-- 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 | -50x2+40xy+17y2 33x2+43xy-44y2 |
| -13x2+34xy+36y2 11x2+11xy-17y2 |
| 24x2-47xy+25y2 -23x2+30xy-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 | 16x2-50xy+19y2 15x2-17xy+11y2 x3 x2y+49xy2+34y3 -8xy2+34y3 y4 0 0 |
| x2-27xy-12y2 -9xy-5y2 0 13xy2+19y3 19xy2-23y3 0 y4 0 |
| -7xy-28y2 x2-31xy+49y2 0 -46y3 xy2+38y3 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
| 16x2-50xy+19y2 15x2-17xy+11y2 x3 x2y+49xy2+34y3 -8xy2+34y3 y4 0 0 |
| x2-27xy-12y2 -9xy-5y2 0 13xy2+19y3 19xy2-23y3 0 y4 0 |
| -7xy-28y2 x2-31xy+49y2 0 -46y3 xy2+38y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 8xy2+12y3 39xy2-35y3 -8y3 21y3 42y3 |
{2} | 6xy2-24y3 -32y3 -6y3 45y3 6y3 |
{3} | -47xy+36y2 3xy-18y2 47y2 -2y2 10y2 |
{3} | 47x2+21xy+2y2 -3x2+46xy-48y2 -47xy+44y2 2xy+46y2 -10xy-17y2 |
{3} | -6x2+30xy+5y2 -35xy-32y2 6xy-6y2 -45xy+11y2 -6xy+45y2 |
{4} | 0 0 x+37y -4y 2y |
{4} | 0 0 11y x-43y -27y |
{4} | 0 0 -y -20y x+6y |
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+27y 9y |
{2} | 0 7y x+31y |
{3} | 1 -16 -15 |
{3} | 0 18 14 |
{3} | 0 30 -35 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 5 11 0 30y -17x+49y xy-2y2 43xy+24y2 -xy-y2 |
{5} | 17 -25 0 -34x+16y 3x-46y -13y2 xy+y2 -19xy+19y2 |
{5} | 0 0 0 0 0 x2-37xy+10y2 4xy-16y2 -2xy-8y2 |
{5} | 0 0 0 0 0 -11xy-39y2 x2+43xy+22y2 27xy+11y2 |
{5} | 0 0 0 0 0 xy+40y2 20xy+37y2 x2-6xy-32y2 |
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
|