-- 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 | 9x2-45xy+40y2 x2+21xy+35y2 |
| -7x2-16xy+28y2 3x2+39xy-16y2 |
| 28x2-11xy+22y2 38x2+49xy+41y2 |
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 | 32x2+36xy+10y2 48x2+11xy+40y2 x3 x2y+41xy2+45y3 -21xy2+36y3 y4 0 0 |
| x2-24xy+28y2 -21xy+8y2 0 -21xy2-25y3 4xy2-9y3 0 y4 0 |
| 8xy-17y2 x2-2xy+29y2 0 42y3 xy2-46y3 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
| 32x2+36xy+10y2 48x2+11xy+40y2 x3 x2y+41xy2+45y3 -21xy2+36y3 y4 0 0 |
| x2-24xy+28y2 -21xy+8y2 0 -21xy2-25y3 4xy2-9y3 0 y4 0 |
| 8xy-17y2 x2-2xy+29y2 0 42y3 xy2-46y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -45xy2-11y3 -30xy2+26y3 45y3 -38y3 -14y3 |
{2} | 18xy2-38y3 -49y3 -18y3 7y3 -2y3 |
{3} | 20xy-50y2 -13xy-y2 -20y2 32y2 -43y2 |
{3} | -20x2-43xy+14y2 13x2+24xy-23y2 20xy-8y2 -32xy+25y2 43xy+35y2 |
{3} | -18x2+42xy+39y2 46xy-11y2 18xy-4y2 -7xy+27y2 2xy+9y2 |
{4} | 0 0 x-34y 23y 38y |
{4} | 0 0 -39y x-43y -50y |
{4} | 0 0 25y 50y x-24y |
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+24y 21y |
{2} | 0 -8y x+2y |
{3} | 1 -32 -48 |
{3} | 0 -16 29 |
{3} | 0 11 9 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -5 -50 0 33y 28x-5y xy+35y2 -6xy-44y2 45xy-10y2 |
{5} | -22 -17 0 -31x+7y 12x-46y 21y2 xy-10y2 -4xy+15y2 |
{5} | 0 0 0 0 0 x2+34xy-3y2 -23xy+28y2 -38xy-21y2 |
{5} | 0 0 0 0 0 39xy+36y2 x2+43xy-33y2 50xy+50y2 |
{5} | 0 0 0 0 0 -25xy+34y2 -50xy-48y2 x2+24xy+36y2 |
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
|