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 |