-- 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 | -17x2-50xy+21y2 8x2+15xy-20y2 |
| -37x2+26xy-32y2 34x2+39xy-8y2 |
| -7x2+15xy-y2 -37x2-20xy-21y2 |
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 | -20x2-4xy-44y2 36x2-15xy+42y2 x3 x2y-13xy2+20y3 -12xy2-47y3 y4 0 0 |
| x2-2xy+43y2 -22xy+37y2 0 41xy2+8y3 -32xy2-39y3 0 y4 0 |
| -35xy+y2 x2-48xy-34y2 0 31y3 xy2+41y3 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
| -20x2-4xy-44y2 36x2-15xy+42y2 x3 x2y-13xy2+20y3 -12xy2-47y3 y4 0 0 |
| x2-2xy+43y2 -22xy+37y2 0 41xy2+8y3 -32xy2-39y3 0 y4 0 |
| -35xy+y2 x2-48xy-34y2 0 31y3 xy2+41y3 0 0 y4 |
8 5
1 : A <--------------------------------------------------------------------------- A : 2
{2} | 28xy2-30y3 -7xy2+13y3 -28y3 36y3 8y3 |
{2} | -2xy2-18y3 -11y3 2y3 34y3 y3 |
{3} | 41xy 22xy+9y2 -41y2 -12y2 33y2 |
{3} | -41x2+22xy-40y2 -22x2-31xy-19y2 41xy-22y2 12xy-49y2 -33xy-14y2 |
{3} | 2x2-30xy-48y2 44xy -2xy+48y2 -34xy-25y2 -xy-23y2 |
{4} | 0 0 x-34y -39y 14y |
{4} | 0 0 47y x+45y 46y |
{4} | 0 0 22y 28y x-11y |
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+2y 22y |
{2} | 0 35y x+48y |
{3} | 1 20 -36 |
{3} | 0 -4 40 |
{3} | 0 32 -23 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -16 -39 0 y -50x+28y xy+23y2 17xy-40y2 -50xy-11y2 |
{5} | 48 24 0 -23x+3y -17x+9y -41y2 xy-y2 32xy+4y2 |
{5} | 0 0 0 0 0 x2+34xy+35y2 39xy-37y2 -14xy |
{5} | 0 0 0 0 0 -47xy+14y2 x2-45xy-35y2 -46xy |
{5} | 0 0 0 0 0 -22xy+23y2 -28xy-7y2 x2+11xy |
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
|