-- 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 | 2x2-11xy-49y2 -41x2+26xy-45y2 |
| 13x2+27xy-10y2 -28x2+38xy-25y2 |
| 2x2-31xy+45y2 8x2-16xy-21y2 |
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 | -36x2+37xy+12y2 33x2-44xy+49y2 x3 x2y+10xy2-45y3 25xy2-12y3 y4 0 0 |
| x2-37xy+44y2 -49xy+12y2 0 -21xy2+33y3 xy2+25y3 0 y4 0 |
| 34xy-24y2 x2+16xy+27y2 0 47y3 xy2-17y3 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
| -36x2+37xy+12y2 33x2-44xy+49y2 x3 x2y+10xy2-45y3 25xy2-12y3 y4 0 0 |
| x2-37xy+44y2 -49xy+12y2 0 -21xy2+33y3 xy2+25y3 0 y4 0 |
| 34xy-24y2 x2+16xy+27y2 0 47y3 xy2-17y3 0 0 y4 |
8 5
1 : A <---------------------------------------------------------------------- A : 2
{2} | -10xy2-2y3 -21xy2-9y3 10y3 -37y3 -11y3 |
{2} | 47xy2-21y3 -36y3 -47y3 10y3 -47y3 |
{3} | 46xy-49y2 xy-35y2 -46y2 -17y2 -45y2 |
{3} | -46x2-25xy+9y2 -x2-4xy-8y2 46xy-27y2 17xy+34y2 45xy+35y2 |
{3} | -47x2-38xy+30y2 -11xy-25y2 47xy-42y2 -10xy+28y2 47xy+11y2 |
{4} | 0 0 x-41y 2y y |
{4} | 0 0 45y x-11y 22y |
{4} | 0 0 44y 43y x-49y |
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+37y 49y |
{2} | 0 -34y x-16y |
{3} | 1 36 -33 |
{3} | 0 -7 13 |
{3} | 0 17 -23 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 24 21 0 -32y -43x+36y xy+50y2 -26xy-41y2 xy+33y2 |
{5} | 0 -44 0 -x-39y -42x-9y 21y2 xy+19y2 -xy-22y2 |
{5} | 0 0 0 0 0 x2+41xy-3y2 -2xy+40y2 -xy-46y2 |
{5} | 0 0 0 0 0 -45xy+42y2 x2+11xy+46y2 -22xy+38y2 |
{5} | 0 0 0 0 0 -44xy-5y2 -43xy+33y2 x2+49xy-43y2 |
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
|