-- 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 | -31x2-6xy+38y2 -20x2-4xy+42y2 |
| 25x2+42xy-5y2 12x2-38xy-12y2 |
| -42x2-21xy+30y2 24x2+17xy-45y2 |
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 | -19x2-42xy-36y2 -25x2+4xy+45y2 x3 x2y-2xy2-9y3 12xy2-36y3 y4 0 0 |
| x2-17xy+17y2 49xy-9y2 0 -17xy2-41y3 -45xy2-34y3 0 y4 0 |
| -30xy-6y2 x2-39xy+39y2 0 12y3 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
| -19x2-42xy-36y2 -25x2+4xy+45y2 x3 x2y-2xy2-9y3 12xy2-36y3 y4 0 0 |
| x2-17xy+17y2 49xy-9y2 0 -17xy2-41y3 -45xy2-34y3 0 y4 0 |
| -30xy-6y2 x2-39xy+39y2 0 12y3 xy2+36y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | 46xy2+6y3 -46xy2+34y3 -46y3 -32y3 21y3 |
{2} | -4xy2 34y3 4y3 -2y3 -37y3 |
{3} | -4xy-36y2 -27xy+3y2 4y2 25y2 -34y2 |
{3} | 4x2-38xy-17y2 27x2-15xy+29y2 -4xy-27y2 -25xy-25y2 34xy+7y2 |
{3} | 4x2+22xy-27y2 -21xy-24y2 -4xy-22y2 2xy-2y2 37xy-29y2 |
{4} | 0 0 x-43y -46y 26y |
{4} | 0 0 -27y x+39y 25y |
{4} | 0 0 -23y -45y x+4y |
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+17y -49y |
{2} | 0 30y x+39y |
{3} | 1 19 25 |
{3} | 0 4 40 |
{3} | 0 -31 12 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 33 -3 0 3y -25x-17y xy+44y2 50xy-14y2 16xy-23y2 |
{5} | -20 -16 0 15x-37y -15x-41y 17y2 xy-27y2 45xy+10y2 |
{5} | 0 0 0 0 0 x2+43xy-32y2 46xy+24y2 -26xy-43y2 |
{5} | 0 0 0 0 0 27xy+38y2 x2-39xy+22y2 -25xy-31y2 |
{5} | 0 0 0 0 0 23xy-9y2 45xy+32y2 x2-4xy+10y2 |
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
|