-- 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 | 42x2-40xy-15y2 39x2-36xy-42y2 |
| -15x2+17xy-18y2 -23x2+7xy+33y2 |
| -37x2-17xy+38y2 -49x2-11xy+16y2 |
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 | 41x2+12xy-6y2 15x2-42xy-19y2 x3 x2y+29xy2+2y3 45xy2-25y3 y4 0 0 |
| x2+45xy-19y2 -46xy 0 -44xy2-40y3 -34xy2+43y3 0 y4 0 |
| 12xy-50y2 x2+42xy-49y2 0 22y3 xy2+45y3 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
| 41x2+12xy-6y2 15x2-42xy-19y2 x3 x2y+29xy2+2y3 45xy2-25y3 y4 0 0 |
| x2+45xy-19y2 -46xy 0 -44xy2-40y3 -34xy2+43y3 0 y4 0 |
| 12xy-50y2 x2+42xy-49y2 0 22y3 xy2+45y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 28xy2+25y3 -4xy2-24y3 -28y3 32y3 25y3 |
{2} | -40xy2+32y3 -3y3 40y3 -40y3 -29y3 |
{3} | -34xy-23y2 46xy-27y2 34y2 21y2 -8y2 |
{3} | 34x2+22xy-26y2 -46x2+10xy+9y2 -34xy+y2 -21xy+16y2 8xy-6y2 |
{3} | 40x2-24xy+8y2 -48xy+33y2 -40xy-8y2 40xy-42y2 29xy+42y2 |
{4} | 0 0 x-14y -34y -46y |
{4} | 0 0 -47y x+24y 45y |
{4} | 0 0 -11y -34y x-10y |
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-45y 46y |
{2} | 0 -12y x-42y |
{3} | 1 -41 -15 |
{3} | 0 -7 -2 |
{3} | 0 -17 49 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------------- A : 1
{5} | 8 -29 0 7y 48x+38y xy+28y2 -xy+35y2 22xy-35y2 |
{5} | -14 40 0 -11x-6y -26x+41y 44y2 xy-35y2 34xy-22y2 |
{5} | 0 0 0 0 0 x2+14xy-23y2 34xy+12y2 46xy-22y2 |
{5} | 0 0 0 0 0 47xy+45y2 x2-24xy+38y2 -45xy-36y2 |
{5} | 0 0 0 0 0 11xy+44y2 34xy-y2 x2+10xy-15y2 |
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
|