-- 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 | -23x2-50xy-25y2 32x2+13xy-16y2 |
| 42x2-40xy-22y2 x2+28xy-28y2 |
| 19x2-7xy-27y2 -31x2-16xy+23y2 |
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 | 50x2-24xy+4y2 -32x2-24xy+43y2 x3 x2y-40xy2-y3 28xy2+2y3 y4 0 0 |
| x2-38xy+25y2 50xy+18y2 0 -44xy2+44y3 -18xy2+16y3 0 y4 0 |
| -2xy-50y2 x2+20xy+40y2 0 31y3 xy2+19y3 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
| 50x2-24xy+4y2 -32x2-24xy+43y2 x3 x2y-40xy2-y3 28xy2+2y3 y4 0 0 |
| x2-38xy+25y2 50xy+18y2 0 -44xy2+44y3 -18xy2+16y3 0 y4 0 |
| -2xy-50y2 x2+20xy+40y2 0 31y3 xy2+19y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 32xy2+36y3 -5xy2-21y3 -32y3 -16y3 2y3 |
{2} | -27xy2+6y3 14y3 27y3 -17y3 49y3 |
{3} | -31xy-40y2 7xy-42y2 31y2 -34y2 35y2 |
{3} | 31x2-21xy-43y2 -7x2+12xy-18y2 -31xy-40y2 34xy+29y2 -35xy+26y2 |
{3} | 27x2+33xy+30y2 -9xy+35y2 -27xy-39y2 17xy+42y2 -49xy+29y2 |
{4} | 0 0 x+15y 33y -26y |
{4} | 0 0 -29y x-30y -15y |
{4} | 0 0 8y y x+15y |
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+38y -50y |
{2} | 0 2y x-20y |
{3} | 1 -50 32 |
{3} | 0 6 -35 |
{3} | 0 -15 -43 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | 23 39 0 24y 15x-y xy+31y2 48xy-50y2 28xy-25y2 |
{5} | 45 9 0 -29x-30y -49x-18y 44y2 xy-5y2 18xy-12y2 |
{5} | 0 0 0 0 0 x2-15xy-31y2 -33xy-16y2 26xy+38y2 |
{5} | 0 0 0 0 0 29xy+12y2 x2+30xy+29y2 15xy-31y2 |
{5} | 0 0 0 0 0 -8xy+9y2 -xy+47y2 x2-15xy+2y2 |
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
|