity(Complex,Complex) -- make the canonical isomorphism arising from commutativity

Description

The commutativity of tensor products of modules induces the commutativity of tensor products of chain complexes. This method implements this isomorphism for chain complexes.

Using two term complexes of small rank, we see that this isomorphism need not be the identity map.

i1 : S = ZZ/101[x_0..x_8];
 
i2 : C = complex{genericMatrix(S,x_0,2,1)}

      2      1
o2 = S  <-- S
             
     0      1

o2 : Complex
 
i3 : D = complex{genericMatrix(S,x_2,1,2)}

      1      2
o3 = S  <-- S
             
     0      1

o3 : Complex
 
i4 : F = C ** D

      2      5      2
o4 = S  <-- S  <-- S
                    
     0      1      2

o4 : Complex
 
i5 : G = D ** C

      2      5      2
o5 = S  <-- S  <-- S
                    
     0      1      2

o5 : Complex
 
i6 : f = tensorCommutativity(C,D)

          2               2
o6 = 0 : S  <----------- S  : 0
               | 1 0 |
               | 0 1 |

          5                         5
     1 : S  <--------------------- S  : 1
               {1} | 0 0 0 0 1 |
               {1} | 1 0 0 0 0 |
               {1} | 0 0 1 0 0 |
               {1} | 0 1 0 0 0 |
               {1} | 0 0 0 1 0 |

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

o6 : ComplexMap
 
i7 : assert isWellDefined f
 
i8 : assert isComplexMorphism f
 
i9 : assert(source f === F)
 
i10 : assert(target f === G)
 
i11 : assert(f_1 != id_(source f_1))
 
i12 : assert(prune ker f == 0)
 
i13 : assert(prune coker f == 0)
 
i14 : g = f^-1

           2               2
o14 = 0 : S  <----------- S  : 0
                | 1 0 |
                | 0 1 |

           5                         5
      1 : S  <--------------------- S  : 1
                {1} | 0 1 0 0 0 |
                {1} | 0 0 0 1 0 |
                {1} | 0 0 1 0 0 |
                {1} | 0 0 0 0 1 |
                {1} | 1 0 0 0 0 |

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

o14 : ComplexMap
 
i15 : assert isWellDefined g
 
i16 : assert(g * f == 1)
 
i17 : assert(f * g == 1)

We illustrate this isomorphism on complexes, none of whose terms are free modules.

i18 : ses = (I,J) -> (
          complex{
              map(S^1/(I+J), S^1/I ++ S^1/J, {{1,1}}),
              map(S^1/I ++ S^1/J, S^1/(intersect(I,J)), {{1},{-1}})
              }
          )

o18 = ses

o18 : FunctionClosure
 
i19 : C = ses(ideal(x_0,x_1), ideal(x_1,x_2))

o19 = cokernel | x_0 x_1 x_1 x_2 | <-- cokernel | x_0 x_1 0   0   | <-- cokernel | x_1 x_0x_2 |
                                                | 0   0   x_1 x_2 |      
      0                                                                 2
                                       1

o19 : Complex
 
i20 : D = ses(ideal(x_3,x_4,x_5), ideal(x_6,x_7,x_8))

o20 = cokernel | x_3 x_4 x_5 x_6 x_7 x_8 | <-- cokernel | x_3 x_4 x_5 0   0   0   | <-- cokernel | x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 |
                                                        | 0   0   0   x_6 x_7 x_8 |      
      0                                                                                 2
                                               1

o20 : Complex
 
i21 : h = tensorCommutativity(C, D);
 
i22 : assert isWellDefined h
 
i23 : assert isComplexMorphism h
 
i24 : assert(ker h == 0)
 
i25 : assert(coker h == 0)
 
i26 : k = h^-1;
 
i27 : assert(h*k == 1)
 
i28 : assert(k*h == 1)
 
i29 : h_2

o29 = | 0 0  0  0  0  1 |
      | 0 -1 0  0  0  0 |
      | 0 0  0  -1 0  0 |
      | 0 0  -1 0  0  0 |
      | 0 0  0  0  -1 0 |
      | 1 0  0  0  0  0 |

o29 : Matrix
 
i30 : assert(source h_2 != target h_2)

Interchanging the arguments gives the inverse map.

i31 : h1 = tensorCommutativity(D, C)

o31 = 0 : cokernel | x_3 x_4 x_5 x_6 x_7 x_8 x_0 x_1 x_1 x_2 | <--------- cokernel | x_0 x_1 x_1 x_2 x_3 x_4 x_5 x_6 x_7 x_8 | : 0
                                                                  | 1 |

      1 : cokernel | x_3 x_4 x_5 0   0   0   x_0 x_1 x_1 x_2 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   | <--------------- cokernel | x_0 x_1 0   0   x_3 x_4 x_5 x_6 x_7 x_8 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   | : 1
                   | 0   0   0   x_6 x_7 x_8 0   0   0   0   x_0 x_1 x_1 x_2 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |    | 0 0 1 0 |            | 0   0   x_1 x_2 0   0   0   0   0   0   x_3 x_4 x_5 x_6 x_7 x_8 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   x_3 x_4 x_5 x_6 x_7 x_8 0   0   0   0   0   0   x_0 x_1 0   0   |    | 0 0 0 1 |            | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   x_0 x_1 x_1 x_2 0   0   0   0   x_3 x_4 x_5 0   0   0   |
                   | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   x_3 x_4 x_5 x_6 x_7 x_8 0   0   x_1 x_2 |    | 1 0 0 0 |            | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   x_0 x_1 x_1 x_2 0   0   0   x_6 x_7 x_8 |
                                                                                                                                                  | 0 1 0 0 |

