-- 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 | 28x2-43xy-35y2 44x2+39xy-24y2 |
| 26x2-40xy-5y2 -4x2-26xy+46y2 |
| -11x2-28xy-40y2 -11x2+14xy-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 | -14x2+4xy+3y2 47x2-5xy+47y2 x3 x2y+4xy2-15y3 -3xy2-38y3 y4 0 0 |
| x2+13xy-32y2 16xy+35y2 0 -3xy2-35y3 -10xy2-38y3 0 y4 0 |
| 39xy-48y2 x2-43xy-18y2 0 -43y3 xy2-19y3 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
| -14x2+4xy+3y2 47x2-5xy+47y2 x3 x2y+4xy2-15y3 -3xy2-38y3 y4 0 0 |
| x2+13xy-32y2 16xy+35y2 0 -3xy2-35y3 -10xy2-38y3 0 y4 0 |
| 39xy-48y2 x2-43xy-18y2 0 -43y3 xy2-19y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | -6xy2-41y3 38xy2-48y3 6y3 46y3 -27y3 |
{2} | 18xy2+y3 -47y3 -18y3 -50y3 -49y3 |
{3} | 43xy+44y2 21xy+4y2 -43y2 -17y2 4y2 |
{3} | -43x2-48xy-45y2 -21x2+6xy-9y2 43xy+4y2 17xy+46y2 -4xy-33y2 |
{3} | -18x2+28xy+45y2 -15xy+32y2 18xy-29y2 50xy-41y2 49xy+7y2 |
{4} | 0 0 x-12y 31y 34y |
{4} | 0 0 -39y x+42y -38y |
{4} | 0 0 -11y -18y x-30y |
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-13y -16y |
{2} | 0 -39y x+43y |
{3} | 1 14 -47 |
{3} | 0 31 -18 |
{3} | 0 -11 -34 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | 5 43 0 -20y 28x+24y xy+38y2 14xy+21y2 42xy+3y2 |
{5} | -28 -36 0 24x+43y 10x+34y 3y2 xy+43y2 10xy-30y2 |
{5} | 0 0 0 0 0 x2+12xy-25y2 -31xy+15y2 -34xy+20y2 |
{5} | 0 0 0 0 0 39xy-45y2 x2-42xy+27y2 38xy+36y2 |
{5} | 0 0 0 0 0 11xy-48y2 18xy+49y2 x2+30xy-2y2 |
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
|