-- 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 | -43x2-47xy-15y2 35x2+18xy+48y2 |
| 20x2-10xy+41y2 -36x2-47xy+28y2 |
| 31x2+18xy+5y2 -12x2-37xy+5y2 |
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 | 5x2+9xy-43y2 41x2+22xy+24y2 x3 x2y+49xy2+50y3 -27xy2+30y3 y4 0 0 |
| x2-8xy-20y2 -31xy+24y2 0 -48xy2-44y3 -43xy2+50y3 0 y4 0 |
| 33xy+31y2 x2-37xy+16y2 0 31y3 xy2+47y3 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
| 5x2+9xy-43y2 41x2+22xy+24y2 x3 x2y+49xy2+50y3 -27xy2+30y3 y4 0 0 |
| x2-8xy-20y2 -31xy+24y2 0 -48xy2-44y3 -43xy2+50y3 0 y4 0 |
| 33xy+31y2 x2-37xy+16y2 0 31y3 xy2+47y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | -4xy2-7y3 49xy2-20y3 4y3 -20y3 -45y3 |
{2} | 7xy2-32y3 -7y3 -7y3 25y3 -23y3 |
{3} | 38xy-28y2 -41xy-27y2 -38y2 18y2 -50y2 |
{3} | -38x2+20xy+30y2 41x2-5xy+43y2 38xy+8y2 -18xy-11y2 50xy+46y2 |
{3} | -7x2+11xy+43y2 48xy+13y2 7xy+21y2 -25xy-15y2 23xy+22y2 |
{4} | 0 0 x+17y 45y 0 |
{4} | 0 0 -46y x-27y -30y |
{4} | 0 0 -35y -47y x+10y |
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+8y 31y |
{2} | 0 -33y x+37y |
{3} | 1 -5 -41 |
{3} | 0 -9 23 |
{3} | 0 -1 27 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------------- A : 1
{5} | -29 -25 0 -3y -29x+14y xy-4y2 -18xy-25y2 -40xy+23y2 |
{5} | 30 28 0 -32x-19y 15x-8y 48y2 xy-35y2 43xy-43y2 |
{5} | 0 0 0 0 0 x2-17xy+37y2 -45xy-46y2 -37y2 |
{5} | 0 0 0 0 0 46xy-5y2 x2+27xy-32y2 30xy+5y2 |
{5} | 0 0 0 0 0 35xy+5y2 47xy+32y2 x2-10xy-5y2 |
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
|