-- produce a nullhomotopy for a map f of chain complexes.
Whether f is null homotopic is not checked.
Here is part of an example provided by Luchezar Avramov. We construct a random module over a complete intersection, resolve it over the polynomial ring, and produce a null homotopy for the map that is multiplication by one of the defining equations for the complete intersection.
i1 : A = ZZ/101[x,y];
|
i2 : M = cokernel random(A^3, A^{-2,-2})
o2 = cokernel | 32x2-31xy+28y2 25x2+46xy+9y2 |
| 11x2+13xy+32y2 35x2-9xy-4y2 |
| -17x2-8xy+8y2 24x2+xy+41y2 |
3
o2 : A-module, quotient of A
|
i3 : R = cokernel matrix {{x^3,y^4}}
o3 = cokernel | x3 y4 |
1
o3 : A-module, quotient of A
|
i4 : N = prune (M**R)
o4 = cokernel | -38x2-25xy+34y2 27x2-50xy-45y2 x3 x2y-2xy2+y3 -44xy2+41y3 y4 0 0 |
| x2+15xy-14y2 3xy-5y2 0 -28xy2+14y3 -5xy2+32y3 0 y4 0 |
| -47xy-40y2 x2-24xy+39y2 0 -4y3 xy2-17y3 0 0 y4 |
3
o4 : A-module, quotient of A
|
i5 : C = resolution N
3 8 5
o5 = A <-- A <-- A <-- 0
0 1 2 3
o5 : ChainComplex
|
i6 : d = C.dd
3 8
o6 = 0 : A <-------------------------------------------------------------------------- A : 1
| -38x2-25xy+34y2 27x2-50xy-45y2 x3 x2y-2xy2+y3 -44xy2+41y3 y4 0 0 |
| x2+15xy-14y2 3xy-5y2 0 -28xy2+14y3 -5xy2+32y3 0 y4 0 |
| -47xy-40y2 x2-24xy+39y2 0 -4y3 xy2-17y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | -48xy2+36y3 3xy2-47y3 48y3 -34y3 23y3 |
{2} | 14xy2+5y3 45y3 -14y3 -12y3 31y3 |
{3} | -8xy-13y2 -47xy-10y2 8y2 25y2 -28y2 |
{3} | 8x2+39xy+28y2 47x2+16xy-40y2 -8xy-26y2 -25xy+15y2 28xy+42y2 |
{3} | -14x2-10xy+39y2 -19xy+45y2 14xy+5y2 12xy+36y2 -31xy-5y2 |
{4} | 0 0 x-17y 34y -32y |
{4} | 0 0 -2y x+21y 49y |
{4} | 0 0 23y -18y x-4y |
5
2 : A <----- 0 : 3
0
o6 : ChainComplexMap
|
i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------- A : 0
{2} | 0 x-15y -3y |
{2} | 0 47y x+24y |
{3} | 1 38 -27 |
{3} | 0 4 -5 |
{3} | 0 -42 -8 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | -3 -15 0 21y 36x+44y xy+26y2 -28xy-49y2 5xy-44y2 |
{5} | -49 22 0 43x-44y 39x-27y 28y2 xy-29y2 5xy-33y2 |
{5} | 0 0 0 0 0 x2+17xy-10y2 -34xy+5y2 32xy+15y2 |
{5} | 0 0 0 0 0 2xy+8y2 x2-21xy-4y2 -49xy-12y2 |
{5} | 0 0 0 0 0 -23xy-43y2 18xy-29y2 x2+4xy+14y2 |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap
|
i8 : s*d + d*s
3 3
o8 = 0 : A <---------------- A : 0
| x3 0 0 |
| 0 x3 0 |
| 0 0 x3 |
8 8
1 : A <----------------------------------- A : 1
{2} | x3 0 0 0 0 0 0 0 |
{2} | 0 x3 0 0 0 0 0 0 |
{3} | 0 0 x3 0 0 0 0 0 |
{3} | 0 0 0 x3 0 0 0 0 |
{3} | 0 0 0 0 x3 0 0 0 |
{4} | 0 0 0 0 0 x3 0 0 |
{4} | 0 0 0 0 0 0 x3 0 |
{4} | 0 0 0 0 0 0 0 x3 |
5 5
2 : A <-------------------------- A : 2
{5} | x3 0 0 0 0 |
{5} | 0 x3 0 0 0 |
{5} | 0 0 x3 0 0 |
{5} | 0 0 0 x3 0 |
{5} | 0 0 0 0 x3 |
3 : 0 <----- 0 : 3
0
o8 : ChainComplexMap
|
i9 : s^2
5 3
o9 = 2 : A <----- A : 0
0
8
3 : 0 <----- A : 1
0
o9 : ChainComplexMap
|