-- 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 | 42x2+37xy -39x2-19xy+16y2 |
| -37xy-27y2 39x2+20xy+10y2 |
| 29xy+30y2 10x2-41xy-32y2 |
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 | x2-40xy 34xy+42y2 0 -35xy2-15y3 -48xy2-3y3 y4 0 0 |
| 40xy+21y2 14x2-44xy-8y2 x3 x2y-37xy2+33y3 -49xy2-7y3 0 y4 0 |
| -45xy+44y2 x2-18xy+27y2 0 25y3 xy2-36y3 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
| x2-40xy 34xy+42y2 0 -35xy2-15y3 -48xy2-3y3 y4 0 0 |
| 40xy+21y2 14x2-44xy-8y2 x3 x2y-37xy2+33y3 -49xy2-7y3 0 y4 0 |
| -45xy+44y2 x2-18xy+27y2 0 25y3 xy2-36y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | -22xy2-25y3 -xy2-28y3 22y3 -17y3 21y3 |
{2} | -15xy2+11y3 -19y3 15y3 -49y3 36y3 |
{3} | xy+26y2 26xy+41y2 -y2 -16y2 -24y2 |
{3} | -x2-27xy-39y2 -26x2+7xy-18y2 xy+y2 16xy+43y2 24xy+34y2 |
{3} | 15x2+xy+46y2 18xy+26y2 -15xy-12y2 49xy+20y2 -36xy-y2 |
{4} | 0 0 x-45y 36y 5y |
{4} | 0 0 46y x-y 31y |
{4} | 0 0 -12y -y x+46y |
5
2 : A <----- 0 : 3
0
o6 : ChainComplexMap
|
i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------- A : 0
{2} | x+40y 0 -34y |
{2} | 45y 0 x+18y |
{3} | 0 1 -14 |
{3} | 37 0 -6 |
{3} | 41 0 -21 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 4 -39 0 42y 27x+y xy+43y2 -10xy-32y2 -38xy-3y2 |
{5} | 44 44 0 -35x-46y -36x-30y 44y2 xy+37y2 49xy+4y2 |
{5} | 0 0 0 0 0 x2+45xy-15y2 -36xy-45y2 -5xy+10y2 |
{5} | 0 0 0 0 0 -46xy+37y2 x2+xy+10y2 -31xy+9y2 |
{5} | 0 0 0 0 0 12xy+43y2 xy+28y2 x2-46xy+5y2 |
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
|