-- 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 | -41x2+38xy+13y2 31x2+40xy-2y2 |
| -27x2+11xy-49y2 30x2+21xy-2y2 |
| 49x2+3xy+23y2 8x2+34xy+50y2 |
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 | -27x2+15xy 42x2-26xy+25y2 x3 x2y-23xy2+30y3 -22xy2-15y3 y4 0 0 |
| x2+12xy-16y2 46xy-16y2 0 2xy2+16y3 25xy2-26y3 0 y4 0 |
| 35xy-34y2 x2-26xy-43y2 0 3y3 xy2+36y3 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
| -27x2+15xy 42x2-26xy+25y2 x3 x2y-23xy2+30y3 -22xy2-15y3 y4 0 0 |
| x2+12xy-16y2 46xy-16y2 0 2xy2+16y3 25xy2-26y3 0 y4 0 |
| 35xy-34y2 x2-26xy-43y2 0 3y3 xy2+36y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 24xy2-46y3 -9xy2+23y3 -24y3 35y3 11y3 |
{2} | 42xy2+39y3 27y3 -42y3 -5y3 9y3 |
{3} | -8xy+45y2 46xy+28y2 8y2 -21y2 -6y2 |
{3} | 8x2+18xy-27y2 -46x2-16xy+7y2 -8xy+38y2 21xy+31y2 6xy-40y2 |
{3} | -42x2-16xy+11y2 22xy-42y2 42xy-23y2 5xy+18y2 -9xy-48y2 |
{4} | 0 0 x-31y -30y -48y |
{4} | 0 0 -40y x+48y 15y |
{4} | 0 0 11y 32y x-17y |
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-12y -46y |
{2} | 0 -35y x+26y |
{3} | 1 27 -42 |
{3} | 0 20 15 |
{3} | 0 49 6 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------------- A : 1
{5} | 18 29 0 -25y 12x-43y xy+42y2 41xy-22y2 7xy-8y2 |
{5} | -14 -10 0 -11x+39y 46x-10y -2y2 xy+y2 -25xy-6y2 |
{5} | 0 0 0 0 0 x2+31xy+17y2 30xy-26y2 48xy+36y2 |
{5} | 0 0 0 0 0 40xy-10y2 x2-48xy+45y2 -15xy-39y2 |
{5} | 0 0 0 0 0 -11xy+10y2 -32xy-45y2 x2+17xy+39y2 |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap
|
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
|