-- 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 | -45x2+29xy+50y2 -10x2-11xy-7y2 |
| 48x2+26xy-18y2 -20x2-2xy+10y2 |
| 31x2+5xy-7y2 23x2+47xy-24y2 |
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 | -17x2+43xy-50y2 -24x2-7xy-35y2 x3 x2y+20xy2+12y3 15xy2+28y3 y4 0 0 |
| x2-50xy+12y2 -26xy+46y2 0 8xy2+34y3 -37xy2+34y3 0 y4 0 |
| -33xy+40y2 x2+48xy+3y2 0 34y3 xy2-8y3 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
| -17x2+43xy-50y2 -24x2-7xy-35y2 x3 x2y+20xy2+12y3 15xy2+28y3 y4 0 0 |
| x2-50xy+12y2 -26xy+46y2 0 8xy2+34y3 -37xy2+34y3 0 y4 0 |
| -33xy+40y2 x2+48xy+3y2 0 34y3 xy2-8y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | -10xy2+3y3 36xy2+3y3 10y3 4y3 -2y3 |
{2} | -8xy2-15y3 19y3 8y3 32y3 -47y3 |
{3} | -13xy+32y2 -46xy 13y2 -3y2 -3y2 |
{3} | 13x2+34xy+4y2 46x2-5xy-12y2 -13xy+35y2 3xy+44y2 3xy-8y2 |
{3} | 8x2-6xy+39y2 9xy+9y2 -8xy+21y2 -32xy-45y2 47xy+39y2 |
{4} | 0 0 x-26y 32y -14y |
{4} | 0 0 32y x+29y 21y |
{4} | 0 0 -32y 9y x-3y |
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+50y 26y |
{2} | 0 33y x-48y |
{3} | 1 17 24 |
{3} | 0 -17 4 |
{3} | 0 26 28 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 22 47 0 -25y 38x+6y xy+49y2 4xy+26y2 32xy-47y2 |
{5} | -30 2 0 11x+39y 20x-32y -8y2 xy+24y2 37xy-23y2 |
{5} | 0 0 0 0 0 x2+26xy+27y2 -32xy-30y2 14xy-33y2 |
{5} | 0 0 0 0 0 -32xy+30y2 x2-29xy+34y2 -21xy-3y2 |
{5} | 0 0 0 0 0 32xy+4y2 -9xy+18y2 x2+3xy+40y2 |
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
|