-- 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 | 43x2+10xy-19y2 -29x2-37xy+34y2 |
| -39x2+48xy-37y2 -28x2-21xy-31y2 |
| -47x2+21xy+42y2 30x2+34xy+50y2 |
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 | -20x2-11xy+16y2 -23x2+39xy+43y2 x3 x2y+26xy2-11y3 -4xy2+35y3 y4 0 0 |
| x2-35xy+30y2 -30xy-37y2 0 11xy2-7y3 37xy2-50y3 0 y4 0 |
| 39xy+33y2 x2-50xy+19y2 0 -10y3 xy2-10y3 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
| -20x2-11xy+16y2 -23x2+39xy+43y2 x3 x2y+26xy2-11y3 -4xy2+35y3 y4 0 0 |
| x2-35xy+30y2 -30xy-37y2 0 11xy2-7y3 37xy2-50y3 0 y4 0 |
| 39xy+33y2 x2-50xy+19y2 0 -10y3 xy2-10y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | -6xy2-38y3 40xy2-41y3 6y3 27y3 35y3 |
{2} | 10xy2-30y3 -50y3 -10y3 23y3 18y3 |
{3} | -25xy+24y2 22xy+36y2 25y2 -29y2 -39y2 |
{3} | 25x2+2xy -22x2+23xy+2y2 -25xy-26y2 29xy+21y2 39xy+28y2 |
{3} | -10x2+5xy-41y2 -13xy 10xy+25y2 -23xy-45y2 -18xy+47y2 |
{4} | 0 0 x-19y -19y -45y |
{4} | 0 0 13y x-42y 41y |
{4} | 0 0 -18y 48y x-40y |
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+35y 30y |
{2} | 0 -39y x+50y |
{3} | 1 20 23 |
{3} | 0 16 -6 |
{3} | 0 -15 -2 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 49 20 0 40y 10x+4y xy-31y2 28xy+17y2 -22xy-2y2 |
{5} | -40 50 0 -23x+26y -7x-34y -11y2 xy+46y2 -37xy+48y2 |
{5} | 0 0 0 0 0 x2+19xy+15y2 19xy+9y2 45xy-43y2 |
{5} | 0 0 0 0 0 -13xy-16y2 x2+42xy-50y2 -41xy-8y2 |
{5} | 0 0 0 0 0 18xy-31y2 -48xy+42y2 x2+40xy+35y2 |
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
|