-- 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 | 37x2-34xy-25y2 -42xy-5y2 |
| -5x2+22xy+26y2 18x2+48xy+6y2 |
| 49x2-40xy+32y2 -20x2+10xy+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 | -44x2-19xy+17y2 21x2-15xy-40y2 x3 x2y-48xy2+15y3 -50xy2-38y3 y4 0 0 |
| x2+48xy-8y2 -40xy+43y2 0 -34xy2-43y3 -36xy2-50y3 0 y4 0 |
| 43xy+36y2 x2+18xy+27y2 0 36y3 xy2+32y3 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
| -44x2-19xy+17y2 21x2-15xy-40y2 x3 x2y-48xy2+15y3 -50xy2-38y3 y4 0 0 |
| x2+48xy-8y2 -40xy+43y2 0 -34xy2-43y3 -36xy2-50y3 0 y4 0 |
| 43xy+36y2 x2+18xy+27y2 0 36y3 xy2+32y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -41xy2 -15xy2+32y3 41y3 12y3 -43y3 |
{2} | -39xy2-49y3 0 39y3 7y3 -20y3 |
{3} | -8xy+40y2 45xy+2y2 8y2 16y2 -34y2 |
{3} | 8x2-4xy+43y2 -45x2+6xy-6y2 -8xy-36y2 -16xy-30y2 34xy+49y2 |
{3} | 39x2-32xy+34y2 43xy-4y2 -39xy-20y2 -7xy-44y2 20xy+41y2 |
{4} | 0 0 x+37y -35y 47y |
{4} | 0 0 42y x+42y 27y |
{4} | 0 0 13y 49y x+22y |
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-48y 40y |
{2} | 0 -43y x-18y |
{3} | 1 44 -21 |
{3} | 0 22 32 |
{3} | 0 -26 -9 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------- A : 1
{5} | -33 8 0 41y -44x+28y xy-21y2 -5xy-24y2 -29xy-30y2 |
{5} | 44 -45 0 -9x+42y -37x-14y 34y2 xy+y2 36xy-9y2 |
{5} | 0 0 0 0 0 x2-37xy+5y2 35xy+43y2 -47xy+10y2 |
{5} | 0 0 0 0 0 -42xy+33y2 x2-42xy+y2 -27xy-35y2 |
{5} | 0 0 0 0 0 -13xy-3y2 -49xy-46y2 x2-22xy-6y2 |
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
|