-- 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 | 49x2+36xy-40y2 39x2-33xy+9y2 |
| 6x2-5xy+24y2 -46x2-10xy-33y2 |
| -43x2-25xy+41y2 -25x2+39xy-12y2 |
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 | 22x2-15xy-25y2 -38x2+37xy-19y2 x3 x2y+24xy2+6y3 35xy2+11y3 y4 0 0 |
| x2-43xy-29y2 27xy-14y2 0 50xy2+16y3 6xy2+33y3 0 y4 0 |
| 3xy+47y2 x2+xy+15y2 0 -19y3 xy2-4y3 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
| 22x2-15xy-25y2 -38x2+37xy-19y2 x3 x2y+24xy2+6y3 35xy2+11y3 y4 0 0 |
| x2-43xy-29y2 27xy-14y2 0 50xy2+16y3 6xy2+33y3 0 y4 0 |
| 3xy+47y2 x2+xy+15y2 0 -19y3 xy2-4y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | -26xy2+18y3 -41xy2+46y3 26y3 47y3 20y3 |
{2} | 31xy2+3y3 9y3 -31y3 -46y3 48y3 |
{3} | 20xy -19xy+14y2 -20y2 -40y2 31y2 |
{3} | -20x2-18xy+24y2 19x2+28xy-11y2 20xy+18y2 40xy+y2 -31xy-30y2 |
{3} | -31x2+45xy+33y2 -30xy-30y2 31xy-48y2 46xy+41y2 -48xy+19y2 |
{4} | 0 0 x-24y 46y -31y |
{4} | 0 0 0 x-44y -6y |
{4} | 0 0 -y -23y x-33y |
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+43y -27y |
{2} | 0 -3y x-y |
{3} | 1 -22 38 |
{3} | 0 -35 14 |
{3} | 0 29 -34 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------- A : 1
{5} | 27 38 0 -22y 13x+7y xy-20y2 8xy+38y2 18xy-21y2 |
{5} | 9 18 0 16x+33y 19x-3y -50y2 xy+50y2 -6xy-2y2 |
{5} | 0 0 0 0 0 x2+24xy+y2 -46xy+9y2 31xy-24y2 |
{5} | 0 0 0 0 0 6y2 x2+44xy-47y2 6xy-43y2 |
{5} | 0 0 0 0 0 xy-44y2 23xy+8y2 x2+33xy+46y2 |
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
|