-- 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 | -31x2-38xy+42y2 40x2+18xy-40y2 |
| 15x2-44xy+11y2 -17x2-37xy-32y2 |
| -27x2+30xy+8y2 41x2-34xy+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 | -20x2-25xy+8y2 5x2-5xy-42y2 x3 x2y-xy2+16y3 -29xy2-3y3 y4 0 0 |
| x2+28xy-2y2 -9xy-9y2 0 -31xy2-42y3 -44xy2-34y3 0 y4 0 |
| 2xy+41y2 x2+15y2 0 49y3 xy2-11y3 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
| -20x2-25xy+8y2 5x2-5xy-42y2 x3 x2y-xy2+16y3 -29xy2-3y3 y4 0 0 |
| x2+28xy-2y2 -9xy-9y2 0 -31xy2-42y3 -44xy2-34y3 0 y4 0 |
| 2xy+41y2 x2+15y2 0 49y3 xy2-11y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 12xy2-41y3 10xy2-4y3 -12y3 -46y3 -49y3 |
{2} | -29xy2+26y3 24y3 29y3 -14y3 23y3 |
{3} | 31xy-3y2 29xy-47y2 -31y2 -21y2 -5y2 |
{3} | -31x2-14xy+15y2 -29x2+16xy-38y2 31xy+17y2 21xy-19y2 5xy-40y2 |
{3} | 29x2-30xy-23y2 -37xy+23y2 -29xy+4y2 14xy+25y2 -23xy+3y2 |
{4} | 0 0 x+44y -43y -13y |
{4} | 0 0 -24y x+36y 46y |
{4} | 0 0 -25y -31y x+21y |
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 9y |
{2} | 0 -2y x |
{3} | 1 20 -5 |
{3} | 0 -20 -17 |
{3} | 0 15 -33 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | -4 29 0 5y 7x+30y xy-7y2 3xy-16y2 -41xy-34y2 |
{5} | 21 -49 0 -7x+50y -4x+32y 31y2 xy-y2 44xy-45y2 |
{5} | 0 0 0 0 0 x2-44xy-40y2 43xy-7y2 13xy+5y2 |
{5} | 0 0 0 0 0 24xy-40y2 x2-36xy-7y2 -46xy+5y2 |
{5} | 0 0 0 0 0 25xy+28y2 31xy+15y2 x2-21xy+47y2 |
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
|