-- 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 | -7x2+10xy-5y2 x2+31xy-14y2 |
| 10x2-41xy-44y2 -29x2-34xy-23y2 |
| -35x2+17xy-33y2 -32x2+10xy-28y2 |
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 | 2x2-21xy+15y2 -5x2+37xy-10y2 x3 x2y+6xy2-4y3 24xy2-34y3 y4 0 0 |
| x2+22xy-32y2 19xy-5y2 0 21xy2-47y3 13xy2+34y3 0 y4 0 |
| 18xy-31y2 x2-4xy-31y2 0 -36y3 xy2+21y3 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
| 2x2-21xy+15y2 -5x2+37xy-10y2 x3 x2y+6xy2-4y3 24xy2-34y3 y4 0 0 |
| x2+22xy-32y2 19xy-5y2 0 21xy2-47y3 13xy2+34y3 0 y4 0 |
| 18xy-31y2 x2-4xy-31y2 0 -36y3 xy2+21y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -42xy2-49y3 12xy2-37y3 42y3 8y3 -y3 |
{2} | 45xy2-28y3 -44y3 -45y3 -7y3 -41y3 |
{3} | -xy+15y2 15xy-37y2 y2 48y2 -42y2 |
{3} | x2-46xy-19y2 -15x2+2xy-48y2 -xy+31y2 -48xy-32y2 42xy-25y2 |
{3} | -45x2+26xy-26y2 -5xy+29y2 45xy+2y2 7xy-27y2 41xy-y2 |
{4} | 0 0 x+21y -24y -24y |
{4} | 0 0 -17y x+39y 36y |
{4} | 0 0 -29y -49y x+41y |
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-22y -19y |
{2} | 0 -18y x+4y |
{3} | 1 -2 5 |
{3} | 0 -25 21 |
{3} | 0 -49 -15 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | -37 -34 0 3y -9x+42y xy-16y2 -38xy+4y2 -35xy-40y2 |
{5} | 43 -48 0 -27x+37y -41x-2y -21y2 xy-49y2 -13xy+26y2 |
{5} | 0 0 0 0 0 x2-21xy+30y2 24xy+39y2 24xy-29y2 |
{5} | 0 0 0 0 0 17xy-44y2 x2-39xy-37y2 -36xy-45y2 |
{5} | 0 0 0 0 0 29xy+45y2 49xy+8y2 x2-41xy+7y2 |
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
|