-- 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 | 25x2+47xy+31y2 3x2-38xy+36y2 |
| 10x2-39y2 12x2+43xy+28y2 |
| -9x2-37xy-24y2 -6x2-5xy-7y2 |
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 | 29x2-13xy-35y2 7x2+14xy+25y2 x3 x2y-49xy2+29y3 -40xy2-47y3 y4 0 0 |
| x2+27xy+48y2 30xy+24y2 0 -6y3 29xy2+35y3 0 y4 0 |
| 10xy+8y2 x2+4xy+34y2 0 -46y3 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
| 29x2-13xy-35y2 7x2+14xy+25y2 x3 x2y-49xy2+29y3 -40xy2-47y3 y4 0 0 |
| x2+27xy+48y2 30xy+24y2 0 -6y3 29xy2+35y3 0 y4 0 |
| 10xy+8y2 x2+4xy+34y2 0 -46y3 xy2+8y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | 40xy2-43y3 -13y3 -40y3 -21y3 19y3 |
{2} | -23xy2+2y3 17y3 23y3 -46y3 32y3 |
{3} | 24xy+20y2 -27xy-37y2 -24y2 -26y2 -4y2 |
{3} | -24x2+38xy-30y2 27x2+47xy-25y2 24xy+43y2 26xy+5y2 4xy-41y2 |
{3} | 23x2+18xy-37y2 13xy-y2 -23xy-20y2 46xy+10y2 -32xy+20y2 |
{4} | 0 0 x-21y 33y -26y |
{4} | 0 0 3y x-26y 0 |
{4} | 0 0 -41y -37y x+47y |
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 -30y |
{2} | 0 -10y x-4y |
{3} | 1 -29 -7 |
{3} | 0 -43 -25 |
{3} | 0 -1 -21 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 22 -43 0 -48y 22x-19y xy+49y2 -2xy-11y2 -3xy+18y2 |
{5} | -20 45 0 15x+21y 42x-5y 0 xy-40y2 -29xy-15y2 |
{5} | 0 0 0 0 0 x2+21xy-10y2 -33xy+17y2 26xy+31y2 |
{5} | 0 0 0 0 0 -3xy-40y2 x2+26xy-33y2 23y2 |
{5} | 0 0 0 0 0 41xy+35y2 37xy-9y2 x2-47xy+43y2 |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap
|
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
|