If $f : C \to D$ is a map of chain complexes of degree $d$, then the cokernel is the complex $E$ whose $i-th$ is $cokernel(f_{i-d})$, and whose differential is induced from the differential on the target.
In the following example, we first construct a random complex morphism $f : C \to D$. We consider the exact sequence $0 \to D \to cone(f) \to C[-1] \to 0$. For the maps $g : D \to cone(f)$ and $h : cone(f) \to C[-1]$, we compute the kernel.
i1 : S = ZZ/101[a,b,c,d]; |
i2 : C = freeResolution ideal(b^2-a*c, b*c-a*d, c^2-b*d)
1 3 2
o2 = S <-- S <-- S
0 1 2
o2 : Complex
|
i3 : D = freeResolution ideal(a,b,c)
1 3 3 1
o3 = S <-- S <-- S <-- S
0 1 2 3
o3 : Complex
|
i4 : f = randomComplexMap(D, C, Cycle => true, InternalDegree => 0)
1 1
o4 = 0 : S <----------- S : 0
| -22 |
3 3
1 : S <------------------------------------------------ S : 1
{1} | 36b+3c 30b-19c+22d -29b-10c |
{1} | -36a-22b+29c -30a-14c 29a+29c+22d |
{1} | 19a-29b 19a-8b 10a-29b-22c |
3 2
2 : S <--------------------------------------------- S : 2
{2} | -29a-30b+31c-22d 29b+6c-36d |
{2} | -10a+24b+3c 34b-19c+19d |
{2} | 24a-8b+29c -24a-29b-14c-29d |
o4 : ComplexMap
|
i5 : Cf = cone f
1 4 6 3
o5 = S <-- S <-- S <-- S
0 1 2 3
o5 : Complex
|
i6 : g = canonicalMap(Cf, D)
1 1
o6 = 0 : S <--------- S : 0
| 1 |
4 3
1 : S <----------------- S : 1
{0} | 0 0 0 |
{1} | 1 0 0 |
{1} | 0 1 0 |
{1} | 0 0 1 |
6 3
2 : S <----------------- S : 2
{2} | 0 0 0 |
{2} | 0 0 0 |
{2} | 0 0 0 |
{2} | 1 0 0 |
{2} | 0 1 0 |
{2} | 0 0 1 |
3 1
3 : S <------------- S : 3
{3} | 0 |
{3} | 0 |
{3} | 1 |
o6 : ComplexMap
|
i7 : h = canonicalMap(C[-1], Cf)
1 4
o7 = 1 : S <--------------- S : 1
| 1 0 0 0 |
3 6
2 : S <----------------------- S : 2
{2} | 1 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 |
{2} | 0 0 1 0 0 0 |
2 3
3 : S <----------------- S : 3
{3} | 1 0 0 |
{3} | 0 1 0 |
o7 : ComplexMap
|
i8 : prune coker g == C[-1] o8 = true |
i9 : coker h == 0 o9 = true |
There is a canonical map of complexes from the target to the cokernel.
i10 : g1 = canonicalMap(coker g, target g)
1
o10 = 0 : cokernel | 1 | <----- S : 0
0
4
1 : cokernel {0} | 0 0 0 | <------------------- S : 1
{1} | 1 0 0 | {0} | 1 0 0 0 |
{1} | 0 1 0 | {1} | 0 0 0 0 |
{1} | 0 0 1 | {1} | 0 0 0 0 |
{1} | 0 0 0 0 |
6
2 : cokernel {2} | 0 0 0 | <----------------------- S : 2
{2} | 0 0 0 | {2} | 1 0 0 0 0 0 |
{2} | 0 0 0 | {2} | 0 1 0 0 0 0 |
{2} | 1 0 0 | {2} | 0 0 1 0 0 0 |
{2} | 0 1 0 | {2} | 0 0 0 0 0 0 |
{2} | 0 0 1 | {2} | 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 |
3
3 : cokernel {3} | 0 | <----------------- S : 3
{3} | 0 | {3} | 1 0 0 |
{3} | 1 | {3} | 0 1 0 |
{3} | 0 0 0 |
o10 : ComplexMap
|
i11 : coker g == image g1 o11 = true |
i12 : coker g1 == 0 o12 = true |