The direct sum is an n-ary operator with projection and inclusion maps from each component satisfying appropriate identities.
One can access these maps as follows. First, we define some non-trivial maps of chain complexes.
i1 : R = ZZ/101[a..d]; |
i2 : C1 = (freeResolution coker matrix{{a,b,c}})[1]
1 3 3 1
o2 = R <-- R <-- R <-- R
-1 0 1 2
o2 : Complex
|
i3 : C2 = freeResolution coker matrix{{a*b,a*c,b*c}}
1 3 2
o3 = R <-- R <-- R
0 1 2
o3 : Complex
|
i4 : D1 = (freeResolution coker matrix{{a,b,c}})
1 3 3 1
o4 = R <-- R <-- R <-- R
0 1 2 3
o4 : Complex
|
i5 : D2 = freeResolution coker matrix{{a^2, b^2, c^2}}[-1]
1 3 3 1
o5 = R <-- R <-- R <-- R
1 2 3 4
o5 : Complex
|
i6 : f = randomComplexMap(D1, C1, Cycle => true)
1
o6 = -1 : 0 <----- R : -1
0
1 3
0 : R <---------------------------------------------------- R : 0
| -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d |
3 3
1 : R <------------------------------------------------------------ R : 1
{1} | -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c |
{1} | 46a+17b-29c-24d -39a+39c -24a-3b-24c+29d |
{1} | -18a+21b -21a-22b-8c-24d -19a-32b+28c+29d |
3 1
2 : R <--------------------------- R : 2
{2} | 24a-36b-30c-29d |
{2} | 19a+19b-10c-29d |
{2} | -8a-22b-29c-24d |
o6 : ComplexMap
|
i7 : g = randomComplexMap(D2, C2, Cycle => true)
1
o7 = 0 : 0 <----- R : 0
0
1 3
1 : R <-------------------------------------------------------------------------------------------------------------- R : 1
| 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |
3 2
2 : R <-------------------------------------------- R : 2
{2} | -7b+19c -38a+5b-16c-22d |
{2} | -45a+34b+32c+47d 30a-34b-48c-47d |
{2} | -43a+8b-28c-47d -39a-23b |
o7 : ComplexMap
|
i8 : h = f ++ g
1
o8 = -1 : 0 <----- R : -1
0
1 4
0 : R <------------------------------------------------------ R : 0
| -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0 |
4 6
1 : R <------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 1
{1} | -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c 0 0 0 |
{1} | 46a+17b-29c-24d -39a+39c -24a-3b-24c+29d 0 0 0 |
{1} | -18a+21b -21a-22b-8c-24d -19a-32b+28c+29d 0 0 0 |
{0} | 0 0 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |
6 3
2 : R <------------------------------------------------------------ R : 2
{2} | 24a-36b-30c-29d 0 0 |
{2} | 19a+19b-10c-29d 0 0 |
{2} | -8a-22b-29c-24d 0 0 |
{2} | 0 -7b+19c -38a+5b-16c-22d |
{2} | 0 -45a+34b+32c+47d 30a-34b-48c-47d |
{2} | 0 -43a+8b-28c-47d -39a-23b |
o8 : ComplexMap
|
The four basic maps are the inclusion from each summand of the source and the projection to each summand of the target.
i9 : h_[0] == h * (C1 ++ C2)_[0] o9 = true |
i10 : h_[1] == h * (C1 ++ C2)_[1] o10 = true |
i11 : h^[0] == (D1 ++ D2)^[0] * h o11 = true |
i12 : h^[1] == (D1 ++ D2)^[1] * h o12 = true |
These can be combined to obtain the blocks of the map of chain complexes.
i13 : h_[0]^[0] == f o13 = true |
i14 : h_[1]^[1] == g o14 = true |
i15 : h_[0]^[1] == 0 o15 = true |
i16 : h_[1]^[0] == 0 o16 = true |
i17 : assert(h == map(D1 ++ D2, C1 ++ C2, {{f,0},{0,g}}))
|
The default names for the components are the non-negative integers. However, one can choose any name.
i18 : h = (mike => f) ++ (greg => g)
1
o18 = -1 : 0 <----- R : -1
0
1 4
0 : R <------------------------------------------------------ R : 0
| -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0 |
4 6
1 : R <------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 1
{1} | -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c 0 0 0 |
{1} | 46a+17b-29c-24d -39a+39c -24a-3b-24c+29d 0 0 0 |
{1} | -18a+21b -21a-22b-8c-24d -19a-32b+28c+29d 0 0 0 |
{0} | 0 0 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |
6 3
2 : R <------------------------------------------------------------ R : 2
{2} | 24a-36b-30c-29d 0 0 |
{2} | 19a+19b-10c-29d 0 0 |
{2} | -8a-22b-29c-24d 0 0 |
{2} | 0 -7b+19c -38a+5b-16c-22d |
{2} | 0 -45a+34b+32c+47d 30a-34b-48c-47d |
{2} | 0 -43a+8b-28c-47d -39a-23b |
o18 : ComplexMap
|
i19 : h_[mike]^[mike] == f o19 = true |
i20 : h_[greg]^[greg] == g o20 = true |