-- 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 | -37x2-18xy-36y2 34x2+39xy+32y2 |
| -32x2-48xy+47y2 -8x2+39xy-48y2 |
| 43x2+13xy-13y2 -17x2+43xy-40y2 |
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 | 6x2+22xy+46y2 13x2+23xy+24y2 x3 x2y-26xy2-46y3 -27xy2-35y3 y4 0 0 |
| x2+41xy-41y2 20xy+34y2 0 25xy2-31y3 -30xy2-27y3 0 y4 0 |
| 12xy+23y2 x2-26xy+45y2 0 -28y3 xy2-27y3 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
| 6x2+22xy+46y2 13x2+23xy+24y2 x3 x2y-26xy2-46y3 -27xy2-35y3 y4 0 0 |
| x2+41xy-41y2 20xy+34y2 0 25xy2-31y3 -30xy2-27y3 0 y4 0 |
| 12xy+23y2 x2-26xy+45y2 0 -28y3 xy2-27y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 27xy2+37y3 -15xy2+6y3 -27y3 -29y3 17y3 |
{2} | -48xy2+20y3 27y3 48y3 -34y3 -27y3 |
{3} | -4xy-23y2 -41xy-36y2 4y2 -46y2 41y2 |
{3} | 4x2-34xy-21y2 41x2-20xy-25y2 -4xy-44y2 46xy+3y2 -41xy+13y2 |
{3} | 48x2+18xy+38y2 -12xy-10y2 -48xy-38y2 34xy-47y2 27xy-13y2 |
{4} | 0 0 x-32y 37y 9y |
{4} | 0 0 22y x+3y -50y |
{4} | 0 0 41y 2y x+29y |
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-41y -20y |
{2} | 0 -12y x+26y |
{3} | 1 -6 -13 |
{3} | 0 -24 -39 |
{3} | 0 -45 -38 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | -18 5 0 -8y 40x-48y xy-11y2 -47xy-17y2 31xy+48y2 |
{5} | -42 -2 0 -32x-37y -31x-29y -25y2 xy-49y2 30xy-26y2 |
{5} | 0 0 0 0 0 x2+32xy-15y2 -37xy-45y2 -9xy+42y2 |
{5} | 0 0 0 0 0 -22xy+39y2 x2-3xy+16y2 50xy+12y2 |
{5} | 0 0 0 0 0 -41xy+22y2 -2xy-35y2 x2-29xy-y2 |
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
|