-- 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 | -19x2-50xy+49y2 18x2-13xy+35y2 |
| -35x2-50xy-32y2 -44x2+2xy+12y2 |
| -3x2+3xy+17y2 -38x2-40xy+35y2 |
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 | -41x2+33xy-26y2 47x2+2xy-16y2 x3 x2y-16xy2+23y3 -22xy2+10y3 y4 0 0 |
| x2+37xy+26y2 -11xy+44y2 0 -29xy2-26y3 33xy2-25y3 0 y4 0 |
| -41xy+17y2 x2+6xy-2y2 0 -47y3 xy2+24y3 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
| -41x2+33xy-26y2 47x2+2xy-16y2 x3 x2y-16xy2+23y3 -22xy2+10y3 y4 0 0 |
| x2+37xy+26y2 -11xy+44y2 0 -29xy2-26y3 33xy2-25y3 0 y4 0 |
| -41xy+17y2 x2+6xy-2y2 0 -47y3 xy2+24y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | -28xy2+12y3 -40xy2+46y3 28y3 20y3 -44y3 |
{2} | 9xy2+19y3 5y3 -9y3 -8y3 -35y3 |
{3} | -48xy-38y2 -23xy+18y2 48y2 32y2 0 |
{3} | 48x2+47xy-10y2 23x2+23xy-3y2 -48xy-9y2 -32xy-49y2 5y2 |
{3} | -9x2+39xy+24y2 42xy-17y2 9xy+43y2 8xy-20y2 35xy-11y2 |
{4} | 0 0 x-43y 2y 8y |
{4} | 0 0 4y x-23y 14y |
{4} | 0 0 -30y 43y x-35y |
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-37y 11y |
{2} | 0 41y x-6y |
{3} | 1 41 -47 |
{3} | 0 -43 24 |
{3} | 0 38 -16 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -7 -13 0 -32y -45x-29y xy-48y2 -44xy-34y2 -41xy-4y2 |
{5} | -45 -31 0 22x+36y 50x+41y 29y2 xy+y2 -33xy-43y2 |
{5} | 0 0 0 0 0 x2+43xy+y2 -2xy+10y2 -8xy+10y2 |
{5} | 0 0 0 0 0 -4xy+23y2 x2+23xy+28y2 -14xy+28y2 |
{5} | 0 0 0 0 0 30xy-13y2 -43xy-29y2 x2+35xy-29y2 |
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
|