-- 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 | -10x2-46xy-38y2 -41x2+49xy-43y2 |
| -x2-23xy-37y2 -27x2-23xy+8y2 |
| -34x2+28xy-15y2 42x2-27xy-32y2 |
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 | -8x2+27xy-45y2 -47x2-44xy+47y2 x3 x2y-20xy2+21y3 -35xy2+39y3 y4 0 0 |
| x2-39xy+39y2 -16xy-6y2 0 21xy2+25y3 -26xy2+18y3 0 y4 0 |
| -11xy+2y2 x2+50xy+39y2 0 -37y3 xy2+y3 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
| -8x2+27xy-45y2 -47x2-44xy+47y2 x3 x2y-20xy2+21y3 -35xy2+39y3 y4 0 0 |
| x2-39xy+39y2 -16xy-6y2 0 21xy2+25y3 -26xy2+18y3 0 y4 0 |
| -11xy+2y2 x2+50xy+39y2 0 -37y3 xy2+y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 46xy2-42y3 32xy2+37y3 -46y3 -15y3 20y3 |
{2} | 20xy2+37y3 45y3 -20y3 3y3 -16y3 |
{3} | -50xy+39y2 40xy-5y2 50y2 3y2 10y2 |
{3} | 50x2-47xy+34y2 -40x2-34xy+42y2 -50xy+8y2 -3xy-43y2 -10xy-30y2 |
{3} | -20x2+26xy 39xy+28y2 20xy+38y2 -3xy-19y2 16xy+27y2 |
{4} | 0 0 x+48y 20y 17y |
{4} | 0 0 -18y x -6y |
{4} | 0 0 19y -43y x-48y |
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+39y 16y |
{2} | 0 11y x-50y |
{3} | 1 8 47 |
{3} | 0 -6 44 |
{3} | 0 -22 11 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------- A : 1
{5} | 9 46 0 -33y 5x+25y xy+8y2 15xy+45y2 21xy+14y2 |
{5} | 1 -2 0 -48x+45y -19x+39y -21y2 xy-45y2 26xy-49y2 |
{5} | 0 0 0 0 0 x2-48xy+45y2 -20xy+27y2 -17xy-19y2 |
{5} | 0 0 0 0 0 18xy+32y2 x2-y2 6xy-18y2 |
{5} | 0 0 0 0 0 -19xy-34y2 43xy+20y2 x2+48xy-44y2 |
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
|