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

## Synopsis

• Function: tensorCommutativity
• Usage:
tensorCommutativity(C, D)
• Inputs:
• C, ,
• D, , both over the same ring $R$
• Outputs:
• , that is an isomorphism from $C \otimes_R D$ to $D \otimes_R C$

## 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)