-- 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 | -36x2+50xy+47y2 50x2+30xy+29y2 |
| 29x2+3xy-23y2 32x2-50xy-37y2 |
| 21x2-22xy+26y2 -7x2+21xy-46y2 |
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 | 30x2-11xy+14y2 29x2+32xy+31y2 x3 x2y+44xy2-36y3 -49xy2+9y3 y4 0 0 |
| x2-44xy-14y2 8xy-y2 0 26xy2+21y3 -29xy2-17y3 0 y4 0 |
| 48xy+29y2 x2-34y2 0 7y3 xy2+28y3 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
| 30x2-11xy+14y2 29x2+32xy+31y2 x3 x2y+44xy2-36y3 -49xy2+9y3 y4 0 0 |
| x2-44xy-14y2 8xy-y2 0 26xy2+21y3 -29xy2-17y3 0 y4 0 |
| 48xy+29y2 x2-34y2 0 7y3 xy2+28y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | -14xy2-44y3 -39xy2+28y3 14y3 -45y3 -13y3 |
{2} | 15xy2+12y3 -26y3 -15y3 -11y3 -44y3 |
{3} | -11xy+35y2 49xy-24y2 11y2 -14y2 32y2 |
{3} | 11x2+44xy+4y2 -49x2+17xy+25y2 -11xy+22y2 14xy-24y2 -32xy-22y2 |
{3} | -15x2-7xy-13y2 19xy+45y2 15xy-5y2 11xy+37y2 44xy+19y2 |
{4} | 0 0 x-5y -24y -23y |
{4} | 0 0 38y x-13y -47y |
{4} | 0 0 -21y -38y x+18y |
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+44y -8y |
{2} | 0 -48y x |
{3} | 1 -30 -29 |
{3} | 0 -18 6 |
{3} | 0 -20 14 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------- A : 1
{5} | -35 -26 0 39y -27x+21y xy+11y2 -6xy-37y2 -24xy |
{5} | 33 -49 0 -33x-13y -4x+10y -26y2 xy+38y2 29xy+39y2 |
{5} | 0 0 0 0 0 x2+5xy 24xy-7y2 23xy+21y2 |
{5} | 0 0 0 0 0 -38xy x2+13xy+33y2 47xy+2y2 |
{5} | 0 0 0 0 0 21xy 38xy+11y2 x2-18xy-33y2 |
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
|