tensorAssociativity(Complex,Complex,Complex) -- make the canonical isomorphism arising from associativity

Synopsis

Function: tensorAssociativity
Usage:

tensorAssociativity(C, D, E)
Inputs:
- C, a complex,
- D, a complex,
- E, a complex,
Outputs:
- a map of complexes, which is an isomorphism from C ** (D ** E) to (C ** D) ** E

Description

The associativity of tensor products of modules induces the associativity 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_11]

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_4,1,2)}

      1      2
o3 = S  <-- S
             
     0      1

o3 : Complex

i4 : E = complex{genericMatrix(S,x_8,2,2)}

      2      2
o4 = S  <-- S
             
     0      1

o4 : Complex

i5 : F = (C ** D) ** E

      4      14      14      4
o5 = S  <-- S   <-- S   <-- S
                             
     0      1       2       3

o5 : Complex

i6 : G = C ** (D ** E)

      4      14      14      4
o6 = S  <-- S   <-- S   <-- S
                             
     0      1       2       3

o6 : Complex

i7 : f = tensorAssociativity(C,D,E)

          4                   4
o7 = 0 : S  <--------------- S  : 0
               | 1 0 0 0 |
               | 0 1 0 0 |
               | 0 0 1 0 |
               | 0 0 0 1 |

          14                                           14
     1 : S   <--------------------------------------- S   : 1
                {1} | 1 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 |
                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                {1} | 0 0 1 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 |
                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                {1} | 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 1 0 0 0 |
                {1} | 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 1 0 |
                {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

          14                                           14
     2 : S   <--------------------------------------- S   : 2
                {2} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                {2} | 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 1 0 0 0 |
                {2} | 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 1 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

          4                       4
     3 : S  <------------------- S  : 3
               {3} | 1 0 0 0 |
               {3} | 0 1 0 0 |
               {3} | 0 0 1 0 |
               {3} | 0 0 0 1 |

o7 : ComplexMap

i8 : assert isWellDefined f

i9 : assert(source f === G)

i10 : assert(target f === F)

i11 : f_1

o11 = {1} | 1 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 |
      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
      {1} | 0 0 1 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 |
      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
      {1} | 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 1 0 0 0 |
      {1} | 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 1 0 |
      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

              14      14
o11 : Matrix S   <-- S

i12 : assert(f_1 != id_(source f_1))

i13 : assert(prune ker f == 0)

i14 : assert(prune coker f == 0)

i15 : g = f^-1

           4                   4
o15 = 0 : S  <--------------- S  : 0
                | 1 0 0 0 |
                | 0 1 0 0 |
                | 0 0 1 0 |
                | 0 0 0 1 |

           14                                           14
      1 : S   <--------------------------------------- S   : 1
                 {1} | 1 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 |
                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                 {1} | 0 0 1 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 |
                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                 {1} | 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 1 0 0 0 |
                 {1} | 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 1 0 |
                 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

           14                                           14
      2 : S   <--------------------------------------- S   : 2
                 {2} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                 {2} | 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 1 0 0 0 |
                 {2} | 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 1 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

           4                       4
      3 : S  <------------------- S  : 3
                {3} | 1 0 0 0 |
                {3} | 0 1 0 0 |
                {3} | 0 0 1 0 |
                {3} | 0 0 0 1 |

o15 : ComplexMap

i16 : assert isWellDefined g

i17 : assert(g * f == 1)

i18 : assert(f * g == 1)

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

i19 : 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}})
              }
          )

o19 = ses

o19 : FunctionClosure

i20 : C = ses(ideal(x_0,x_1), ideal(x_1,x_2))

o20 = 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

o20 : Complex

i21 : D = ses(ideal(x_3,x_4,x_5), ideal(x_6,x_7,x_8))

o21 = 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

o21 : Complex

i22 : E = ses(ideal(x_1^2, x_1*x_2), ideal(x_1*x_3,x_9^2))

o22 = cokernel | x_1^2 x_1x_2 x_1x_3 x_9^2 | <-- cokernel | x_1^2 x_1x_2 0      0     | <-- cokernel | x_1x_2x_3 x_1^2x_3 x_1x_2x_9^2 x_1^2x_9^2 |
                                                          | 0     0      x_1x_3 x_9^2 |      
      0                                                                                     2
                                                 1

o22 : Complex

i23 : h = tensorAssociativity(C, D, E);

i24 : assert isWellDefined h

i25 : assert(ker h == 0)

i26 : assert(coker h == 0)

i27 : k = h^-1;

i28 : assert(h*k == 1)

i29 : assert(k*h == 1)

i30 : h_2

o30 = | 1 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 1 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 1 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 1 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 1 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 1 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 1 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 1 |

o30 : Matrix

i31 : assert(source h_2 != target h_2)

Ways to use this method: