-- 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 | -36x2-50xy+35y2 -32xy+25y2 |
| -37x2+47xy-43y2 3x2+23xy+47y2 |
| -20x2-49xy-38y2 8x2+25xy-49y2 |
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 | -45x2-11xy-17y2 -21x2+38xy-41y2 x3 x2y+36xy2+14y3 50xy2-24y3 y4 0 0 |
| x2+17xy+30y2 47xy-18y2 0 -47xy2+41y3 -47xy2-29y3 0 y4 0 |
| -21xy+14y2 x2+11xy-24y2 0 -12y3 xy2+4y3 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
| -45x2-11xy-17y2 -21x2+38xy-41y2 x3 x2y+36xy2+14y3 50xy2-24y3 y4 0 0 |
| x2+17xy+30y2 47xy-18y2 0 -47xy2+41y3 -47xy2-29y3 0 y4 0 |
| -21xy+14y2 x2+11xy-24y2 0 -12y3 xy2+4y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 25xy2+5y3 27xy2-45y3 -25y3 -32y3 49y3 |
{2} | -16xy2+37y3 19y3 16y3 -17y3 17y3 |
{3} | -49xy-50y2 -50xy-46y2 49y2 -21y2 -3y2 |
{3} | 49x2-8xy+41y2 50x2-34xy-29y2 -49xy-43y2 21xy+31y2 3xy-29y2 |
{3} | 16x2-24xy-26y2 37xy+33y2 -16xy-13y2 17xy+2y2 -17xy+36y2 |
{4} | 0 0 x+16y -11y -28y |
{4} | 0 0 0 x+47y -42y |
{4} | 0 0 -33y 0 x+38y |
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-17y -47y |
{2} | 0 21y x-11y |
{3} | 1 45 21 |
{3} | 0 -10 40 |
{3} | 0 4 -34 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | 25 40 0 -8y 19x-48y xy+34y2 -20xy-22y2 20xy-18y2 |
{5} | 6 -40 0 -2x-42y 44x-21y 47y2 xy-46y2 47xy+19y2 |
{5} | 0 0 0 0 0 x2-16xy-32y2 11xy+14y2 28xy-40y2 |
{5} | 0 0 0 0 0 -28y2 x2-47xy-13y2 42xy-35y2 |
{5} | 0 0 0 0 0 33xy+36y2 -41y2 x2-38xy+45y2 |
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
|