-- 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 | 23x2+5xy+18y2 46x2-25xy-5y2 |
| 8x2+2xy+25y2 11x2-20xy-22y2 |
| 41x2-26xy-2y2 -49x2-10xy+37y2 |
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 | -38x2+21xy+28y2 40x2-36xy+27y2 x3 x2y-15xy2-23y3 -46xy2+49y3 y4 0 0 |
| x2+14xy+15y2 -15xy+10y2 0 -21xy2-46y3 33xy2+13y3 0 y4 0 |
| 50xy+29y2 x2-3xy+14y2 0 9y3 xy2+19y3 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
| -38x2+21xy+28y2 40x2-36xy+27y2 x3 x2y-15xy2-23y3 -46xy2+49y3 y4 0 0 |
| x2+14xy+15y2 -15xy+10y2 0 -21xy2-46y3 33xy2+13y3 0 y4 0 |
| 50xy+29y2 x2-3xy+14y2 0 9y3 xy2+19y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | -21xy2-47y3 35xy2+5y3 21y3 -30y3 37y3 |
{2} | -20xy2-46y3 -11y3 20y3 24y3 27y3 |
{3} | -16xy-46y2 32xy-32y2 16y2 -33y2 7y2 |
{3} | 16x2-4xy+31y2 -32x2-27xy-19y2 -16xy+50y2 33xy+18y2 -7xy-11y2 |
{3} | 20x2+7xy-26y2 -37xy-49y2 -20xy+39y2 -24xy+14y2 -27xy+18y2 |
{4} | 0 0 x+30y 21y 29y |
{4} | 0 0 -35y x+48y -50y |
{4} | 0 0 41y 5y x+23y |
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-14y 15y |
{2} | 0 -50y x+3y |
{3} | 1 38 -40 |
{3} | 0 33 -19 |
{3} | 0 16 -48 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -21 -38 0 -10y -5x+35y xy+20y2 -19xy-22y2 -34xy-40y2 |
{5} | 0 -48 0 41x+33y 48x-35y 21y2 xy+10y2 -33xy-15y2 |
{5} | 0 0 0 0 0 x2-30xy+41y2 -21xy-35y2 -29xy-18y2 |
{5} | 0 0 0 0 0 35xy-33y2 x2-48xy+6y2 50xy-20y2 |
{5} | 0 0 0 0 0 -41xy-22y2 -5xy+4y2 x2-23xy-47y2 |
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
|