-- 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 | -36x2+45xy+23y2 39x2-47xy-39y2 |
| 19x2+21xy-21y2 -25x2+12xy-44y2 |
| 4x2+32xy+11y2 3x2-3xy+47y2 |
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 | -48x2+22xy-5y2 17x2+33xy-21y2 x3 x2y-48xy2-15y3 -44xy2+34y3 y4 0 0 |
| x2-34xy-7y2 -10xy+28y2 0 26xy2-12y3 14y3 0 y4 0 |
| 38xy-19y2 x2-21xy-8y2 0 -38y3 xy2-36y3 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
| -48x2+22xy-5y2 17x2+33xy-21y2 x3 x2y-48xy2-15y3 -44xy2+34y3 y4 0 0 |
| x2-34xy-7y2 -10xy+28y2 0 26xy2-12y3 14y3 0 y4 0 |
| 38xy-19y2 x2-21xy-8y2 0 -38y3 xy2-36y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | -9xy2+40y3 37xy2-14y3 9y3 23y3 -8y3 |
{2} | -45xy2+y3 7y3 45y3 -3y3 -40y3 |
{3} | -12xy-14y2 -18xy+50y2 12y2 -3y2 23y2 |
{3} | 12x2-26xy+5y2 18x2-36xy+9y2 -12xy+40y2 3xy+17y2 -23xy+43y2 |
{3} | 45x2-43xy-21y2 -22xy+30y2 -45xy+42y2 3xy-8y2 40xy+29y2 |
{4} | 0 0 x-40y -27y -9y |
{4} | 0 0 8y x-45y -38y |
{4} | 0 0 28y -37y x-16y |
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+34y 10y |
{2} | 0 -38y x+21y |
{3} | 1 48 -17 |
{3} | 0 34 -11 |
{3} | 0 50 -32 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -46 -15 0 11y 9x-28y xy+43y2 -27xy-43y2 44xy-9y2 |
{5} | 40 5 0 -28x+4y -6x-14y -26y2 xy-2y2 48y2 |
{5} | 0 0 0 0 0 x2+40xy+21y2 27xy+2y2 9xy+15y2 |
{5} | 0 0 0 0 0 -8xy-27y2 x2+45xy-17y2 38xy+24y2 |
{5} | 0 0 0 0 0 -28xy-46y2 37xy-14y2 x2+16xy-4y2 |
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
|