-- 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 | -35x2+37xy-13y2 17x2+49xy-15y2 |
| -7x2+41xy-25y2 10x2+40xy-4y2 |
| -43x2-32xy-29y2 38x2-16xy+48y2 |
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 | 45x2-26xy+22y2 -30x2+48xy+39y2 x3 x2y-34xy2+27y3 15xy2+25y3 y4 0 0 |
| x2+36xy-13y2 -35xy-2y2 0 -29xy2-21y3 5xy2+31y3 0 y4 0 |
| -19xy-20y2 x2-22xy-36y2 0 -5y3 xy2-49y3 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
| 45x2-26xy+22y2 -30x2+48xy+39y2 x3 x2y-34xy2+27y3 15xy2+25y3 y4 0 0 |
| x2+36xy-13y2 -35xy-2y2 0 -29xy2-21y3 5xy2+31y3 0 y4 0 |
| -19xy-20y2 x2-22xy-36y2 0 -5y3 xy2-49y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -7xy2-23y3 19xy2-10y3 7y3 -42y3 4y3 |
{2} | 21xy2+21y3 28y3 -21y3 -32y3 -30y3 |
{3} | -24xy-13y2 -32xy+33y2 24y2 -18y2 33y2 |
{3} | 24x2-32xy-3y2 32x2-2xy+22y2 -24xy+45y2 18xy-49y2 -33xy-26y2 |
{3} | -21x2+5xy+19y2 -12xy+6y2 21xy-26y2 32xy-46y2 30xy+13y2 |
{4} | 0 0 x-y -y 45y |
{4} | 0 0 -18y x-11y -48y |
{4} | 0 0 -49y -47y x+12y |
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-36y 35y |
{2} | 0 19y x+22y |
{3} | 1 -45 30 |
{3} | 0 -6 47 |
{3} | 0 -44 -27 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------- A : 1
{5} | 46 42 0 9y 24x+16y xy-18y2 40xy-43y2 -13xy+21y2 |
{5} | -28 27 0 -41x-3y -18x-27y 29y2 xy-22y2 -5xy-13y2 |
{5} | 0 0 0 0 0 x2+xy+36y2 xy+18y2 -45xy+38y2 |
{5} | 0 0 0 0 0 18xy+43y2 x2+11xy-29y2 48xy-50y2 |
{5} | 0 0 0 0 0 49xy+4y2 47xy+2y2 x2-12xy-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
|