-- 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 | 4x2+xy-37y2 -29x2+38xy-32y2 |
| -12x2+45xy+11y2 -31x2+5xy-2y2 |
| -10x2-15xy-33y2 5x2+3xy+19y2 |
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 | -2x2+44xy 2x2+38xy+34y2 x3 x2y+28xy2-17y3 -18xy2+4y3 y4 0 0 |
| x2+17xy+4y2 -35xy-16y2 0 -38xy2+31y3 -35xy2+26y3 0 y4 0 |
| -34xy-26y2 x2+12xy-16y2 0 10y3 xy2+7y3 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
| -2x2+44xy 2x2+38xy+34y2 x3 x2y+28xy2-17y3 -18xy2+4y3 y4 0 0 |
| x2+17xy+4y2 -35xy-16y2 0 -38xy2+31y3 -35xy2+26y3 0 y4 0 |
| -34xy-26y2 x2+12xy-16y2 0 10y3 xy2+7y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | 22xy2-40y3 45xy2+34y3 -22y3 -30y3 -32y3 |
{2} | 27xy2+49y3 -31y3 -27y3 23y3 16y3 |
{3} | 41xy+33y2 44xy+17y2 -41y2 -39y2 18y2 |
{3} | -41x2+13xy-36y2 -44x2-8xy+11y2 41xy-46y2 39xy+4y2 -18xy+19y2 |
{3} | -27x2-36xy-14y2 -19xy-18y2 27xy-13y2 -23xy-10y2 -16xy+21y2 |
{4} | 0 0 x-14y 33y -2y |
{4} | 0 0 6y x+18y -44y |
{4} | 0 0 -49y 22y x-4y |
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-17y 35y |
{2} | 0 34y x-12y |
{3} | 1 2 -2 |
{3} | 0 -45 -45 |
{3} | 0 50 37 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 43 50 0 -5y -15x-36y xy-8y2 -42xy-32y2 -38xy-26y2 |
{5} | 14 -4 0 39x+32y -48x+8y 38y2 xy+37y2 35xy+44y2 |
{5} | 0 0 0 0 0 x2+14xy-13y2 -33xy-13y2 2xy-2y2 |
{5} | 0 0 0 0 0 -6xy-42y2 x2-18xy-42y2 44xy-22y2 |
{5} | 0 0 0 0 0 49xy+4y2 -22xy+4y2 x2+4xy-46y2 |
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
|