-- 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 | 9x2+31xy+10y2 -37x2-8xy-44y2 |
| 34x2-7xy-20y2 -8x2+22xy+48y2 |
| -20x2-35xy -39x2-49xy+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 | 39x2+12xy-2y2 -20x2+34xy-14y2 x3 x2y+44xy2+34y3 -41xy2+22y3 y4 0 0 |
| x2+42xy-41y2 -4xy+2y2 0 -26xy2+5y3 -30xy2+19y3 0 y4 0 |
| -32xy+36y2 x2+13xy+41y2 0 -18y3 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
| 39x2+12xy-2y2 -20x2+34xy-14y2 x3 x2y+44xy2+34y3 -41xy2+22y3 y4 0 0 |
| x2+42xy-41y2 -4xy+2y2 0 -26xy2+5y3 -30xy2+19y3 0 y4 0 |
| -32xy+36y2 x2+13xy+41y2 0 -18y3 xy2-10y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 40xy2+17y3 -xy2+49y3 -40y3 -48y3 -50y3 |
{2} | 2xy2-25y3 -3y3 -2y3 43y3 -4y3 |
{3} | 35xy+y2 35xy-14y2 -35y2 -40y2 -9y2 |
{3} | -35x2+38xy+6y2 -35x2-23xy-11y2 35xy-39y2 40xy+17y2 9xy+22y2 |
{3} | -2x2+23xy-42y2 48xy-8y2 2xy+2y2 -43xy+13y2 4xy-34y2 |
{4} | 0 0 x-38y 46y 46y |
{4} | 0 0 36y x+38y 9y |
{4} | 0 0 32y -3y x |
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-42y 4y |
{2} | 0 32y x-13y |
{3} | 1 -39 20 |
{3} | 0 44 -46 |
{3} | 0 22 -47 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | 11 27 0 -18y 50x-19y xy+3y2 29xy+30y2 2xy+15y2 |
{5} | 22 -11 0 -26x+12y -50x+34y 26y2 xy+5y2 30xy+33y2 |
{5} | 0 0 0 0 0 x2+38xy+27y2 -46xy-37y2 -46xy-21y2 |
{5} | 0 0 0 0 0 -36xy-15y2 x2-38xy+43y2 -9xy-22y2 |
{5} | 0 0 0 0 0 -32xy-11y2 3xy+45y2 x2+31y2 |
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
|