      2 : cokernel | x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 x_0 x_1 x_1 x_2 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      | <----------------------- cokernel | x_1 x_0x_2 x_3 x_4 x_5 x_6 x_7 x_8 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      | : 2
                   | 0      0      0      0      0      0      0      0      0      0   0   0   0   x_3 x_4 x_5 0   0   0   0   0   0   0   0   0   x_0 x_1 0   0   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   x_0 x_1 0   0   0   0   0   0   x_3 x_4 x_5 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   x_6 x_7 x_8 0   0   0   0   0   0   0   0   0   0   x_0 x_1 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   x_1 x_2 0   0   0   0   0   0   0   0   0   0   x_3 x_4 x_5 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   x_3 x_4 x_5 0   0   0   0   0   x_1 x_2 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   x_0 x_1 0   0   0   0   0   x_6 x_7 x_8 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   x_6 x_7 x_8 0   0   0   0   0   0   x_1 x_2 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   x_1 x_2 0   0   0   0   0   0   0   0   0   x_6 x_7 x_8 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   x_3 x_4 x_5 x_6 x_7 x_8 x_1 x_0x_2 |    | 0 0  0  0  -1 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   x_0 x_1 x_1 x_2 x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 |
                                                                                                                                                                                                                            | 1 0  0  0  0  0 |

      3 : cokernel | x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 0      0      0      0      0      0      0      0      0      x_0 x_1 0   0   0   0   0   0   0   0   0   0      0   0      | <--------------- cokernel | x_1 x_0x_2 0   0      x_3 x_4 x_5 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      | : 3
                   | 0      0      0      0      0      0      0      0      0      x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 0   0   x_1 x_2 0   0   0   0   0   0   0   0      0   0      |    | 0 0 1 0 |            | 0   0      x_1 x_0x_2 0   0   0   x_6 x_7 x_8 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   x_3 x_4 x_5 0   0   0   x_1 x_0x_2 0   0      |    | 0 0 0 1 |            | 0   0      0   0      0   0   0   0   0   0   x_0 x_1 0   0   x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 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   x_6 x_7 x_8 0   0      x_1 x_0x_2 |    | 1 0 0 0 |            | 0   0      0   0      0   0   0   0   0   0   0   0   x_1 x_2 0      0      0      0      0      0      0      0      0      x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 |
                                                                                                                                                                                                                      | 0 1 0 0 |

      4 : cokernel | x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 x_1 x_0x_2 | <--------- cokernel | x_1 x_0x_2 x_5x_8 x_4x_8 x_3x_8 x_5x_7 x_4x_7 x_3x_7 x_5x_6 x_4x_6 x_3x_6 | : 4
                                                                                                    | 1 |

o31 : ComplexMap
 
i32 : assert isComplexMorphism h1
 
i33 : assert(h1*h == id_(C**D))
 
i34 : assert(h*h1 == id_(D**C))

Interchanging the factors in a tensor product of two complex maps can be accomplished as follows.

i35 : C = freeResolution ideal(x_0^2, x_1^2)

       1      2      1
o35 = S  <-- S  <-- S
                     
      0      1      2

o35 : Complex
 
i36 : D = freeResolution ideal(x_0, x_1)

       1      2      1
o36 = S  <-- S  <-- S
                     
      0      1      2

o36 : Complex
 
i37 : f = extend(D, C, map(D_0, C_0, 1))

           1             1
o37 = 0 : S  <--------- S  : 0
                | 1 |

           2                       2
      1 : S  <------------------- S  : 1
                {1} | x_0 0   |
                {1} | 0   x_1 |

           1                      1
      2 : S  <------------------ S  : 2
                {2} | x_0x_1 |

o37 : ComplexMap
 
i38 : E = freeResolution ideal(x_2^3, x_3^3, x_4^3)

       1      3      3      1
o38 = S  <-- S  <-- S  <-- S
                            
      0      1      2      3

o38 : Complex
 
i39 : F = freeResolution ideal(x_2, x_3, x_4)

       1      3      3      1
o39 = S  <-- S  <-- S  <-- S
                            
      0      1      2      3

o39 : Complex
 
i40 : g = extend(F, E, map(F_0, E_0, 1))

           1             1
o40 = 0 : S  <--------- S  : 0
                | 1 |

           3                                 3
      1 : S  <----------------------------- S  : 1
                {1} | x_2^2 0     0     |
                {1} | 0     x_3^2 0     |
                {1} | 0     0     x_4^2 |

           3                                                3
      2 : S  <-------------------------------------------- S  : 2
                {2} | x_2^2x_3^2 0          0          |
                {2} | 0          x_2^2x_4^2 0          |
                {2} | 0          0          x_3^2x_4^2 |

           1                               1
      3 : S  <--------------------------- S  : 3
                {3} | x_2^2x_3^2x_4^2 |

o40 : ComplexMap
 
i41 : assert(tensorCommutativity(D,F) * (f**g) == (g**f) * tensorCommutativity(C,E))
 
i42 : assert isComplexMorphism tensorCommutativity(D,F)
 
i43 : assert isComplexMorphism tensorCommutativity(C,E)

See also

Ways to use this method:

The source of this document is in Complexes/ChainComplexMapDoc.m2:2969:0.