-- 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 | 5x2+30xy+41y2 7x2+38xy+20y2 |
| 3x2+31xy-38y2 -9x2+15xy-43y2 |
| -23x2-9xy-25y2 -35x2+18xy-38y2 |
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 | 24x2+42xy+41y2 -41x2-9xy-14y2 x3 x2y+31xy2-44y3 -xy2+39y3 y4 0 0 |
| x2-30xy+7y2 -36xy-15y2 0 33xy2+13y3 -4xy2+26y3 0 y4 0 |
| 20xy-45y2 x2+3xy+4y2 0 -35y3 xy2-41y3 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
| 24x2+42xy+41y2 -41x2-9xy-14y2 x3 x2y+31xy2-44y3 -xy2+39y3 y4 0 0 |
| x2-30xy+7y2 -36xy-15y2 0 33xy2+13y3 -4xy2+26y3 0 y4 0 |
| 20xy-45y2 x2+3xy+4y2 0 -35y3 xy2-41y3 0 0 y4 |
8 5
1 : A <--------------------------------------------------------------------------- A : 2
{2} | 3xy2-49y3 -47xy2-26y3 -3y3 39y3 -49y3 |
{2} | -36xy2+46y3 46y3 36y3 -5y3 16y3 |
{3} | -41xy+50y2 20xy+14y2 41y2 22y2 28y2 |
{3} | 41x2-5xy+20y2 -20x2+17xy+39y2 -41xy-45y2 -22xy-50y2 -28xy-43y2 |
{3} | 36x2-16xy-13y2 -8xy+49y2 -36xy-30y2 5xy-17y2 -16xy+2y2 |
{4} | 0 0 x+19y 26y -40y |
{4} | 0 0 7y x+37y -21y |
{4} | 0 0 47y 35y 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+30y 36y |
{2} | 0 -20y x-3y |
{3} | 1 -24 41 |
{3} | 0 34 32 |
{3} | 0 10 -8 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | 39 -11 0 46y -14x+48y xy-18y2 -31xy+y2 -22xy+39y2 |
{5} | 16 -26 0 5x+43y 42x-24y -33y2 xy+35y2 4xy-16y2 |
{5} | 0 0 0 0 0 x2-19xy-24y2 -26xy-45y2 40xy+25y2 |
{5} | 0 0 0 0 0 -7xy+11y2 x2-37xy+8y2 21xy+18y2 |
{5} | 0 0 0 0 0 -47xy+21y2 -35xy-49y2 x2-45xy+16y2 |
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
|