-- 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 | 4x2+38xy+42y2 40x2-47xy+30y2 |
| -15x2+35xy+48y2 45x2-38xy-34y2 |
| 12x2+42xy-28y2 15x2+10xy-49y2 |
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 | -45x2-30xy-36y2 3x2+33xy+9y2 x3 x2y+35xy2-34y3 -18xy2+41y3 y4 0 0 |
| x2-50xy-9y2 33xy+18y2 0 41xy2+19y3 -48xy2-23y3 0 y4 0 |
| 33xy+18y2 x2-31xy+37y2 0 -42y3 xy2+49y3 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
| -45x2-30xy-36y2 3x2+33xy+9y2 x3 x2y+35xy2-34y3 -18xy2+41y3 y4 0 0 |
| x2-50xy-9y2 33xy+18y2 0 41xy2+19y3 -48xy2-23y3 0 y4 0 |
| 33xy+18y2 x2-31xy+37y2 0 -42y3 xy2+49y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -49xy2+19y3 45xy2+24y3 49y3 -49y3 45y3 |
{2} | 22xy2-45y3 43y3 -22y3 -25y3 20y3 |
{3} | -5xy+34y2 -26xy+14y2 5y2 -28y2 -42y2 |
{3} | 5x2-48xy-41y2 26x2-10xy+30y2 -5xy+14y2 28xy-34y2 42xy+34y2 |
{3} | -22x2-4xy-37y2 -32xy-30y2 22xy+49y2 25xy-15y2 -20xy-41y2 |
{4} | 0 0 x+25y 41y 35y |
{4} | 0 0 -19y x+7y -29y |
{4} | 0 0 38y 3y x-32y |
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+50y -33y |
{2} | 0 -33y x+31y |
{3} | 1 45 -3 |
{3} | 0 -45 5 |
{3} | 0 36 -7 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------ A : 1
{5} | 20 41 0 -5y -23x+26y xy+24y2 -31xy-6y2 45xy-10y2 |
{5} | -6 23 0 35x-5y -15x+40y -41y2 xy-46y2 48xy-38y2 |
{5} | 0 0 0 0 0 x2-25xy-36y2 -41xy+3y2 -35xy-20y2 |
{5} | 0 0 0 0 0 19xy+7y2 x2-7xy-9y2 29xy-41y2 |
{5} | 0 0 0 0 0 -38xy-20y2 -3xy-32y2 x2+32xy+45y2 |
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
|