-- 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 | -13x2+4xy-42y2 40x2-37xy+40y2 |
| 25x2-46xy+24y2 18x2-49xy-44y2 |
| 22xy-35y2 30x2+8xy-19y2 |
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 | -49x2-16xy-34y2 24x2+5xy+15y2 x3 x2y-13xy2-33y3 38xy2-23y3 y4 0 0 |
| x2-18xy+5y2 26xy+9y2 0 13xy2+46y3 40xy2-30y3 0 y4 0 |
| 13xy+39y2 x2-21xy+13y2 0 -14y3 xy2+26y3 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
| -49x2-16xy-34y2 24x2+5xy+15y2 x3 x2y-13xy2-33y3 38xy2-23y3 y4 0 0 |
| x2-18xy+5y2 26xy+9y2 0 13xy2+46y3 40xy2-30y3 0 y4 0 |
| 13xy+39y2 x2-21xy+13y2 0 -14y3 xy2+26y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | -22xy2+25y3 -27xy2+26y3 22y3 10y3 -24y3 |
{2} | 23xy2-13y3 8y3 -23y3 4y3 7y3 |
{3} | 13xy-28y2 29xy-7y2 -13y2 4y2 31y2 |
{3} | -13x2+12xy-8y2 -29x2+24xy-25y2 13xy+16y2 -4xy-11y2 -31xy+34y2 |
{3} | -23x2-14xy-12y2 38xy+29y2 23xy+27y2 -4xy+2y2 -7xy+4y2 |
{4} | 0 0 x+20y -37y -10y |
{4} | 0 0 -31y x-25y 27y |
{4} | 0 0 -27y -42y x+5y |
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+18y -26y |
{2} | 0 -13y x+21y |
{3} | 1 49 -24 |
{3} | 0 -2 35 |
{3} | 0 -41 -42 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------- A : 1
{5} | 7 -15 0 6y -22x+10y xy+46y2 13xy-39y2 48xy-35y2 |
{5} | -45 -6 0 -7x-5y -18x-25y -13y2 xy+2y2 -40xy-49y2 |
{5} | 0 0 0 0 0 x2-20xy-y2 37xy-y2 10xy-37y2 |
{5} | 0 0 0 0 0 31xy+32y2 x2+25xy+32y2 -27xy-28y2 |
{5} | 0 0 0 0 0 27xy+21y2 42xy+21y2 x2-5xy-31y2 |
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
|