-- 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 | 50x2-40xy-12y2 -35x2-37xy-15y2 |
| 16x2-32xy+50y2 -43x2+48xy+5y2 |
| -29x2-48xy-8y2 -6x2+xy |
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 | 27x2-16xy+2y2 48x2+3xy+5y2 x3 x2y+7xy2-18y3 27xy2+49y3 y4 0 0 |
| x2+28xy+22y2 27xy-7y2 0 22xy2+22y3 -9xy2+18y3 0 y4 0 |
| 13xy-42y2 x2-26xy-2y2 0 6y3 xy2-25y3 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
| 27x2-16xy+2y2 48x2+3xy+5y2 x3 x2y+7xy2-18y3 27xy2+49y3 y4 0 0 |
| x2+28xy+22y2 27xy-7y2 0 22xy2+22y3 -9xy2+18y3 0 y4 0 |
| 13xy-42y2 x2-26xy-2y2 0 6y3 xy2-25y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 48xy2+11y3 -41xy2+40y3 -48y3 -13y3 -33y3 |
{2} | -46xy2-50y3 -42y3 46y3 -24y3 -44y3 |
{3} | -35xy+3y2 -34xy+26y2 35y2 -15y2 31y2 |
{3} | 35x2+28xy-35y2 34x2+35xy+23y2 -35xy-31y2 15xy+43y2 -31xy+12y2 |
{3} | 46x2-22xy-35y2 -33xy-26y2 -46xy-29y2 24xy-46y2 44xy-36y2 |
{4} | 0 0 x+22y 43y 44y |
{4} | 0 0 -44y x -6y |
{4} | 0 0 -39y 18y x-22y |
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-28y -27y |
{2} | 0 -13y x+26y |
{3} | 1 -27 -48 |
{3} | 0 -18 -17 |
{3} | 0 -33 19 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------- A : 1
{5} | -41 -7 0 -22y 11x+22y xy+44y2 39xy-12y2 21xy+4y2 |
{5} | 16 -22 0 3x-25y -44x-28y -22y2 xy+45y2 9xy-50y2 |
{5} | 0 0 0 0 0 x2-22xy+7y2 -43xy+21y2 -44xy+45y2 |
{5} | 0 0 0 0 0 44xy-27y2 x2+20y2 6xy+14y2 |
{5} | 0 0 0 0 0 39xy+16y2 -18xy+48y2 x2+22xy-27y2 |
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
|