-- 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 | 15x2-25xy-45y2 -17x2+26xy+4y2 |
| -30x2+9xy+49y2 -2x2-8xy-37y2 |
| -22x2-15xy-37y2 -11x2+12xy+14y2 |
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 | 2x2-32xy+37y2 -8x2+31xy+48y2 x3 x2y-45xy2-6y3 -45xy2+6y3 y4 0 0 |
| x2+34xy+38y2 -33xy+24y2 0 -8xy2+21y3 26xy2+23y3 0 y4 0 |
| -49xy-48y2 x2+17xy+35y2 0 -29y3 xy2+7y3 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
| 2x2-32xy+37y2 -8x2+31xy+48y2 x3 x2y-45xy2-6y3 -45xy2+6y3 y4 0 0 |
| x2+34xy+38y2 -33xy+24y2 0 -8xy2+21y3 26xy2+23y3 0 y4 0 |
| -49xy-48y2 x2+17xy+35y2 0 -29y3 xy2+7y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | 9xy2+34y3 -4xy2+2y3 -9y3 -6y3 -47y3 |
{2} | 14xy2+36y3 -11y3 -14y3 -38y3 -46y3 |
{3} | -44xy+37y2 -50xy+27y2 44y2 -47y2 -30y2 |
{3} | 44x2-7xy-36y2 50x2+9xy+42y2 -44xy-30y2 47xy-2y2 30xy+15y2 |
{3} | -14x2+26xy+5y2 -48xy-17y2 14xy+39y2 38xy+35y2 46xy+36y2 |
{4} | 0 0 x-15y 6y -17y |
{4} | 0 0 7y x-27y 30y |
{4} | 0 0 26y 32y x+42y |
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-34y 33y |
{2} | 0 49y x-17y |
{3} | 1 -2 8 |
{3} | 0 -13 -31 |
{3} | 0 -27 -48 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -38 11 0 -22y 36x+41y xy+9y2 46xy+25y2 -40xy+17y2 |
{5} | -37 -3 0 -2x+48y 37x+33y 8y2 xy-11y2 -26xy+34y2 |
{5} | 0 0 0 0 0 x2+15xy+27y2 -6xy+12y2 17xy+24y2 |
{5} | 0 0 0 0 0 -7xy-19y2 x2+27xy+14y2 -30xy+28y2 |
{5} | 0 0 0 0 0 -26xy+17y2 -32xy+30y2 x2-42xy-41y2 |
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
|