-- 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 | 14x2-23xy-32y2 32x2+38xy+50y2 |
| 42x2+33xy+48y2 28x2+23xy+16y2 |
| 18x2+19xy-33y2 20x2+45xy-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 | 40x2-xy-13y2 -14x2-27xy-50y2 x3 x2y+18xy2+27y3 8xy2-50y3 y4 0 0 |
| x2+35xy+2y2 38xy-2y2 0 -27xy2+3y3 -39xy2-10y3 0 y4 0 |
| 39xy-30y2 x2-17xy-16y2 0 13y3 xy2+50y3 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
| 40x2-xy-13y2 -14x2-27xy-50y2 x3 x2y+18xy2+27y3 8xy2-50y3 y4 0 0 |
| x2+35xy+2y2 38xy-2y2 0 -27xy2+3y3 -39xy2-10y3 0 y4 0 |
| 39xy-30y2 x2-17xy-16y2 0 13y3 xy2+50y3 0 0 y4 |
8 5
1 : A <--------------------------------------------------------------------------- A : 2
{2} | -41xy2-15y3 -50xy2+38y3 41y3 3y3 -9y3 |
{2} | 36xy2-6y3 5y3 -36y3 -8y3 -24y3 |
{3} | -43xy-2y2 43xy+25y2 43y2 -11y2 35y2 |
{3} | 43x2+44xy+27y2 -43x2+22xy+39y2 -43xy-42y2 11xy+11y2 -35xy-43y2 |
{3} | -36x2+24xy-24y2 -21xy-12y2 36xy-18y2 8xy+12y2 24xy+5y2 |
{4} | 0 0 x-23y 43y -7y |
{4} | 0 0 -6y x-36y 48y |
{4} | 0 0 -21y 27y 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-35y -38y |
{2} | 0 -39y x+17y |
{3} | 1 -40 14 |
{3} | 0 47 -33 |
{3} | 0 25 -31 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -47 30 0 19y 14x+18y xy-12y2 -11xy+32y2 -33xy+19y2 |
{5} | 41 -6 0 -47x+2y 14x+16y 27y2 xy-10y2 39xy-7y2 |
{5} | 0 0 0 0 0 x2+23xy+14y2 -43xy+y2 7xy-6y2 |
{5} | 0 0 0 0 0 6xy-48y2 x2+36xy+11y2 -48xy+35y2 |
{5} | 0 0 0 0 0 21xy-9y2 -27xy+21y2 x2+42xy-25y2 |
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
|