-- 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 | -14x2+47xy-32y2 -17x2+30xy+16y2 |
| -14x2+40xy+40y2 49x2+41xy-42y2 |
| -43x2-46xy-35y2 -39x2-24xy-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 | -48x2+36xy-48y2 23x2-41xy-40y2 x3 x2y+2xy2-21y3 -43xy2-38y3 y4 0 0 |
| x2+15xy+10y2 -34xy+24y2 0 13xy2+49y3 -31xy2-20y3 0 y4 0 |
| 17xy-39y2 x2-35xy+37y2 0 20y3 xy2+36y3 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
| -48x2+36xy-48y2 23x2-41xy-40y2 x3 x2y+2xy2-21y3 -43xy2-38y3 y4 0 0 |
| x2+15xy+10y2 -34xy+24y2 0 13xy2+49y3 -31xy2-20y3 0 y4 0 |
| 17xy-39y2 x2-35xy+37y2 0 20y3 xy2+36y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 10xy2-37y3 10xy2+29y3 -10y3 7y3 -11y3 |
{2} | 8xy2+6y3 -7y3 -8y3 -47y3 17y3 |
{3} | -19xy+27y2 -7xy+40y2 19y2 5y2 -38y2 |
{3} | 19x2-12xy+2y2 7x2+22xy-17y2 -19xy-15y2 -5xy+14y2 38xy-9y2 |
{3} | -8x2+12xy+34y2 20y2 8xy-18y2 47xy-23y2 -17xy+27y2 |
{4} | 0 0 x+19y -3y -21y |
{4} | 0 0 -40y x+13y -24y |
{4} | 0 0 46y 35y x-32y |
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-15y 34y |
{2} | 0 -17y x+35y |
{3} | 1 48 -23 |
{3} | 0 39 -41 |
{3} | 0 2 4 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------------- A : 1
{5} | -25 -50 0 0 -38x+44y xy+10y2 -32xy-12y2 -40xy+50y2 |
{5} | 19 -46 0 29x-19y 31x+21y -13y2 xy+46y2 31xy+4y2 |
{5} | 0 0 0 0 0 x2-19xy+20y2 3xy-23y2 21xy+42y2 |
{5} | 0 0 0 0 0 40xy+40y2 x2-13xy-46y2 24xy-17y2 |
{5} | 0 0 0 0 0 -46xy+22y2 -35xy+5y2 x2+32xy+26y2 |
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
|