-- 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 | 19x2-37xy-5y2 16x2+10xy+8y2 |
| 13x2+18xy-13y2 42x2-13xy-23y2 |
| 37x2-10xy-47y2 -9x2+17xy-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 | -10x2-3xy-34y2 -26x2+41xy+20y2 x3 x2y+34xy2+31y3 -10xy2-36y3 y4 0 |
| x2-26xy+43y2 26xy-10y2 0 -26xy2+31y3 -49xy2-35y3 0 y4 |
| 30xy+37y2 x2-19xy-17y2 0 -14y3 xy2+44y3 0 0 |
3
o4 : A-module, quotient of A
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i5 : C = resolution N
3 7 4
o5 = A <-- A <-- A <-- 0
0 1 2 3
o5 : ChainComplex
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i6 : d = C.dd
3 7
o6 = 0 : A <-------------------------------------------------------------------------- A : 1
| -10x2-3xy-34y2 -26x2+41xy+20y2 x3 x2y+34xy2+31y3 -10xy2-36y3 y4 0 |
| x2-26xy+43y2 26xy-10y2 0 -26xy2+31y3 -49xy2-35y3 0 y4 |
| 30xy+37y2 x2-19xy-17y2 0 -14y3 xy2+44y3 0 0 |
7 4
1 : A <------------------------------------------------------------------ A : 2
{2} | 38xy2-28y3 25y3 -49xy2-35y3 23y4 |
{2} | 29y3 -44y3 23xy2+2y3 49y4 |
{3} | -17xy-7y2 47y2 44xy+17y2 0 |
{3} | 17x2+11xy-28y2 -47xy+33y2 -44x2+44xy-43y2 3y3 |
{3} | -22xy-4y2 44xy-39y2 -23x2-22xy+28y2 -49xy2-27y3 |
{4} | 0 -26x+10y 0 x2-36xy+50y2 |
{4} | 0 x+36y 0 -22y2 |
4
2 : A <----- 0 : 3
0
o6 : ChainComplexMap
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i7 : s = nullhomotopy (x^3 * id_C)
7 3
o7 = 1 : A <------------------------- A : 0
{2} | 0 x+26y -26y |
{2} | 0 -30y x+19y |
{3} | 1 10 26 |
{3} | 0 -12 -9 |
{3} | 0 27 47 |
{4} | 0 0 0 |
{4} | 0 0 0 |
4 7
2 : A <------------------------------------------------------- A : 1
{5} | 35 27 0 6x+43y 50x-22y 26y2 xy+28y2 |
{5} | 0 0 0 0 0 22y2 x2-36xy+50y2 |
{5} | -12 24 0 25y -22x-18y 33y2 -42xy+45y2 |
{6} | 0 0 0 0 0 x+36y 26x-10y |
4
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 |
7 7
1 : A <-------------------------------- A : 1
{2} | x3 0 0 0 0 0 0 |
{2} | 0 x3 0 0 0 0 0 |
{3} | 0 0 x3 0 0 0 0 |
{3} | 0 0 0 x3 0 0 0 |
{3} | 0 0 0 0 x3 0 0 |
{4} | 0 0 0 0 0 x3 0 |
{4} | 0 0 0 0 0 0 x3 |
4 4
2 : A <----------------------- A : 2
{5} | x3 0 0 0 |
{5} | 0 x3 0 0 |
{5} | 0 0 x3 0 |
{6} | 0 0 0 x3 |
3 : 0 <----- 0 : 3
0
o8 : ChainComplexMap
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i9 : s^2
4 3
o9 = 2 : A <----- A : 0
0
7
3 : 0 <----- A : 1
0
o9 : ChainComplexMap
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