-- 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 | 47x2-20xy-5y2 -11x2-13xy-4y2 |
| -22x2+11xy+7y2 -39x2+43xy+9y2 |
| -33x2-19xy+18y2 34x2+2xy-3y2 |
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 | -15x2+47xy-27y2 30x2+6xy-40y2 x3 x2y+27xy2+6y3 -15xy2-5y3 y4 0 0 |
| x2-32xy-2y2 21xy-5y2 0 xy2+42y3 19xy2-21y3 0 y4 0 |
| 5xy-14y2 x2-15xy-31y2 0 -30y3 xy2-11y3 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
| -15x2+47xy-27y2 30x2+6xy-40y2 x3 x2y+27xy2+6y3 -15xy2-5y3 y4 0 0 |
| x2-32xy-2y2 21xy-5y2 0 xy2+42y3 19xy2-21y3 0 y4 0 |
| 5xy-14y2 x2-15xy-31y2 0 -30y3 xy2-11y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | -8xy2+30y3 -10xy2-18y3 8y3 -37y3 15y3 |
{2} | -43xy2-47y3 5y3 43y3 -38y3 -47y3 |
{3} | xy+14y2 -10xy-11y2 -y2 -22y2 -y2 |
{3} | -x2+10xy+13y2 10x2-5xy+48y2 xy-24y2 22xy-45y2 xy-10y2 |
{3} | 43x2-14xy-33y2 42xy+34y2 -43xy-40y2 38xy-14y2 47xy-32y2 |
{4} | 0 0 x-39y 4y 41y |
{4} | 0 0 -5y x+19y 48y |
{4} | 0 0 -18y 32y 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+32y -21y |
{2} | 0 -5y x+15y |
{3} | 1 15 -30 |
{3} | 0 -23 37 |
{3} | 0 -19 -43 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------------- A : 1
{5} | -16 1 0 45y 47x+14y xy+42y2 32xy+27y2 13xy-26y2 |
{5} | 31 -44 0 -10x+50y 35x-18y -y2 xy+11y2 -19xy+27y2 |
{5} | 0 0 0 0 0 x2+39xy-45y2 -4xy+20y2 -41xy+19y2 |
{5} | 0 0 0 0 0 5xy+44y2 x2-19xy-42y2 -48xy-50y2 |
{5} | 0 0 0 0 0 18xy-20y2 -32xy-36y2 x2-20xy-14y2 |
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
|