-- 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 | -19x2-41xy+20y2 -27x2+43xy-9y2 |
| -37x2-23xy+27y2 -50x2-14xy+2y2 |
| -3x2-37xy-18y2 18x2+xy+19y2 |
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+13xy-29y2 -44x2-12xy+48y2 x3 x2y+20xy2-47y3 -15xy2-35y3 y4 0 0 |
| x2-44xy-11y2 -22xy-8y2 0 -23xy2+41y3 21xy2-43y3 0 y4 0 |
| 45xy+46y2 x2-4xy+11y2 0 -15y3 xy2+18y3 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+13xy-29y2 -44x2-12xy+48y2 x3 x2y+20xy2-47y3 -15xy2-35y3 y4 0 0 |
| x2-44xy-11y2 -22xy-8y2 0 -23xy2+41y3 21xy2-43y3 0 y4 0 |
| 45xy+46y2 x2-4xy+11y2 0 -15y3 xy2+18y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | -13xy2-24y3 -30xy2-35y3 13y3 -22y3 -50y3 |
{2} | 15xy2-2y3 -12y3 -15y3 -41y3 30y3 |
{3} | 45xy+8y2 -47xy-49y2 -45y2 25y2 12y2 |
{3} | -45x2+7xy-44y2 47x2-37xy+y2 45xy-15y2 -25xy+45y2 -12xy+42y2 |
{3} | -15x2+40xy-19y2 47xy-9y2 15xy-38y2 41xy+16y2 -30xy+2y2 |
{4} | 0 0 x-29y -35y -38y |
{4} | 0 0 14y x-10y -27y |
{4} | 0 0 26y 32y x+39y |
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+44y 22y |
{2} | 0 -45y x+4y |
{3} | 1 -15 44 |
{3} | 0 -27 -41 |
{3} | 0 16 25 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <---------------------------------------------------------------------------- A : 1
{5} | 28 -32 0 27y -27x+49y xy-11y2 -4xy-35y2 -2xy+38y2 |
{5} | -7 39 0 43x-37y -28x+16y 23y2 xy+38y2 -21xy+41y2 |
{5} | 0 0 0 0 0 x2+29xy-31y2 35xy+48y2 38xy-41y2 |
{5} | 0 0 0 0 0 -14xy-36y2 x2+10xy-42y2 27xy-2y2 |
{5} | 0 0 0 0 0 -26xy+y2 -32xy+18y2 x2-39xy-28y2 |
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
|