-- 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 | -6x2-32xy+22y2 -41x2+2xy+42y2 |
| 20x2-30xy+18y2 -28x2+32xy-17y2 |
| 26x2+11xy-29y2 50x2-4xy+34y2 |
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 | -13x2-40xy-36y2 2x2+14xy+13y2 x3 x2y-26xy2-36y3 -5xy2-45y3 y4 0 0 |
| x2-10xy+21y2 -9xy-31y2 0 -42xy2-13y3 49xy2-22y3 0 y4 0 |
| -14xy-y2 x2-16xy-12y2 0 14y3 xy2+49y3 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
| -13x2-40xy-36y2 2x2+14xy+13y2 x3 x2y-26xy2-36y3 -5xy2-45y3 y4 0 0 |
| x2-10xy+21y2 -9xy-31y2 0 -42xy2-13y3 49xy2-22y3 0 y4 0 |
| -14xy-y2 x2-16xy-12y2 0 14y3 xy2+49y3 0 0 y4 |
8 5
1 : A <--------------------------------------------------------------------------- A : 2
{2} | 37xy2-7y3 38xy2-45y3 -37y3 -11y3 -50y3 |
{2} | 46xy2+39y3 -20y3 -46y3 9y3 36y3 |
{3} | -41xy+12y2 -49xy+21y2 41y2 30y2 48y2 |
{3} | 41x2+xy+30y2 49x2+30xy+21y2 -41xy-13y2 -30xy+24y2 -48xy-29y2 |
{3} | -46x2-34xy+40y2 -33xy-18y2 46xy-5y2 -9xy+43y2 -36xy-12y2 |
{4} | 0 0 x-13y -37y -14y |
{4} | 0 0 -19y x-50y 10y |
{4} | 0 0 40y -23y 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+10y 9y |
{2} | 0 14y x+16y |
{3} | 1 13 -2 |
{3} | 0 41 -30 |
{3} | 0 -39 -10 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -25 -9 0 40y -11x+35y xy+9y2 2xy-14y2 8xy+13y2 |
{5} | -50 37 0 33x-31y 36x-14y 42y2 xy-29y2 -49xy+50y2 |
{5} | 0 0 0 0 0 x2+13xy+9y2 37xy+27y2 14xy+41y2 |
{5} | 0 0 0 0 0 19xy-19y2 x2+50xy+44y2 -10xy-8y2 |
{5} | 0 0 0 0 0 -40xy+13y2 23xy+39y2 x2+38xy+48y2 |
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
|