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

Synopsis

Function: tensorCommutativity
Usage:

tensorCommutativity(M, N)
Inputs:
- M, a module,
- N, a module, both over the same ring $R$
Outputs:
- a matrix, that is an isomorphism from $M \otimes_R N$ to $N \otimes_R M$.

Description

Given $R$-modules $M$ and $N$, there exists a canonical isomorphism $f \colon M \otimes_R N \to N \otimes_R M$ interchanging the factors. This method implements this isomorphism.

Even for free modules, this map is not simply given by the identity matrix.

i1 : R = ZZ/101[x,y];

i2 : M = R^2

      2
o2 = R

o2 : R-module, free

i3 : N = R^3

      3
o3 = R

o3 : R-module, free

i4 : f = tensorCommutativity(M, N)

o4 = | 1 0 0 0 0 0 |
     | 0 0 0 1 0 0 |
     | 0 1 0 0 0 0 |
     | 0 0 0 0 1 0 |
     | 0 0 1 0 0 0 |
     | 0 0 0 0 0 1 |

             6      6
o4 : Matrix R  <-- R

i5 : assert isWellDefined f

i6 : assert isIsomorphism f

By giving the generators of $M$ and $N$ sufficiently different degrees, we see that the canonical generators for the two tensor products come in different orders. The isomorphism is given by the corresponding permutation matrix.

i7 : M = R^{1,2}

      2
o7 = R

o7 : R-module, free, degrees {-1, -2}

i8 : N = R^{100,200,300}

      3
o8 = R

o8 : R-module, free, degrees {-100, -200, -300}

i9 : M ** N

      6
o9 = R

o9 : R-module, free, degrees {-101, -201, -301, -102, -202, -302}

i10 : N ** M

       6
o10 = R

o10 : R-module, free, degrees {-101, -102, -201, -202, -301, -302}

i11 : tensorCommutativity(M, N)

o11 = {-101} | 1 0 0 0 0 0 |
      {-102} | 0 0 0 1 0 0 |
      {-201} | 0 1 0 0 0 0 |
      {-202} | 0 0 0 0 1 0 |
      {-301} | 0 0 1 0 0 0 |
      {-302} | 0 0 0 0 0 1 |

              6      6
o11 : Matrix R  <-- R

For completeness, we include an example when neither module is free.

i12 : g = tensorCommutativity(coker vars R ++ coker vars R, image vars R)

o12 = {1} | 1 0 0 0 |
      {1} | 0 0 1 0 |
      {1} | 0 1 0 0 |
      {1} | 0 0 0 1 |

o12 : Matrix

i13 : source g

o13 = cokernel {1} | -y 0  x y 0 0 0 0 0 0 |
               {1} | x  0  0 0 0 0 x y 0 0 |
               {1} | 0  -y 0 0 x y 0 0 0 0 |
               {1} | 0  x  0 0 0 0 0 0 x y |

                             4
o13 : R-module, quotient of R

i14 : target g

o14 = cokernel {1} | x y 0 0 0 0 0 0 -y 0  |
               {1} | 0 0 x y 0 0 0 0 0  -y |
               {1} | 0 0 0 0 x y 0 0 x  0  |
               {1} | 0 0 0 0 0 0 x y 0  x  |

                             4
o14 : R-module, quotient of R

i15 : assert isWellDefined g

i16 : assert isIsomorphism g

Ways to use this method: