-- 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 | -9x2+31xy+6y2 x2-6xy-12y2 |
| 28x2+33xy+34y2 17x2+36xy-16y2 |
| -19x2-35xy+9y2 13x2-45xy-27y2 |
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 | 15x2+31xy-43y2 -4x2-41xy+32y2 x3 x2y+38xy2+38y3 26xy2-35y3 y4 0 0 |
| x2+10xy-17y2 13xy+21y2 0 -4xy2-37y3 34xy2+11y3 0 y4 0 |
| 15xy+40y2 x2+8xy 0 -45y3 xy2-46y3 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
| 15x2+31xy-43y2 -4x2-41xy+32y2 x3 x2y+38xy2+38y3 26xy2-35y3 y4 0 0 |
| x2+10xy-17y2 13xy+21y2 0 -4xy2-37y3 34xy2+11y3 0 y4 0 |
| 15xy+40y2 x2+8xy 0 -45y3 xy2-46y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 46xy2-13y3 43xy2-27y3 -46y3 43y3 37y3 |
{2} | -16xy2-15y3 11y3 16y3 14y3 17y3 |
{3} | 4xy+2y2 -36xy-y2 -4y2 -18y2 9y2 |
{3} | -4x2-10xy+22y2 36x2+8xy-47y2 4xy+8y2 18xy -9xy-6y2 |
{3} | 16x2+9xy-46y2 -46xy-28y2 -16xy+6y2 -14xy+15y2 -17xy+40y2 |
{4} | 0 0 x+42y 7y 49y |
{4} | 0 0 21y x-31y 14y |
{4} | 0 0 -49y -20y x-11y |
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 -13y |
{2} | 0 -15y x-8y |
{3} | 1 -15 4 |
{3} | 0 -42 2 |
{3} | 0 28 -44 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------------- A : 1
{5} | -27 28 0 -15y 19x-46y xy+16y2 -37xy+y2 20xy+22y2 |
{5} | 15 -25 0 -14x-21y 47x+29y 4y2 xy+36y2 -34xy-28y2 |
{5} | 0 0 0 0 0 x2-42xy+15y2 -7xy+6y2 -49xy+y2 |
{5} | 0 0 0 0 0 -21xy+50y2 x2+31xy+20y2 -14xy+37y2 |
{5} | 0 0 0 0 0 49xy-20y2 20xy-8y2 x2+11xy-35y2 |
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
|