-- 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 | -50x2-35xy+23y2 11x2-41xy-22y2 |
| -26x2-xy+35y2 -48x2+29xy+27y2 |
| -40x2+29xy-30y2 -28x2+32xy+2y2 |
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 | -9x2-33xy+45y2 -3x2+40xy+3y2 x3 x2y+25xy2-2y3 46xy2+13y3 y4 0 0 |
| x2+19xy+5y2 -30xy-42y2 0 -48xy2-49y3 -17xy2+6y3 0 y4 0 |
| -17xy+46y2 x2+28xy-14y2 0 25y3 xy2+34y3 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
| -9x2-33xy+45y2 -3x2+40xy+3y2 x3 x2y+25xy2-2y3 46xy2+13y3 y4 0 0 |
| x2+19xy+5y2 -30xy-42y2 0 -48xy2-49y3 -17xy2+6y3 0 y4 0 |
| -17xy+46y2 x2+28xy-14y2 0 25y3 xy2+34y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -4xy2+41y3 -24xy2-34y3 4y3 -17y3 32y3 |
{2} | 21xy2+21y3 -13y3 -21y3 -12y3 -46y3 |
{3} | 20xy+38y2 -50xy-41y2 -20y2 -48y2 3y2 |
{3} | -20x2+41xy+24y2 50x2-11xy-34y2 20xy+22y2 48xy-22y2 -3xy+28y2 |
{3} | -21x2+32xy-11y2 -29xy+30y2 21xy+48y2 12xy-46y2 46xy+39y2 |
{4} | 0 0 x+10y 42y -36y |
{4} | 0 0 -11y x-9y -45y |
{4} | 0 0 -34y y x-y |
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-19y 30y |
{2} | 0 17y x-28y |
{3} | 1 9 3 |
{3} | 0 14 45 |
{3} | 0 -37 -5 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------ A : 1
{5} | -21 19 0 22y 24x+6y xy-12y2 15xy-39y2 7xy-24y2 |
{5} | -41 37 0 -2x-28y 50x+22y 48y2 xy-5y2 17xy+y2 |
{5} | 0 0 0 0 0 x2-10xy-47y2 -42xy+6y2 36xy+8y2 |
{5} | 0 0 0 0 0 11xy+4y2 x2+9xy-22y2 45xy+38y2 |
{5} | 0 0 0 0 0 34xy-14y2 -xy-24y2 x2+xy-32y2 |
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
|