-- 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];
|
i2 : M = cokernel random(A^3, A^{-2,-2})
o2 = cokernel | -27x2-42xy-41y2 50x2-43xy+33y2 |
| -49x2+19xy+4y2 -24x2+28xy+28y2 |
| 49x2-15xy-43y2 13x2+15xy-15y2 |
3
o2 : A-module, quotient of A
|
i3 : R = cokernel matrix {{x^3,y^4}}
o3 = cokernel | x3 y4 |
1
o3 : A-module, quotient of A
|
i4 : N = prune (M**R)
o4 = cokernel | -20x2+37xy+14y2 -2x2-36xy-26y2 x3 x2y-38xy2-28y3 49xy2-2y3 y4 0 0 |
| x2-42xy-21y2 -21xy+10y2 0 -30xy2-7y3 24xy2-4y3 0 y4 0 |
| 38xy+23y2 x2+48xy+18y2 0 -27y3 xy2-15y3 0 0 y4 |
3
o4 : A-module, quotient of A
|
i5 : C = resolution N
3 8 5
o5 = A <-- A <-- A <-- 0
0 1 2 3
o5 : ChainComplex
|
i6 : d = C.dd
3 8
o6 = 0 : A <--------------------------------------------------------------------------- A : 1
| -20x2+37xy+14y2 -2x2-36xy-26y2 x3 x2y-38xy2-28y3 49xy2-2y3 y4 0 0 |
| x2-42xy-21y2 -21xy+10y2 0 -30xy2-7y3 24xy2-4y3 0 y4 0 |
| 38xy+23y2 x2+48xy+18y2 0 -27y3 xy2-15y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 41xy2+4y3 -12xy2+40y3 -41y3 -y3 13y3 |
{2} | -13xy2-14y3 48y3 13y3 46y3 -24y3 |
{3} | 32xy-45y2 -40xy -32y2 20y2 -23y2 |
{3} | -32x2-4xy-36y2 40x2-33xy+4y2 32xy+49y2 -20xy-50y2 23xy+11y2 |
{3} | 13x2+27xy+7y2 -27xy+4y2 -13xy-13y2 -46xy+34y2 24xy+22y2 |
{4} | 0 0 x+36y -21y -50y |
{4} | 0 0 7y x+12y -29y |
{4} | 0 0 19y -29y x-48y |
5
2 : A <----- 0 : 3
0
o6 : ChainComplexMap
|
i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------- A : 0
{2} | 0 x+42y 21y |
{2} | 0 -38y x-48y |
{3} | 1 20 2 |
{3} | 0 20 -44 |
{3} | 0 3 -27 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | -8 -29 0 22y -31x+2y xy-19y2 -12xy-4y2 37xy-10y2 |
{5} | -17 -12 0 48x+37y -45x-26y 30y2 xy+39y2 -24xy+21y2 |
{5} | 0 0 0 0 0 x2-36xy-3y2 21xy+38y2 50xy-3y2 |
{5} | 0 0 0 0 0 -7xy-13y2 x2-12xy+30y2 29xy-13y2 |
{5} | 0 0 0 0 0 -19xy-27y2 29xy+39y2 x2+48xy-27y2 |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap
|
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
|
i9 : s^2
5 3
o9 = 2 : A <----- A : 0
0
8
3 : 0 <----- A : 1
0
o9 : ChainComplexMap
|