-- 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 | -33x2+41xy-4y2 -3x2-2xy+40y2 |
| -47x2+48xy-18y2 -8x2-24xy-10y2 |
| 26x2-29xy+27y2 30x2-43xy+10y2 |
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 | 30x2+7xy-15y2 18x2+22xy+31y2 x3 x2y-50xy2+5y3 39xy2+3y3 y4 0 0 |
| x2-36xy-28y2 30xy-28y2 0 22xy2+33y3 -50xy2-8y3 0 y4 0 |
| 45xy-46y2 x2-13xy+15y2 0 -16y3 xy2+42y3 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
| 30x2+7xy-15y2 18x2+22xy+31y2 x3 x2y-50xy2+5y3 39xy2+3y3 y4 0 0 |
| x2-36xy-28y2 30xy-28y2 0 22xy2+33y3 -50xy2-8y3 0 y4 0 |
| 45xy-46y2 x2-13xy+15y2 0 -16y3 xy2+42y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | 30xy2+43y3 -9xy2+10y3 -30y3 35y3 -9y3 |
{2} | -19xy2-11y3 25y3 19y3 -3y3 18y3 |
{3} | 50xy-46y2 -5xy+16y2 -50y2 -19y2 -10y2 |
{3} | -50x2-16xy-30y2 5x2-xy-17y2 50xy-39y2 19xy+39y2 10xy+37y2 |
{3} | 19x2+48xy+42y2 -45xy+22y2 -19xy-37y2 3xy-22y2 -18xy+39y2 |
{4} | 0 0 x-26y -16y 15y |
{4} | 0 0 -24y x+39y 50y |
{4} | 0 0 -28y -29y x-13y |
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+36y -30y |
{2} | 0 -45y x+13y |
{3} | 1 -30 -18 |
{3} | 0 26 38 |
{3} | 0 -38 -11 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 2 -26 0 43y 16x-7y xy-5y2 -48xy-y2 -15xy |
{5} | 35 -5 0 -20x+22y -42x-7y -22y2 xy-39y2 50xy+6y2 |
{5} | 0 0 0 0 0 x2+26xy+34y2 16xy-37y2 -15xy+29y2 |
{5} | 0 0 0 0 0 24xy+5y2 x2-39xy-50y2 -50xy+31y2 |
{5} | 0 0 0 0 0 28xy-30y2 29xy-3y2 x2+13xy+16y2 |
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
|