-- 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 | -40x2+18xy-34y2 14x2+5xy+25y2 |
| 33x2+14xy+y2 -37x2+35xy+3y2 |
| -9x2+30y2 16x2+13xy-50y2 |
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 | -42x2+39xy-41y2 30x2+40xy+33y2 x3 x2y-44xy2+30y3 33xy2+6y3 y4 0 0 |
| x2+24xy-35y2 -37xy-5y2 0 28xy2-23y3 6xy2+47y3 0 y4 0 |
| 41xy+39y2 x2-18xy+5y2 0 34y3 xy2+14y3 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
| -42x2+39xy-41y2 30x2+40xy+33y2 x3 x2y-44xy2+30y3 33xy2+6y3 y4 0 0 |
| x2+24xy-35y2 -37xy-5y2 0 28xy2-23y3 6xy2+47y3 0 y4 0 |
| 41xy+39y2 x2-18xy+5y2 0 34y3 xy2+14y3 0 0 y4 |
8 5
1 : A <---------------------------------------------------------------------- A : 2
{2} | -49xy2+35y3 -9xy2-5y3 49y3 -8y3 47y3 |
{2} | 33xy2-41y3 -3y3 -33y3 30y3 0 |
{3} | 2xy-8y2 -40xy-50y2 -2y2 -50y2 -38y2 |
{3} | -2x2-19xy-19y2 40x2+18xy+6y2 2xy+27y2 50xy+35y2 38xy+10y2 |
{3} | -33x2+43xy+21y2 22xy-14y2 33xy-2y2 -30xy-8y2 12y2 |
{4} | 0 0 x-23y 3y 40y |
{4} | 0 0 43y x+41y -2y |
{4} | 0 0 -10y -7y x-18y |
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-24y 37y |
{2} | 0 -41y x+18y |
{3} | 1 42 -30 |
{3} | 0 -19 -36 |
{3} | 0 5 14 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 43 -21 0 32y -49x-26y xy-4y2 45xy+26y2 28y2 |
{5} | -10 15 0 48x-20y 43x-y -28y2 xy-45y2 -6xy-39y2 |
{5} | 0 0 0 0 0 x2+23xy-45y2 -3xy-24y2 -40xy-30y2 |
{5} | 0 0 0 0 0 -43xy-14y2 x2-41xy+6y2 2xy-43y2 |
{5} | 0 0 0 0 0 10xy+8y2 7xy+11y2 x2+18xy+39y2 |
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
|