-- 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 | 27x2+xy-3y2 2x2-37xy-15y2 |
| 37x2-20xy-6y2 26x2-21xy-18y2 |
| 2x2-11xy-41y2 -16x2-49xy-7y2 |
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 | -47x2+41xy-18y2 -26x2+50xy-22y2 x3 x2y+45xy2-26y3 -33xy2+31y3 y4 0 0 |
| x2+33xy-46y2 36xy+40y2 0 25xy2-y3 39xy2-12y3 0 y4 0 |
| 30xy-7y2 x2-5xy+8y2 0 -2y3 xy2-23y3 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
| -47x2+41xy-18y2 -26x2+50xy-22y2 x3 x2y+45xy2-26y3 -33xy2+31y3 y4 0 0 |
| x2+33xy-46y2 36xy+40y2 0 25xy2-y3 39xy2-12y3 0 y4 0 |
| 30xy-7y2 x2-5xy+8y2 0 -2y3 xy2-23y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | 44xy2+8y3 31xy2+8y3 -44y3 35y3 11y3 |
{2} | -11xy2+18y3 37y3 11y3 34y3 47y3 |
{3} | 27xy+49y2 -23xy-21y2 -27y2 13y2 16y2 |
{3} | -27x2-22xy+30y2 23x2+39xy+29y2 27xy-27y2 -13xy-28y2 -16xy-y2 |
{3} | 11x2+18xy+10y2 -12xy-36y2 -11xy-36y2 -34xy+29y2 -47xy+3y2 |
{4} | 0 0 x-35y -47y 2y |
{4} | 0 0 6y x-36y -26y |
{4} | 0 0 35y -22y x-30y |
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-33y -36y |
{2} | 0 -30y x+5y |
{3} | 1 47 26 |
{3} | 0 -49 4 |
{3} | 0 39 -14 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 24 38 0 24y 46x+2y xy+43y2 49xy+42y2 41xy+14y2 |
{5} | -18 -39 0 22x+26y -47x+47y -25y2 xy-38y2 -39xy+28y2 |
{5} | 0 0 0 0 0 x2+35xy+3y2 47xy-40y2 -2xy-19y2 |
{5} | 0 0 0 0 0 -6xy-23y2 x2+36xy-30y2 26xy+11y2 |
{5} | 0 0 0 0 0 -35xy+17y2 22xy+9y2 x2+30xy+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
|