-- 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 | -27x2+11xy+y2 -6x2-27xy+48y2 |
| -44x2-22xy-15y2 47x2+12xy-31y2 |
| -6x2+8xy+20y2 17x2-23xy+25y2 |
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+11xy+13y2 16x2-32xy-45y2 x3 x2y-5xy2-15y3 26xy2-5y3 y4 0 0 |
| x2-44xy+12y2 -44xy-35y2 0 -34xy2+20y3 41xy2+33y3 0 y4 0 |
| -13xy-47y2 x2-7xy-29y2 0 -35y3 xy2-26y3 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+11xy+13y2 16x2-32xy-45y2 x3 x2y-5xy2-15y3 26xy2-5y3 y4 0 0 |
| x2-44xy+12y2 -44xy-35y2 0 -34xy2+20y3 41xy2+33y3 0 y4 0 |
| -13xy-47y2 x2-7xy-29y2 0 -35y3 xy2-26y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | -40xy2-9y3 -46xy2-8y3 40y3 -22y3 -37y3 |
{2} | -48xy2+33y3 30y3 48y3 -21y3 17y3 |
{3} | -27xy+46y2 37xy-26y2 27y2 8y2 -20y2 |
{3} | 27x2-13xy-23y2 -37x2+3xy+9y2 -27xy-33y2 -8xy-32y2 20xy+49y2 |
{3} | 48x2-9xy-9y2 -4xy-35y2 -48xy-24y2 21xy+35y2 -17xy-4y2 |
{4} | 0 0 x+15y 46y 42y |
{4} | 0 0 26y x+24y -11y |
{4} | 0 0 -22y -35y x-39y |
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+44y 44y |
{2} | 0 13y x+7y |
{3} | 1 36 -16 |
{3} | 0 -49 -11 |
{3} | 0 3 44 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -19 -43 0 -48y 40x-16y xy+28y2 -32xy+46y2 -27xy+7y2 |
{5} | -7 12 0 30x+22y 21x+6y 34y2 xy+28y2 -41xy+21y2 |
{5} | 0 0 0 0 0 x2-15xy-8y2 -46xy+21y2 -42xy+y2 |
{5} | 0 0 0 0 0 -26xy+44y2 x2-24xy+36y2 11xy+45y2 |
{5} | 0 0 0 0 0 22xy+22y2 35xy+18y2 x2+39xy-28y2 |
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
|