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) |