-- 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 which 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 | 42x2-50xy+39y2 -39x2+30xy+19y2 |
| 9x2-15xy-22y2 -38x2+2xy-4y2 |
| 50x2+45xy-29y2 -36x2-16xy-6y2 |
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 | -16x2+43xy-27y2 32x2-35xy+42y2 x2y-19xy2+5y3 42xy2+42y3 x3 y4
| x2-11xy-45y2 34xy+42y2 20xy2-13y3 49xy2-39y3 0 0
| -30xy+27y2 x2-24xy+39y2 -y3 xy2+10y3 0 0
------------------------------------------------------------------------
0 0 |
y4 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
o6 = 0 : A <----------------------------------------------------------------
| -16x2+43xy-27y2 32x2-35xy+42y2 x3 x2y-19xy2+5y3 42xy2+42y3 y
| x2-11xy-45y2 34xy+42y2 0 20xy2-13y3 49xy2-39y3 0
| -30xy+27y2 x2-24xy+39y2 0 -y3 xy2+10y3 0
8
1 : A <----------------------------------------------------------------
{2} | -29xy2-y3 -24xy2+4y3 29y3 43y3 8y3
{2} | 34xy2-21y3 -10y3 -34y3 23y3 -2y
{3} | -9xy+11y2 19xy+50y2 9y2 -34y2 -25
{3} | 9x2+36xy-32y2 -19x2+13xy-45y2 -9xy-47y2 34xy+49y2 25x
{3} | -34x2+13xy+41y2 -22xy-47y2 34xy+8y2 -23xy-15y2 2xy
{4} | 0 0 x-11y -26y -40
{4} | 0 0 10y x+11y 12y
{4} | 0 0 12y -41y x
5
2 : A <----- 0 : 3
0
------------------------------------------------------------------------
8
----------- A : 1
4 0 0 |
y4 0 |
0 y4 |
5
---------- A : 2
|
3 |
y2 |
y-23y2 |
-6y2 |
y |
|
|
o6 : ChainComplexMap
|
i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------- A : 0
{2} | 0 x+11y -34y |
{2} | 0 30y x+24y |
{3} | 1 16 -32 |
{3} | 0 -19 36 |
{3} | 0 13 22 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5
2 : A <----------------------------------------------------------------
{5} | 39 -35 0 46y -3x+14y xy-15y2 32xy+47y2 6x
{5} | -41 -20 0 -16x-37y -28x+19y -20y2 xy+39y2 -4
{5} | 0 0 0 0 0 x2+11xy-13y2 26xy+24y2 40
{5} | 0 0 0 0 0 -10xy+43y2 x2-11xy-25y2 -1
{5} | 0 0 0 0 0 -12xy-37y2 41xy+45y2 x2
5
3 : 0 <----- A : 2
0
------------------------------------------------------------------------
8
------------ A : 1
y-25y2 |
9xy+36y2 |
xy+27y2 |
2xy+35y2 |
+38y2 |
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
|