-- 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 | 2x2-3xy-24y2 -19x2-23xy+40y2 |
| 43x2-9xy+5y2 -5x2+38xy-45y2 |
| 21x2+17xy-37y2 42x2-4xy-22y2 |
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 | 28x2-32xy+29y2 -38x2-26xy+2y2 x3 x2y+37xy2+34y3 -40xy2-y3 y4 0 0 |
| x2+46xy+45y2 16xy-39y2 0 -5xy2+46y3 -6xy2+24y3 0 y4 0 |
| -24xy-15y2 x2-27xy-48y2 0 -42y3 xy2-2y3 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
| 28x2-32xy+29y2 -38x2-26xy+2y2 x3 x2y+37xy2+34y3 -40xy2-y3 y4 0 0 |
| x2+46xy+45y2 16xy-39y2 0 -5xy2+46y3 -6xy2+24y3 0 y4 0 |
| -24xy-15y2 x2-27xy-48y2 0 -42y3 xy2-2y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -39xy2+22y3 -47xy2-44y3 39y3 -28y3 -22y3 |
{2} | 7xy2+31y3 30y3 -7y3 29y3 -5y3 |
{3} | -41xy+19y2 -31xy+13y2 41y2 -9y2 -30y2 |
{3} | 41x2+47xy+24y2 31x2-46xy+4y2 -41xy+35y2 9xy-11y2 30xy+21y2 |
{3} | -7x2+21xy+y2 43xy+31y2 7xy+49y2 -29xy+27y2 5xy |
{4} | 0 0 x-36y 44y 35y |
{4} | 0 0 34y x+27y 31y |
{4} | 0 0 -y -42y x+9y |
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-46y -16y |
{2} | 0 24y x+27y |
{3} | 1 -28 38 |
{3} | 0 10 -15 |
{3} | 0 -37 -11 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------------- A : 1
{5} | 38 -16 0 50y -29x-25y xy -33xy-20y2 44xy+25y2 |
{5} | -18 -37 0 -13x-17y -4x+32y 5y2 xy-42y2 6xy+33y2 |
{5} | 0 0 0 0 0 x2+36xy+30y2 -44xy-48y2 -35xy+15y2 |
{5} | 0 0 0 0 0 -34xy-34y2 x2-27xy+14y2 -31xy-17y2 |
{5} | 0 0 0 0 0 xy+13y2 42xy-41y2 x2-9xy-44y2 |
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
|