-- 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];
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i2 : M = cokernel random(A^3, A^{-2,-2})
o2 = cokernel | 39x2+40xy+32y2 49x2+49xy-22y2 |
| 2x2-11xy+35y2 31x2-38xy-17y2 |
| -25x2+xy+3y2 -26x2+8xy+44y2 |
3
o2 : A-module, quotient of A
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i3 : R = cokernel matrix {{x^3,y^4}}
o3 = cokernel | x3 y4 |
1
o3 : A-module, quotient of A
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i4 : N = prune (M**R)
o4 = cokernel | -31x2-20xy+2y2 17xy+11y2 x3 x2y-44xy2+3y3 13xy2-23y3 y4 0 0 |
| x2+9xy-14y2 -15xy+50y2 0 -25xy2-48y3 34xy2+12y3 0 y4 0 |
| -27xy+26y2 x2+18xy+6y2 0 22y3 xy2-15y3 0 0 y4 |
3
o4 : A-module, quotient of A
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i5 : C = resolution N
3 8 5
o5 = A <-- A <-- A <-- 0
0 1 2 3
o5 : ChainComplex
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i6 : d = C.dd
3 8
o6 = 0 : A <----------------------------------------------------------------------- A : 1
| -31x2-20xy+2y2 17xy+11y2 x3 x2y-44xy2+3y3 13xy2-23y3 y4 0 0 |
| x2+9xy-14y2 -15xy+50y2 0 -25xy2-48y3 34xy2+12y3 0 y4 0 |
| -27xy+26y2 x2+18xy+6y2 0 22y3 xy2-15y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -22xy2-44y3 32xy2-3y3 22y3 -36y3 29y3 |
{2} | -45xy2+36y3 -47y3 45y3 y3 -13y3 |
{3} | -28xy-22y2 27xy+21y2 28y2 23y2 -35y2 |
{3} | 28x2-13xy-30y2 -27x2-15xy-3y2 -28xy+35y2 -23xy+6y2 35xy+48y2 |
{3} | 45x2+37xy+49y2 -10xy+12y2 -45xy+28y2 -xy+6y2 13xy+37y2 |
{4} | 0 0 x -21y -16y |
{4} | 0 0 8y x-35y -13y |
{4} | 0 0 20y -21y x+35y |
5
2 : A <----- 0 : 3
0
o6 : ChainComplexMap
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i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------ A : 0
{2} | 0 x-9y 15y |
{2} | 0 27y x-18y |
{3} | 1 31 0 |
{3} | 0 44 44 |
{3} | 0 -48 16 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 28 -43 0 -37y 9x-36y xy-34y2 9xy-32y2 -16xy+2y2 |
{5} | -3 -28 0 -15x-45y 43x+13y 25y2 xy+50y2 -34xy+21y2 |
{5} | 0 0 0 0 0 x2+17y2 21xy-40y2 16xy+16y2 |
{5} | 0 0 0 0 0 -8xy-35y2 x2+35xy+17y2 13xy-27y2 |
{5} | 0 0 0 0 0 -20xy+27y2 21xy-16y2 x2-35xy-34y2 |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap
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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
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i9 : s^2
5 3
o9 = 2 : A <----- A : 0
0
8
3 : 0 <----- A : 1
0
o9 : ChainComplexMap
|