-- 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 | -29x2+21xy+16y2 27x2+21y2 |
| -46x2+35xy+41y2 -13x2-32xy+4y2 |
| 12x2+27xy-44y2 -37x2-2xy+32y2 |
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 | -4x2-27xy-11y2 -43x2-26xy-24y2 x3 x2y-3xy2-36y3 -35xy2-29y3 y4 0 0 |
| x2+5xy+24y2 -20xy-14y2 0 -28xy2+27y3 30xy2+40y3 0 y4 0 |
| -37xy+21y2 x2-47xy-41y2 0 -16y3 xy2+18y3 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
| -4x2-27xy-11y2 -43x2-26xy-24y2 x3 x2y-3xy2-36y3 -35xy2-29y3 y4 0 0 |
| x2+5xy+24y2 -20xy-14y2 0 -28xy2+27y3 30xy2+40y3 0 y4 0 |
| -37xy+21y2 x2-47xy-41y2 0 -16y3 xy2+18y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 14xy2+4y3 -11xy2+10y3 -14y3 -43y3 -8y3 |
{2} | 30xy2-40y3 -45y3 -30y3 43y3 -17y3 |
{3} | 10xy-43y2 4xy-45y2 -10y2 44y2 47y2 |
{3} | -10x2+6xy+26y2 -4x2-11xy-8y2 10xy+37y2 -44xy+40y2 -47xy-6y2 |
{3} | -30x2+25xy-39y2 -22xy+19y2 30xy+15y2 -43xy-5y2 17xy-33y2 |
{4} | 0 0 x-16y 36y 48y |
{4} | 0 0 34y x+47y 22y |
{4} | 0 0 -8y -38y x-31y |
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-5y 20y |
{2} | 0 37y x+47y |
{3} | 1 4 43 |
{3} | 0 -18 6 |
{3} | 0 18 -40 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 41 -35 0 49y 37x-36y xy+10y2 -25xy+48y2 -23xy-14y2 |
{5} | -6 -12 0 25x+36y -42x-42y 28y2 xy+7y2 -30xy+16y2 |
{5} | 0 0 0 0 0 x2+16xy-15y2 -36xy-y2 -48xy-50y2 |
{5} | 0 0 0 0 0 -34xy-31y2 x2-47xy-29y2 -22xy-36y2 |
{5} | 0 0 0 0 0 8xy-7y2 38xy+13y2 x2+31xy+44y2 |
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
|