-- 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 | -37x2+2xy+41y2 -34x2-5xy+11y2 |
| 49x2+24xy-7y2 -29x2-40xy-49y2 |
| 26x2-46xy-18y2 4x2+40xy+18y2 |
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 | -19x2+28xy-35y2 -20x2+25xy-49y2 x3 x2y+14xy2+48y3 2xy2+40y3 y4 0 0 |
| x2+8xy+23y2 48xy+3y2 0 -25xy2+13y3 6xy2+49y3 0 y4 0 |
| -20xy+9y2 x2-34xy-6y2 0 27y3 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
| -19x2+28xy-35y2 -20x2+25xy-49y2 x3 x2y+14xy2+48y3 2xy2+40y3 y4 0 0 |
| x2+8xy+23y2 48xy+3y2 0 -25xy2+13y3 6xy2+49y3 0 y4 0 |
| -20xy+9y2 x2-34xy-6y2 0 27y3 xy2-49y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | -32xy2+32y3 -48xy2+48y3 32y3 -8y3 -16y3 |
{2} | 8xy2-46y3 -17y3 -8y3 14y3 33y3 |
{3} | -17xy+20y2 10xy-2y2 17y2 36y2 -48y2 |
{3} | 17x2+17xy-24y2 -10x2+38xy-22y2 -17xy-37y2 -36xy-47y2 48xy-48y2 |
{3} | -8x2+39xy-45y2 34xy+40y2 8xy+7y2 -14xy-4y2 -33xy+9y2 |
{4} | 0 0 x+2y -6y -29y |
{4} | 0 0 32y x+40y 48y |
{4} | 0 0 -4y 17y 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-8y -48y |
{2} | 0 20y x+34y |
{3} | 1 19 20 |
{3} | 0 18 46 |
{3} | 0 6 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------- A : 1
{5} | 26 0 0 -8y -38x+25y xy+18y2 -27xy-7y2 -42xy+y2 |
{5} | 46 45 0 10x-16y -4x-7y 25y2 xy+44y2 -6xy+46y2 |
{5} | 0 0 0 0 0 x2-2xy+29y2 6xy-38y2 29xy-37y2 |
{5} | 0 0 0 0 0 -32xy+41y2 x2-40xy+2y2 -48xy-14y2 |
{5} | 0 0 0 0 0 4xy-3y2 -17xy-10y2 x2+42xy-31y2 |
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
|