-- 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 | 27x2+48xy-20y2 23x2-22xy-38y2 |
| -38x2+32xy+49y2 -24x2-46xy-21y2 |
| 29x2+37xy-37y2 -24x2+33xy-28y2 |
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 | -25x2+39xy-21y2 3x2+4xy+31y2 x3 x2y+28xy2-12y3 3xy2+25y3 y4 0 0 |
| x2+5xy+33y2 39xy+31y2 0 15xy2+14y3 -43xy2-27y3 0 y4 0 |
| 16xy+40y2 x2-30xy-22y2 0 -y3 xy2-15y3 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
| -25x2+39xy-21y2 3x2+4xy+31y2 x3 x2y+28xy2-12y3 3xy2+25y3 y4 0 0 |
| x2+5xy+33y2 39xy+31y2 0 15xy2+14y3 -43xy2-27y3 0 y4 0 |
| 16xy+40y2 x2-30xy-22y2 0 -y3 xy2-15y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -48xy2-15y3 -48xy2-41y3 48y3 -33y3 -48y3 |
{2} | 11xy2-44y3 9y3 -11y3 -13y3 -21y3 |
{3} | -39xy+21y2 17xy-24y2 39y2 -26y2 -23y2 |
{3} | 39x2+10xy+10y2 -17x2+7xy+12y2 -39xy-31y2 26xy-38y2 23xy-26y2 |
{3} | -11x2+6xy-30y2 35xy+45y2 11xy+38y2 13xy-45y2 21xy-30y2 |
{4} | 0 0 x+27y -25y -20y |
{4} | 0 0 15y x+y -29y |
{4} | 0 0 -7y 18y x-28y |
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-5y -39y |
{2} | 0 -16y x+30y |
{3} | 1 25 -3 |
{3} | 0 -15 42 |
{3} | 0 -36 31 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------ A : 1
{5} | -40 12 0 36y -46x-26y xy-9y2 -8xy-25y2 -44xy+25y2 |
{5} | 44 30 0 -6x-9y 43x+32y -15y2 xy+43y2 43xy-17y2 |
{5} | 0 0 0 0 0 x2-27xy-11y2 25xy-50y2 20xy+38y2 |
{5} | 0 0 0 0 0 -15xy+17y2 x2-xy+13y2 29xy-22y2 |
{5} | 0 0 0 0 0 7xy-26y2 -18xy-8y2 x2+28xy-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
|