-- 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 | 39x2-7xy+40y2 -11x2-43xy+30y2 |
| -42x2-47xy-16y2 -25x2+19xy+5y2 |
| 26x2+35xy+50y2 -43x2-7xy+42y2 |
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 | -34x2-3xy-15y2 -34x2-9xy-46y2 x3 x2y-30xy2+45y3 14xy2+40y3 y4 0 0 |
| x2+13xy+49y2 -8xy-38y2 0 -24xy2-37y3 29xy2+46y3 0 y4 0 |
| 6xy+6y2 x2+46xy-35y2 0 23y3 xy2+7y3 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
| -34x2-3xy-15y2 -34x2-9xy-46y2 x3 x2y-30xy2+45y3 14xy2+40y3 y4 0 0 |
| x2+13xy+49y2 -8xy-38y2 0 -24xy2-37y3 29xy2+46y3 0 y4 0 |
| 6xy+6y2 x2+46xy-35y2 0 23y3 xy2+7y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | -45xy2-26y3 -32xy2-30y3 45y3 14y3 -26y3 |
{2} | 43xy2-31y3 -42y3 -43y3 -3y3 -18y3 |
{3} | 37xy-49y2 35xy+22y2 -37y2 0 13y2 |
{3} | -37x2-22xy+6y2 -35x2-39xy-20y2 37xy-30y2 29y2 -13xy-17y2 |
{3} | -43x2-20xy-37y2 29xy-32y2 43xy-50y2 3xy+33y2 18xy+45y2 |
{4} | 0 0 x+27y -28y -31y |
{4} | 0 0 -23y x-33y 12y |
{4} | 0 0 -28y 24y x+6y |
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-13y 8y |
{2} | 0 -6y x-46y |
{3} | 1 34 34 |
{3} | 0 -37 30 |
{3} | 0 45 -18 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -6 -38 0 13y -47x-41y xy-20y2 40xy-11y2 38xy+22y2 |
{5} | -38 -34 0 -26x+21y -34x+4y 24y2 xy-33y2 -29xy+37y2 |
{5} | 0 0 0 0 0 x2-27xy+19y2 28xy+30y2 31xy-46y2 |
{5} | 0 0 0 0 0 23xy+4y2 x2+33xy+y2 -12xy-15y2 |
{5} | 0 0 0 0 0 28xy+39y2 -24xy+35y2 x2-6xy-20y2 |
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
|