-- 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 | -38x2-48xy+28y2 42x2-49xy-48y2 |
| 23x2+35xy+32y2 -31x2-36xy+22y2 |
| -22x2-45xy+43y2 -13x2+31xy-29y2 |
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 | 28x2-47xy-42y2 31x2-24xy-36y2 x3 x2y+38xy2 -40xy2-26y3 y4 0 0 |
| x2-43xy-29y2 -19xy-18y2 0 -36xy2+45y3 22xy2-26y3 0 y4 0 |
| -19xy+4y2 x2-27xy+16y2 0 32y3 xy2+27y3 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
| 28x2-47xy-42y2 31x2-24xy-36y2 x3 x2y+38xy2 -40xy2-26y3 y4 0 0 |
| x2-43xy-29y2 -19xy-18y2 0 -36xy2+45y3 22xy2-26y3 0 y4 0 |
| -19xy+4y2 x2-27xy+16y2 0 32y3 xy2+27y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | -9xy2+y3 -12xy2-46y3 9y3 -43y3 -7y3 |
{2} | 41xy2-7y3 -47y3 -41y3 -30y3 30y3 |
{3} | -28xy+49y2 34xy+9y2 28y2 -48y2 -46y2 |
{3} | 28x2-35xy+7y2 -34x2+3xy-33y2 -28xy-14y2 48xy-5y2 46xy-46y2 |
{3} | -41x2+43xy-23y2 -2xy-16y2 41xy-36y2 30xy-34y2 -30xy+14y2 |
{4} | 0 0 x-14y -33y 39y |
{4} | 0 0 25y x-22y 44y |
{4} | 0 0 20y 13y x+36y |
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 19y |
{2} | 0 19y x+27y |
{3} | 1 -28 -31 |
{3} | 0 -29 -32 |
{3} | 0 13 -37 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | -12 -28 0 10y 32x+33y xy+45y2 50xy-13y2 -50xy+47y2 |
{5} | 16 -24 0 -3x+5y -39x-30y 36y2 xy+31y2 -22xy+26y2 |
{5} | 0 0 0 0 0 x2+14xy+50y2 33xy-22y2 -39xy+12y2 |
{5} | 0 0 0 0 0 -25xy-20y2 x2+22xy+29y2 -44xy-25y2 |
{5} | 0 0 0 0 0 -20xy-43y2 -13xy+27y2 x2-36xy+22y2 |
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
|