-- 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 | 48x2-10xy-15y2 -8x2-37xy+32y2 |
| 31x2-xy+11y2 -43x2+30xy-24y2 |
| -11x2-43xy+2y2 41x2-46xy+10y2 |
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 | 14x2+11xy-4y2 -20x2-14xy+36y2 x3 x2y-41xy2-37y3 -5xy2-y3 y4 0 0 |
| x2-39xy-49y2 -18xy+17y2 0 -14xy2+34y3 -2xy2+30y3 0 y4 0 |
| 35xy+y2 x2-26xy+21y2 0 -2y3 xy2 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
| 14x2+11xy-4y2 -20x2-14xy+36y2 x3 x2y-41xy2-37y3 -5xy2-y3 y4 0 0 |
| x2-39xy-49y2 -18xy+17y2 0 -14xy2+34y3 -2xy2+30y3 0 y4 0 |
| 35xy+y2 x2-26xy+21y2 0 -2y3 xy2 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 16xy2+40y3 27xy2+29y3 -16y3 21y3 -38y3 |
{2} | -36xy2+42y3 -47y3 36y3 -7y3 -36y3 |
{3} | 4xy+20y2 -38xy+19y2 -4y2 50y2 -21y2 |
{3} | -4x2-39xy-44y2 38x2+50xy+26y2 4xy+19y2 -50xy-25y2 21xy+45y2 |
{3} | 36x2-31xy-50y2 -14xy-43y2 -36xy-11y2 7xy-7y2 36xy+31y2 |
{4} | 0 0 x+39y 10y 12y |
{4} | 0 0 5y x-14y 27y |
{4} | 0 0 5y -25y x-25y |
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+39y 18y |
{2} | 0 -35y x+26y |
{3} | 1 -14 20 |
{3} | 0 -45 -21 |
{3} | 0 47 25 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | -49 47 0 48y -14x-46y xy-y2 -3xy+22y2 -xy+15y2 |
{5} | 4 -45 0 8x-48y -44x+44y 14y2 xy-25y2 2xy+47y2 |
{5} | 0 0 0 0 0 x2-39xy+15y2 -10xy-50y2 -12xy+34y2 |
{5} | 0 0 0 0 0 -5xy-43y2 x2+14xy-25y2 -27xy+17y2 |
{5} | 0 0 0 0 0 -5xy+46y2 25xy+15y2 x2+25xy+10y2 |
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
|