-- 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 | x2-44xy+4y2 17x2-16xy+30y2 |
| 46x2-28xy+23y2 39x2-37xy+14y2 |
| -35x2+44xy+31y2 -37x2+10xy+14y2 |
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 | 43x2+28xy+21y2 -7x2-11xy-36y2 x3 x2y-21xy2+19y3 -32xy2-50y3 y4 0 0 |
| x2-31xy-28y2 12xy-49y2 0 12xy2+37y3 -13xy2+43y3 0 y4 0 |
| -4xy-49y2 x2+31xy-22y2 0 4y3 xy2+20y3 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
| 43x2+28xy+21y2 -7x2-11xy-36y2 x3 x2y-21xy2+19y3 -32xy2-50y3 y4 0 0 |
| x2-31xy-28y2 12xy-49y2 0 12xy2+37y3 -13xy2+43y3 0 y4 0 |
| -4xy-49y2 x2+31xy-22y2 0 4y3 xy2+20y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | -36xy2+19y3 31xy2+36y3 36y3 23y3 37y3 |
{2} | 29xy2-45y3 -32y3 -29y3 7y3 -29y3 |
{3} | 20xy+37y2 11xy-40y2 -20y2 -41y2 39y2 |
{3} | -20x2-38xy+35y2 -11x2-9xy+43y2 20xy+y2 41xy -39xy+32y2 |
{3} | -29x2-35xy-15y2 -2xy+4y2 29xy-21y2 -7xy-48y2 29xy+16y2 |
{4} | 0 0 x-41y -5y -13y |
{4} | 0 0 49y x+21y -35y |
{4} | 0 0 27y 19y x+20y |
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+31y -12y |
{2} | 0 4y x-31y |
{3} | 1 -43 7 |
{3} | 0 -20 7 |
{3} | 0 49 26 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <------------------------------------------------------------------------------ A : 1
{5} | 40 -20 0 38y -7x+31y xy+3y2 -49xy+45y2 xy-17y2 |
{5} | -47 37 0 -46x+42y -24x+15y -12y2 xy-7y2 13xy+6y2 |
{5} | 0 0 0 0 0 x2+41xy-26y2 5xy-46y2 13xy+44y2 |
{5} | 0 0 0 0 0 -49xy-6y2 x2-21xy+36y2 35xy+49y2 |
{5} | 0 0 0 0 0 -27xy-40y2 -19xy+38y2 x2-20xy-10y2 |
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
|