-- 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 | 16x2-18xy-9y2 -46x2+13xy+16y2 |
| 3x2+5xy+32y2 -25x2+7xy+3y2 |
| 31x2+13xy-21y2 -12x2+17xy+30y2 |
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 | 7x2-9xy-50y2 -36x2-16xy-15y2 x3 x2y+44xy2+17y3 17xy2-41y3 y4 0 0 |
| x2+45xy-37y2 -27xy-41y2 0 -18xy2-9y3 -37xy2-12y3 0 y4 0 |
| 26xy+23y2 x2+37xy-42y2 0 21y3 xy2-31y3 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
| 7x2-9xy-50y2 -36x2-16xy-15y2 x3 x2y+44xy2+17y3 17xy2-41y3 y4 0 0 |
| x2+45xy-37y2 -27xy-41y2 0 -18xy2-9y3 -37xy2-12y3 0 y4 0 |
| 26xy+23y2 x2+37xy-42y2 0 21y3 xy2-31y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | -11xy2-21y3 -35xy2-4y3 11y3 -31y3 -21y3 |
{2} | 48xy2-9y3 -8y3 -48y3 -15y3 -42y3 |
{3} | 31xy-14y2 30xy+11y2 -31y2 -46y2 37y2 |
{3} | -31x2-41xy-38y2 -30x2+39xy-19y2 31xy-46y2 46xy+24y2 -37xy-47y2 |
{3} | -48x2+5xy+23y2 33xy-5y2 48xy+4y2 15xy-49y2 42xy+37y2 |
{4} | 0 0 x-32y 50y 30y |
{4} | 0 0 -8y x-13y 47y |
{4} | 0 0 33y -21y x+45y |
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-45y 27y |
{2} | 0 -26y x-37y |
{3} | 1 -7 36 |
{3} | 0 -6 10 |
{3} | 0 -12 2 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------- A : 1
{5} | 25 -21 0 -44y -40x-y xy+50y2 -6xy-18y2 -37xy-7y2 |
{5} | 11 -15 0 37x-31y -26x+32y 18y2 xy-22y2 37xy+13y2 |
{5} | 0 0 0 0 0 x2+32xy-2y2 -50xy+49y2 -30xy+13y2 |
{5} | 0 0 0 0 0 8xy-8y2 x2+13xy-6y2 -47xy-49y2 |
{5} | 0 0 0 0 0 -33xy-9y2 21xy-32y2 x2-45xy+8y2 |
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
|