The complex of homomorphisms is a complex D whose ith component is the direct sum of Hom(C1j, C2j+i) over all j. The differential on Hom(C1j, C2j+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 |
i5 : assert isWellDefined D |
The homology of this complex is Hom(C, ZZ/101)
i6 : prune HH D == Hom(C, coker vars S) o6 = true |
If one of the arguments is a module, it is considered as a complex concentrated in homological degree 0.
i7 : E = Hom(C, S^2) 2 6 6 2 o7 = S <-- S <-- S <-- S -3 -2 -1 0 o7 : Complex |
i8 : prune HH E o8 = cokernel {-3} | c b a 0 0 0 | {-3} | 0 0 0 c b a | -3 o8 : Complex |
There is a simple relationship between Hom complexes and shifts. Specifically, shifting the first argument is the same as the negative shift of the result. But shifting the second argument is only the same as the positive shift of the result up to a sign.
i9 : Hom(C[3], C) == D[-3] o9 = true |
i10 : Hom(C, C[-2]) == D[-2] o10 = true |
i11 : Hom(C, C[-3]) != D[-3] o11 = true |
i12 : Hom(C, C[-3]) == complex(- dd^(D[-3])) o12 = true |
Specific maps and morphisms between complexes can be obtained with homomorphism(ComplexMap) (missing documentation).
Because the Hom complex can be regarded as the total complex of a double complex, each term comes with pairs of indices, labelling the summands.
i13 : indices D_-1 o13 = {{0, -1}, {1, 0}, {2, 1}, {3, 2}} o13 : List |
i14 : components D_-1 3 9 3 o14 = {0, S , S , S } o14 : List |
i15 : indices D_-2 o15 = {{0, -2}, {1, -1}, {2, 0}, {3, 1}} o15 : List |
i16 : components D_-2 3 3 o16 = {0, 0, S , S } o16 : List |