The complex of homomorphisms is a complex D whose ith component is the direct sum of Hom(C1j, C2(j+i)) over all j. The differential on Hom(C1j, C2(j+i)) is the differential Hom(idC1, ddC2) + (-1)j Hom(ddC1, idC2). ddC1 ⊗idC2 + (-1)j idC1 ⊗ddC2.
i1 : S = ZZ/101[a..c] o1 = S o1 : PolynomialRing |
i2 : C = freeResolution coker vars S
1 3 3 1
o2 = S <-- S <-- S <-- S
0 1 2 3
o2 : Complex
|
i3 : D = Hom(C,C)
1 6 15 20 15 6 1
o3 = S <-- S <-- S <-- S <-- S <-- S <-- S
-3 -2 -1 0 1 2 3
o3 : Complex
|
i4 : dd^D
1 6
o4 = -3 : S <---------------------------- S : -2
{-3} | c -b a -a -b -c |
6 15
-2 : S <-------------------------------------------------- S : -1
{-2} | -b a 0 a b c 0 0 0 0 0 0 0 0 0 |
{-2} | -c 0 a 0 0 0 a b c 0 0 0 0 0 0 |
{-2} | 0 -c b 0 0 0 0 0 0 a b c 0 0 0 |
{-2} | 0 0 0 c 0 0 -b 0 0 a 0 0 -b -c 0 |
{-2} | 0 0 0 0 c 0 0 -b 0 0 a 0 a 0 -c |
{-2} | 0 0 0 0 0 c 0 0 -b 0 0 a 0 a b |
15 20
-1 : S <----------------------------------------------------------------------- S : 0
{-1} | a -a -b -c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-1} | b 0 0 0 -a -b -c 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-1} | c 0 0 0 0 0 0 -a -b -c 0 0 0 0 0 0 0 0 0 0 |
{-1} | 0 -b 0 0 a 0 0 0 0 0 b c 0 0 0 0 0 0 0 0 |
{-1} | 0 0 -b 0 0 a 0 0 0 0 -a 0 c 0 0 0 0 0 0 0 |
{-1} | 0 0 0 -b 0 0 a 0 0 0 0 -a -b 0 0 0 0 0 0 0 |
{-1} | 0 -c 0 0 0 0 0 a 0 0 0 0 0 b c 0 0 0 0 0 |
{-1} | 0 0 -c 0 0 0 0 0 a 0 0 0 0 -a 0 c 0 0 0 0 |
{-1} | 0 0 0 -c 0 0 0 0 0 a 0 0 0 0 -a -b 0 0 0 0 |
{-1} | 0 0 0 0 -c 0 0 b 0 0 0 0 0 0 0 0 b c 0 0 |
{-1} | 0 0 0 0 0 -c 0 0 b 0 0 0 0 0 0 0 -a 0 c 0 |
{-1} | 0 0 0 0 0 0 -c 0 0 b 0 0 0 0 0 0 0 -a -b 0 |
{-1} | 0 0 0 0 0 0 0 0 0 0 c 0 0 -b 0 0 a 0 0 -c |
{-1} | 0 0 0 0 0 0 0 0 0 0 0 c 0 0 -b 0 0 a 0 b |
{-1} | 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 -b 0 0 a -a |
20 15
0 : S <------------------------------------------------- S : 1
| a b c 0 0 0 0 0 0 0 0 0 0 0 0 |
| a 0 0 -b -c 0 0 0 0 0 0 0 0 0 0 |
| 0 a 0 a 0 -c 0 0 0 0 0 0 0 0 0 |
| 0 0 a 0 a b 0 0 0 0 0 0 0 0 0 |
| b 0 0 0 0 0 -b -c 0 0 0 0 0 0 0 |
| 0 b 0 0 0 0 a 0 -c 0 0 0 0 0 0 |
| 0 0 b 0 0 0 0 a b 0 0 0 0 0 0 |
| c 0 0 0 0 0 0 0 0 -b -c 0 0 0 0 |
| 0 c 0 0 0 0 0 0 0 a 0 -c 0 0 0 |
| 0 0 c 0 0 0 0 0 0 0 a b 0 0 0 |
| 0 0 0 -b 0 0 a 0 0 0 0 0 c 0 0 |
| 0 0 0 0 -b 0 0 a 0 0 0 0 -b 0 0 |
| 0 0 0 0 0 -b 0 0 a 0 0 0 a 0 0 |
| 0 0 0 -c 0 0 0 0 0 a 0 0 0 c 0 |
| 0 0 0 0 -c 0 0 0 0 0 a 0 0 -b 0 |
| 0 0 0 0 0 -c 0 0 0 0 0 a 0 a 0 |
| 0 0 0 0 0 0 -c 0 0 b 0 0 0 0 c |
| 0 0 0 0 0 0 0 -c 0 0 b 0 0 0 -b |
| 0 0 0 0 0 0 0 0 -c 0 0 b 0 0 a |
| 0 0 0 0 0 0 0 0 0 0 0 0 c -b a |
15 6
1 : S <----------------------------- S : 2
{1} | b c 0 0 0 0 |
{1} | -a 0 c 0 0 0 |
{1} | 0 -a -b 0 0 0 |
{1} | a 0 0 -c 0 0 |
{1} | 0 a 0 b 0 0 |
{1} | 0 0 a -a 0 0 |
{1} | b 0 0 0 -c 0 |
{1} | 0 b 0 0 b 0 |
{1} | 0 0 b 0 -a 0 |
{1} | c 0 0 0 0 -c |
{1} | 0 c 0 0 0 b |
{1} | 0 0 c 0 0 -a |
{1} | 0 0 0 -b a 0 |
{1} | 0 0 0 -c 0 a |
{1} | 0 0 0 0 -c b |
6 1
2 : S <-------------- S : 3
{2} | c |
{2} | -b |
{2} | a |
{2} | a |
{2} | b |
{2} | c |
o4 : ComplexMap
|
The homology of this complex is Hom(C, ZZ/101)
i5 : prune HH D == Hom(C, coker vars S) o5 = true |
If one of the arguments is a module, it is considered as a complex concentrated in homological degree 0.
i6 : E = Hom(C, S^1)
1 3 3 1
o6 = S <-- S <-- S <-- S
-3 -2 -1 0
o6 : Complex
|
i7 : prune HH E
o7 = cokernel {-3} | c b a |
-3
o7 : Complex
|