-- 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 | -9x2-19xy-37y2 35x2+37xy+49y2 |
| -9x2+28y2 35x2+18xy+25y2 |
| -29x2+3xy-35y2 9x2-43xy-16y2 |
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 | x2+28xy-9y2 -15xy+18y2 x3 x2y+43xy2+15y3 20xy2+46y3 y4 0 0 |
| x2+33xy+31y2 42xy-28y2 0 2xy2+17y3 -29xy2-44y3 0 y4 0 |
| 32xy-15y2 x2-17xy-22y2 0 19y3 xy2+33y3 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
| x2+28xy-9y2 -15xy+18y2 x3 x2y+43xy2+15y3 20xy2+46y3 y4 0 0 |
| x2+33xy+31y2 42xy-28y2 0 2xy2+17y3 -29xy2-44y3 0 y4 0 |
| 32xy-15y2 x2-17xy-22y2 0 19y3 xy2+33y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | 44xy2-18y3 46xy2+7y3 -44y3 45y3 -37y3 |
{2} | -30xy2-43y3 -33y3 30y3 16y3 -35y3 |
{3} | -9xy+24y2 23xy-20y2 9y2 2y2 -21y2 |
{3} | 9x2-45xy-13y2 -23x2-47xy+2y2 -9xy+21y2 -2xy-43y2 21xy+50y2 |
{3} | 30x2-6xy-11y2 8xy+29y2 -30xy+49y2 -16xy+4y2 35xy+44y2 |
{4} | 0 0 x+30y -28y 48y |
{4} | 0 0 -37y x-40y 41y |
{4} | 0 0 4y -5y x+10y |
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-33y -42y |
{2} | 0 -32y x+17y |
{3} | 1 -1 0 |
{3} | 0 5 -44 |
{3} | 0 22 39 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 6 29 0 -48y -37x+7y xy+30y2 14xy-45y2 -18xy+30y2 |
{5} | -15 27 0 -22x+13y 47x+23y -2y2 xy-50y2 29xy+47y2 |
{5} | 0 0 0 0 0 x2-30xy+7y2 28xy+40y2 -48xy-36y2 |
{5} | 0 0 0 0 0 37xy+29y2 x2+40xy+7y2 -41xy+24y2 |
{5} | 0 0 0 0 0 -4xy+42y2 5xy+38y2 x2-10xy-14y2 |
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
|