-- 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 | 6x2-20xy+34y2 -43x2-29xy+39y2 |
| 27x2-21xy+47y2 48x2-28xy-9y2 |
| 38x2-35xy-50y2 20x2+42xy-3y2 |
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 | -44x2-35xy+42y2 -43x2+27xy-13y2 x3 x2y-32xy2+26y3 -41xy2+40y3 y4 0 0 |
| x2-37xy-15y2 47xy-20y2 0 34xy2-22y3 -3xy2+41y3 0 y4 0 |
| -13xy-41y2 x2+3xy-28y2 0 -9y3 xy2-35y3 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
| -44x2-35xy+42y2 -43x2+27xy-13y2 x3 x2y-32xy2+26y3 -41xy2+40y3 y4 0 0 |
| x2-37xy-15y2 47xy-20y2 0 34xy2-22y3 -3xy2+41y3 0 y4 0 |
| -13xy-41y2 x2+3xy-28y2 0 -9y3 xy2-35y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | -31xy2-y3 -5xy2-33y3 31y3 19y3 15y3 |
{2} | -50xy2-30y3 12y3 50y3 -10y3 0 |
{3} | -38xy+30y2 -15xy+12y2 38y2 -33y2 45y2 |
{3} | 38x2+25xy+46y2 15x2+46xy+26y2 -38xy+46y2 33xy-47y2 -45xy+28y2 |
{3} | 50x2-50xy+22y2 -43xy-44y2 -50xy-21y2 10xy+15y2 -9y2 |
{4} | 0 0 x+2y -3y 12y |
{4} | 0 0 5y x-49y -2y |
{4} | 0 0 27y -5y x+47y |
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+37y -47y |
{2} | 0 13y x-3y |
{3} | 1 44 43 |
{3} | 0 0 -2 |
{3} | 0 -22 32 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -44 -37 0 y -2x-27y xy-42y2 20xy+38y2 -46y2 |
{5} | -3 40 0 27x-18y 32x+14y -34y2 xy-6y2 3xy-3y2 |
{5} | 0 0 0 0 0 x2-2xy+10y2 3xy-20y2 -12xy-12y2 |
{5} | 0 0 0 0 0 -5xy+14y2 x2+49xy-28y2 2xy-37y2 |
{5} | 0 0 0 0 0 -27xy-15y2 5xy+30y2 x2-47xy+18y2 |
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
|