-- 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 | -4x2-33xy+22y2 36x2+34xy+50y2 |
| -5x2+48xy-44y2 5x2+33xy-3y2 |
| 39x2+33xy-5y2 4x2+45xy+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 | 16x2+50xy+50y2 -11x2+47xy-50y2 x3 x2y+xy2-35y3 16xy2-26y3 y4 0 0 |
| x2-9xy-11y2 -31xy+13y2 0 -15xy2+41y3 43xy2-8y3 0 y4 0 |
| -15xy+11y2 x2+30xy+22y2 0 -39y3 xy2-20y3 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
| 16x2+50xy+50y2 -11x2+47xy-50y2 x3 x2y+xy2-35y3 16xy2-26y3 y4 0 0 |
| x2-9xy-11y2 -31xy+13y2 0 -15xy2+41y3 43xy2-8y3 0 y4 0 |
| -15xy+11y2 x2+30xy+22y2 0 -39y3 xy2-20y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 24xy2+32y3 20xy2+31y3 -24y3 28y3 -12y3 |
{2} | 31xy2+7y3 27y3 -31y3 -39y3 -49y3 |
{3} | -7xy-9y2 -35xy-19y2 7y2 21y2 -5y2 |
{3} | 7x2-50xy-30y2 35x2-33xy+10y2 -7xy-42y2 -21xy+35y2 5xy-25y2 |
{3} | -31x2-15xy+3y2 22xy+2y2 31xy+8y2 39xy+36y2 49xy+40y2 |
{4} | 0 0 x+4y 23y 32y |
{4} | 0 0 6y x-29y 32y |
{4} | 0 0 -27y 9y x+25y |
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+9y 31y |
{2} | 0 15y x-30y |
{3} | 1 -16 11 |
{3} | 0 -29 36 |
{3} | 0 -23 30 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -4 14 0 38y 13x-14y xy-31y2 -2xy-37y2 -31xy-25y2 |
{5} | -33 -50 0 26x+40y -43x+27y 15y2 xy+33y2 -43xy-29y2 |
{5} | 0 0 0 0 0 x2-4xy-3y2 -23xy+16y2 -32xy+48y2 |
{5} | 0 0 0 0 0 -6xy-4y2 x2+29xy-46y2 -32xy-37y2 |
{5} | 0 0 0 0 0 27xy-22y2 -9xy+50y2 x2-25xy+49y2 |
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
|