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