The direct sum of two complexes is another complex in the same category.
i1 : S = ZZ/101[a,b,c]; |
i2 : C1 = freeResolution coker vars S 1 3 3 1 o2 = S <-- S <-- S <-- S 0 1 2 3 o2 : Complex |
i3 : C1 ++ complex(S^13)[-2] 1 3 16 1 o3 = S <-- S <-- S <-- S 0 1 2 3 o3 : Complex |
i4 : C2 = complex (ideal(a,b,c)) o4 = image | a b c | 0 o4 : Complex |
i5 : C1 ++ C2 3 3 1 o5 = image | 1 0 0 0 | <-- S <-- S <-- S | 0 a b c | 1 2 3 0 o5 : Complex |
The direct sum of a sequence of complexes can be computed as follows.
i6 : C3 = directSum(C1,C2,C2[-2]) 3 1 o6 = image | 1 0 0 0 | <-- S <-- image {2} | 1 0 0 0 0 0 | <-- S | 0 a b c | {2} | 0 1 0 0 0 0 | 1 {2} | 0 0 1 0 0 0 | 3 0 {0} | 0 0 0 a b c | 2 o6 : Complex |
The direct sum is an n-ary operator with projection and inclusion maps from each component satisfying appropriate identities.
i7 : C4 = directSum(first => C1, second => C2) 3 3 1 o7 = image | 1 0 0 0 | <-- S <-- S <-- S | 0 a b c | 1 2 3 0 o7 : Complex |
i8 : C4_[first] -- inclusion map C1 --> C4 1 o8 = 0 : image | 1 0 0 0 | <------------- S : 0 | 0 a b c | {0} | 1 | {1} | 0 | {1} | 0 | {1} | 0 | 3 3 1 : S <----------------- S : 1 {1} | 1 0 0 | {1} | 0 1 0 | {1} | 0 0 1 | 3 3 2 : S <----------------- S : 2 {2} | 1 0 0 | {2} | 0 1 0 | {2} | 0 0 1 | 1 1 3 : S <------------- S : 3 {3} | 1 | o8 : ComplexMap |
i9 : C4^[first] -- projection map C4 --> C1 1 o9 = 0 : S <--------------- image | 1 0 0 0 | : 0 | 1 0 0 0 | | 0 a b c | 3 3 1 : S <----------------- S : 1 {1} | 1 0 0 | {1} | 0 1 0 | {1} | 0 0 1 | 3 3 2 : S <----------------- S : 2 {2} | 1 0 0 | {2} | 0 1 0 | {2} | 0 0 1 | 1 1 3 : S <------------- S : 3 {3} | 1 | o9 : ComplexMap |
i10 : C4^[first] * C4_[first] == 1 o10 = true |
i11 : C4^[second] * C4_[second] == 1 o11 = true |
i12 : C4^[first] * C4_[second] == 0 o12 = true |
i13 : C4^[second] * C4_[first] == 0 o13 = true |
i14 : C4_[first] * C4^[first] + C4_[second] * C4^[second] == 1 o14 = true |
Given a complex which is a direct sum, we obtain the component complexes and their names (indices) as follows.
i15 : components C4 1 3 3 1 o15 = {S <-- S <-- S <-- S , image | a b c |} 0 1 2 3 0 o15 : List |
i16 : indices C4 o16 = {first, second} o16 : List |