-- 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 | 24x2-36xy-30y2 -22x2-29xy-24y2 |
| -29x2+19xy+19y2 -38x2-16xy+39y2 |
| -10x2-29xy-8y2 21x2+34xy+19y2 |
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 | -49x2-45xy-27y2 -32x2+23xy-50y2 x3 x2y-44xy2-44y3 20xy2-33y3 y4 0 0 |
| x2+21xy-49y2 18xy+19y2 0 38xy2-30y3 -9xy2+44y3 0 y4 0 |
| -8xy-23y2 x2+16xy+17y2 0 -22y3 xy2+y3 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
| -49x2-45xy-27y2 -32x2+23xy-50y2 x3 x2y-44xy2-44y3 20xy2-33y3 y4 0 0 |
| x2+21xy-49y2 18xy+19y2 0 38xy2-30y3 -9xy2+44y3 0 y4 0 |
| -8xy-23y2 x2+16xy+17y2 0 -22y3 xy2+y3 0 0 y4 |
8 5
1 : A <---------------------------------------------------------------------------- A : 2
{2} | -38xy2+32y3 -9xy2+23y3 38y3 21y3 35y3 |
{2} | 43xy2+2y3 36y3 -43y3 13y3 -33y3 |
{3} | -36xy-4y2 29xy-44y2 36y2 -7y2 25y2 |
{3} | 36x2+43xy+29y2 -29x2+44xy-19y2 -36xy-39y2 7xy-7y2 -25xy+10y2 |
{3} | -43x2+43xy+27y2 -39xy+4y2 43xy-45y2 -13xy+26y2 33xy+22y2 |
{4} | 0 0 x+18y 50y -44y |
{4} | 0 0 -46y x+34y -43y |
{4} | 0 0 -16y -19y x+49y |
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-21y -18y |
{2} | 0 8y x-16y |
{3} | 1 49 32 |
{3} | 0 -21 -3 |
{3} | 0 30 -6 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | -4 21 0 4y -47x-46y xy+32y2 -5xy-14y2 36xy+10y2 |
{5} | -44 19 0 -7x-37y -27x-18y -38y2 xy+7y2 9xy+36y2 |
{5} | 0 0 0 0 0 x2-18xy+41y2 -50xy+2y2 44xy-48y2 |
{5} | 0 0 0 0 0 46xy+13y2 x2-34xy-24y2 43xy-30y2 |
{5} | 0 0 0 0 0 16xy+4y2 19xy+47y2 x2-49xy-17y2 |
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
|