-- 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 | -37x2+48xy+8y2 -34x2+37xy-25y2 |
| -45x2+37xy+10y2 x2-30xy-41y2 |
| -10x2+3xy-19y2 29x2-10xy+2y2 |
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 | -28x2-48xy-y2 -42x2-11xy+34y2 x3 x2y+26xy2-25y3 11xy2-5y3 y4 0 0 |
| x2-30xy+27y2 29y2 0 29xy2-19y3 -41xy2+17y3 0 y4 0 |
| -40xy+21y2 x2+8xy-32y2 0 46y3 xy2-30y3 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
| -28x2-48xy-y2 -42x2-11xy+34y2 x3 x2y+26xy2-25y3 11xy2-5y3 y4 0 0 |
| x2-30xy+27y2 29y2 0 29xy2-19y3 -41xy2+17y3 0 y4 0 |
| -40xy+21y2 x2+8xy-32y2 0 46y3 xy2-30y3 0 0 y4 |
8 5
1 : A <--------------------------------------------------------------------- A : 2
{2} | -15xy2-15y3 -30xy2-38y3 15y3 46y3 4y3 |
{2} | -12xy2+8y3 -37y3 12y3 33y3 9y3 |
{3} | -14xy-31y2 -8xy+30y2 14y2 -4y2 -15y2 |
{3} | 14x2+25xy+5y2 8x2+33xy+5y2 -14xy+6y2 4xy+4y2 15xy-3y2 |
{3} | 12x2+12xy-8y2 -16xy+34y2 -12xy-20y2 -33xy-2y2 -9xy+36y2 |
{4} | 0 0 x-40y 24y 5y |
{4} | 0 0 4y x+32y -28y |
{4} | 0 0 y 48y x+8y |
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+30y 0 |
{2} | 0 40y x-8y |
{3} | 1 28 42 |
{3} | 0 43 -22 |
{3} | 0 -50 -5 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 21 -8 0 17y -42x+39y xy-30y2 28xy-23y2 26xy-43y2 |
{5} | 21 42 0 38x-35y 23x+6y -29y2 xy+50y2 41xy-31y2 |
{5} | 0 0 0 0 0 x2+40xy-16y2 -24xy+48y2 -5xy-24y2 |
{5} | 0 0 0 0 0 -4xy+41y2 x2-32xy-22y2 28xy+11y2 |
{5} | 0 0 0 0 0 -xy-42y2 -48xy+25y2 x2-8xy+38y2 |
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
|