-- 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 | -30x2-16xy-32y2 2x2+31xy-41y2 |
| 6x2-29xy-3y2 -50x2+47xy+39y2 |
| -33x2+28xy+11y2 17x2+18xy-38y2 |
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 | 41x2-35xy+33y2 -10x2-12xy-42y2 x3 x2y+23xy2+40y3 -15xy2+24y3 y4 0 0 |
| x2-29xy+36y2 -35xy+25y2 0 -36xy2-22y3 -9xy2+44y3 0 y4 0 |
| -11xy-35y2 x2+40xy-22y2 0 25y3 xy2+26y3 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
| 41x2-35xy+33y2 -10x2-12xy-42y2 x3 x2y+23xy2+40y3 -15xy2+24y3 y4 0 0 |
| x2-29xy+36y2 -35xy+25y2 0 -36xy2-22y3 -9xy2+44y3 0 y4 0 |
| -11xy-35y2 x2+40xy-22y2 0 25y3 xy2+26y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 22xy2+36y3 -39xy2+10y3 -22y3 13y3 -11y3 |
{2} | -32xy2+11y3 -49y3 32y3 47y3 -6y3 |
{3} | 13xy-29y2 -41xy+40y2 -13y2 44y2 -y2 |
{3} | -13x2-10xy+23y2 41x2+10xy+40y2 13xy+39y2 -44xy+42y2 xy-47y2 |
{3} | 32x2-6xy+24y2 9xy+19y2 -32xy-5y2 -47xy-21y2 6xy+38y2 |
{4} | 0 0 x+24y -35y -32y |
{4} | 0 0 -41y x+3y -39y |
{4} | 0 0 -2y -25y x-27y |
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+29y 35y |
{2} | 0 11y x-40y |
{3} | 1 -41 10 |
{3} | 0 -34 -5 |
{3} | 0 15 -13 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | 9 -28 0 9y -41x+37y xy-8y2 -8xy+9y2 44xy+13y2 |
{5} | 16 -26 0 -32x-46y -13x+32y 36y2 xy-35y2 9xy+35y2 |
{5} | 0 0 0 0 0 x2-24xy-46y2 35xy-44y2 32xy+47y2 |
{5} | 0 0 0 0 0 41xy-19y2 x2-3xy-5y2 39xy+26y2 |
{5} | 0 0 0 0 0 2xy+21y2 25xy-37y2 x2+27xy-50y2 |
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
|