-- 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 | -42x2-43xy-39y2 13x2-12xy+25y2 |
| 50x2-22xy-9y2 11x2+50xy-13y2 |
| -49x2-28xy-5y2 -41x2+49xy+17y2 |
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+16xy-y2 -25x2+44xy+41y2 x3 x2y-43xy2+38y3 -38xy2+13y3 y4 0 0 |
| x2+30xy+34y2 29xy+39y2 0 48xy2-48y3 -18xy2+49y3 0 y4 0 |
| -47xy+7y2 x2+33xy-y2 0 19y3 xy2-20y3 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+16xy-y2 -25x2+44xy+41y2 x3 x2y-43xy2+38y3 -38xy2+13y3 y4 0 0 |
| x2+30xy+34y2 29xy+39y2 0 48xy2-48y3 -18xy2+49y3 0 y4 0 |
| -47xy+7y2 x2+33xy-y2 0 19y3 xy2-20y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | -42xy2+6y3 -23xy2-2y3 42y3 9y3 41y3 |
{2} | 21xy2-31y3 -y3 -21y3 7y3 35y3 |
{3} | 7xy+39y2 -11xy+48y2 -7y2 47y2 -26y2 |
{3} | -7x2-33xy-16y2 11x2+26xy-47y2 7xy-6y2 -47xy+29y2 26xy-9y2 |
{3} | -21x2+6xy+44y2 24xy-15y2 21xy+25y2 -7xy+36y2 -35xy-19y2 |
{4} | 0 0 x-2y -30y 3y |
{4} | 0 0 -y x-42y -38y |
{4} | 0 0 -4y 12y x+44y |
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-30y -29y |
{2} | 0 47y x-33y |
{3} | 1 -9 25 |
{3} | 0 15 -2 |
{3} | 0 -39 30 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | 27 -13 0 -34y 24x+23y xy-42y2 -34xy-43y2 32xy+48y2 |
{5} | 25 47 0 46x-11y -49x+28y -48y2 xy+36y2 18xy-5y2 |
{5} | 0 0 0 0 0 x2+2xy+22y2 30xy+43y2 -3xy-47y2 |
{5} | 0 0 0 0 0 xy-6y2 x2+42xy+25y2 38xy+22y2 |
{5} | 0 0 0 0 0 4xy+22y2 -12xy+43y2 x2-44xy-47y2 |
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
|