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Complexes :: tensorCommutativity(Complex,Complex)

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

Synopsis

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]

o1 = S

o1 : PolynomialRing
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(source f === F)
i9 : assert(target f === G)
i10 : assert(f_1 != id_(source f_1))
i11 : assert(prune ker f == 0)
i12 : assert(prune coker f == 0)
i13 : g = f^-1

           2               2
o13 = 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 |

o13 : ComplexMap
i14 : assert isWellDefined g
i15 : assert(g * f == 1)
i16 : assert(f * g == 1)

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

i17 : 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}})
              }
          )

o17 = ses

o17 : FunctionClosure
i18 : C = ses(ideal(x_0,x_1), ideal(x_1,x_2))

o18 = 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

o18 : Complex
i19 : D = ses(ideal(x_3,x_4,x_5), ideal(x_6,x_7,x_8))

o19 = 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

o19 : Complex
i20 : h = tensorCommutativity(C, D);
i21 : assert isWellDefined h
i22 : assert(ker h == 0)
i23 : assert(coker h == 0)
i24 : k = h^-1;
i25 : assert(h*k == 1)
i26 : assert(k*h == 1)
i27 : h_2

o27 = | 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 |

o27 : Matrix
i28 : assert(source h_2 != target h_2)

Interchanging the arguments gives the inverse map.

i29 : h1 = tensorCommutativity(D, C)

o29 = 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 |

o29 : ComplexMap
i30 : assert(h1*h == id_(C**D))
i31 : assert(h*h1 == id_(D**C))

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

i32 : C = freeResolution ideal(x_0^2, x_1^2)

       1      2      1
o32 = S  <-- S  <-- S
                     
      0      1      2

o32 : Complex
i33 : D = freeResolution ideal(x_0, x_1)

       1      2      1
o33 = S  <-- S  <-- S
                     
      0      1      2

o33 : Complex
i34 : f = extend(D, C, map(D_0, C_0, 1))

           1             1
o34 = 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 |

o34 : ComplexMap
i35 : E = freeResolution ideal(x_2^3, x_3^3, x_4^3)

       1      3      3      1
o35 = S  <-- S  <-- S  <-- S
                            
      0      1      2      3

o35 : Complex
i36 : F = freeResolution ideal(x_2, x_3, x_4)

       1      3      3      1
o36 = S  <-- S  <-- S  <-- S
                            
      0      1      2      3

o36 : Complex
i37 : g = extend(F, E, map(F_0, E_0, 1))

           1             1
o37 = 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 |

o37 : ComplexMap
i38 : assert(tensorCommutativity(D,F) * (f**g) == (g**f) * tensorCommutativity(C,E))

See also