-- 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 | 48x2-47xy+6y2 -31x2+41xy+23y2 |
| 26x2+43xy+18y2 -17x2-4xy-48y2 |
| 19x2+17xy-39y2 -30x2+12xy-46y2 |
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 | -3x2-10xy-36y2 -4x2-26xy+23y2 x3 x2y-10xy2+5y3 28xy2-15y3 y4 0 0 |
| x2+21xy+33y2 32xy-7y2 0 35xy2-18y3 -36xy2-31y3 0 y4 0 |
| -28xy-50y2 x2+2xy-24y2 0 20y3 xy2-12y3 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
| -3x2-10xy-36y2 -4x2-26xy+23y2 x3 x2y-10xy2+5y3 28xy2-15y3 y4 0 0 |
| x2+21xy+33y2 32xy-7y2 0 35xy2-18y3 -36xy2-31y3 0 y4 0 |
| -28xy-50y2 x2+2xy-24y2 0 20y3 xy2-12y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | 7xy2+16y3 11xy2-18y3 -7y3 37y3 13y3 |
{2} | 48xy2-y3 -48y3 -48y3 -46y3 -9y3 |
{3} | -37xy-15y2 -17xy-21y2 37y2 23y2 -6y2 |
{3} | 37x2+23xy-37y2 17x2+22xy-45y2 -37xy-8y2 -23xy+25y2 6xy+13y2 |
{3} | -48x2-3xy+42y2 16xy-5y2 48xy+4y2 46xy+19y2 9xy-35y2 |
{4} | 0 0 x+43y 25y -16y |
{4} | 0 0 -24y x+y -30y |
{4} | 0 0 19y -31y x-44y |
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-21y -32y |
{2} | 0 28y x-2y |
{3} | 1 3 4 |
{3} | 0 -42 23 |
{3} | 0 12 41 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | -25 24 0 2y -40x-17y xy+48y2 22xy-25y2 -44xy+6y2 |
{5} | 45 -12 0 6x-24y -8x+48y -35y2 xy-27y2 36xy-42y2 |
{5} | 0 0 0 0 0 x2-43xy+36y2 -25xy-20y2 16xy-27y2 |
{5} | 0 0 0 0 0 24xy-10y2 x2-xy+28y2 30xy-43y2 |
{5} | 0 0 0 0 0 -19xy+18y2 31xy-10y2 x2+44xy+37y2 |
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
|