-- 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 | -24x2-47xy-32y2 -36x2-34xy-33y2 |
| -32x2+2xy+24y2 25x2+40xy+37y2 |
| -36x2+19xy+21y2 17x2+50xy-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 | -15x2+19xy-6y2 14x2-24xy-5y2 x3 x2y-3xy2+26y3 -6xy2-42y3 y4 0 0 |
| x2+24xy-11y2 -27xy-47y2 0 50xy2+31y3 -28xy2+30y3 0 y4 0 |
| 21xy+43y2 x2-22xy-36y2 0 -36y3 xy2-44y3 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
| -15x2+19xy-6y2 14x2-24xy-5y2 x3 x2y-3xy2+26y3 -6xy2-42y3 y4 0 0 |
| x2+24xy-11y2 -27xy-47y2 0 50xy2+31y3 -28xy2+30y3 0 y4 0 |
| 21xy+43y2 x2-22xy-36y2 0 -36y3 xy2-44y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | -15xy2+17y3 46xy2-33y3 15y3 39y3 -30y3 |
{2} | -34xy2-31y3 16y3 34y3 -43y3 10y3 |
{3} | 15xy-15y2 9xy-24y2 -15y2 7y2 5y2 |
{3} | -15x2+21xy+10y2 -9x2-20xy+37y2 15xy-6y2 -7xy+10y2 -5xy+42y2 |
{3} | 34x2+49xy-19y2 7xy-32y2 -34xy-18y2 43xy-24y2 -10xy+27y2 |
{4} | 0 0 x-37y -37y 13y |
{4} | 0 0 -36y x+30y 48y |
{4} | 0 0 -25y 18y x+7y |
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-24y 27y |
{2} | 0 -21y x+22y |
{3} | 1 15 -14 |
{3} | 0 16 20 |
{3} | 0 -1 -47 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 3 40 0 36y 3x-33y xy-11y2 -28xy-41y2 30xy-19y2 |
{5} | 8 -42 0 -45x+40y -5x+17y -50y2 xy-49y2 28xy+30y2 |
{5} | 0 0 0 0 0 x2+37xy-48y2 37xy-12y2 -13xy-45y2 |
{5} | 0 0 0 0 0 36xy-39y2 x2-30xy-35y2 -48xy-5y2 |
{5} | 0 0 0 0 0 25xy+y2 -18xy-25y2 x2-7xy-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
|