A map of chain complexes $f : C \to D$ of degree $d$ is a sequence of maps $f_i : C_i \to D_{d+i}$. No relationship is required between these maps and the differentials in the source and target.
This routine checks that $C$ and $D$ are well-defined chain complexes, and that, for each $f_i$, the source and target equal $C_i$ and $D_{d+i}$, respectively. If the variable debugLevel is set to a value greater than zero, then information about the nature of any failure is displayed.
Unlike the corresponding function for Complexes, the basic constructors for complex maps are all but assured to be well defined. The only case that could cause a problem is if one constructs the source or target complex, and those are not well defined.
i1 : R = ZZ/101[a,b,c]; |
i2 : C = freeResolution coker matrix{{a^2-b^2,b^3-c^3,c^4}}
1 3 3 1
o2 = R <-- R <-- R <-- R
0 1 2 3
o2 : Complex
|
i3 : D = freeResolution coker vars R
1 3 3 1
o3 = R <-- R <-- R <-- R
0 1 2 3
o3 : Complex
|
i4 : H = hashTable { 0 => map(D_0, C_0, 1),
1 => map(D_1, C_1, {{a, 0, 0}, {-b, b^2, 0}, {0, -c^2, c^3}}),
2 => map(D_2, C_2, {{a*b^2, 0, 0}, {-a*c^2, a*c^3, 0}, {b*c^2, -b*c^3, b^2*c^3}}),
3 => map(D_3, C_3, {{a*b^2*c^3}})
}
o4 = HashTable{0 => | 1 | }
1 => {1} | a 0 0 |
{1} | -b b2 0 |
{1} | 0 -c2 c3 |
2 => {2} | ab2 0 0 |
{2} | -ac2 ac3 0 |
{2} | bc2 -bc3 b2c3 |
3 => {3} | ab2c3 |
o4 : HashTable
|
i5 : f = map(D, C, H)
1 1
o5 = 0 : R <--------- R : 0
| 1 |
3 3
1 : R <--------------------- R : 1
{1} | a 0 0 |
{1} | -b b2 0 |
{1} | 0 -c2 c3 |
3 3
2 : R <-------------------------- R : 2
{2} | ab2 0 0 |
{2} | -ac2 ac3 0 |
{2} | bc2 -bc3 b2c3 |
1 1
3 : R <----------------- R : 3
{3} | ab2c3 |
o5 : ComplexMap
|
i6 : assert isWellDefined f |
i7 : assert isHomogeneous f |
i8 : assert(degree f == 0) |
i9 : assert isComplexMorphism f |
We construct two random maps of chain complexes, and check to see that, as should be the case, both are well defined.
i10 : g = randomComplexMap(D,C)
1 1
o10 = 0 : R <---------- R : 0
| 24 |
3 3
1 : R <---------------------------------------------------------------------------------------------------------------- R : 1
{1} | -36a-30b-29c -29a2-24ab-16b2-38ac+39bc+21c2 16a3+22a2b-34ab2+47b3+45a2c-48abc+19b2c-47ac2-16bc2+7c3 |
{1} | 19a+19b-10c 34a2+19ab-39b2-47ac-18bc-13c2 15a3-23a2b+43ab2+48b3+39a2c-17abc+36b2c-11ac2+35bc2+11c3 |
{1} | -29a-8b-22c -43a2-15ab-47b2-28ac+38bc+2c2 -38a3+33a2b+11ab2+b3+40a2c+46abc-3b2c-28ac2+22bc2-47c3 |
3 3
2 : R <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 2
{2} | -23a3-7a2b+29ab2-37b3+2a2c-47abc-13b2c+15ac2-10bc2+30c3 44a4-39a3b+9a2b2+13ab3-11b4+36a3c-39a2bc-26ab2c-8b3c+4a2c2+22abc2+43b2c2-49ac3-8bc3+36c4 -29a5-48a4b-37a3b2+28a2b3+40ab4+13b5+30a4c+47a3bc-18a2b2c-22ab3c-17b4c-49a3c2+46a2bc2+10ab2c2-13b3c2+a2c3+7abc3+3b2c3+30ac4-41bc4+8c5 |
{2} | -18a3+39a2b-22ab2-32b3+27a2c+32abc-20b2c-9ac2+24bc2-30c3 -3a4-22a3b+41a2b2-6ab3+6b4-30a3c+16a2bc+35ab2c+40b3c-28a2c2-9abc2+3b2c2-35ac3-31bc3+25c4 8a5-29a4b-46a3b2+42a2b3+18ab4+27b5+30a4c+49a3bc+23a2b2c-16ab3c-21b4c-18a3c2-28a2bc2-46ab2c2+23b3c2+15a2c3+12abc3-37b2c3-18ac4-23bc4+44c5 |
{2} | -48a3-15a2b-33b3+39a2c+33abc-19b2c-49ac2+17bc2-20c3 -2a4-41a3b-13a2b2-47ab3-35b4-49a3c+4a2bc+27ab2c-31b3c+30a2c2-40abc2-39b2c2+37ac3-31bc3-48c4 -39a5+20a4b+47a2b3-33ab4-37b5+19a4c-47a3bc-28a2b2c+28ab3c-33b4c-28a3c2+6a2bc2-29ab2c2-28b3c2-9a2c3+26abc3+42b2c3+5ac4+44bc4+30c5 |
1 1
3 : R <-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 3
{3} | 4a6+22a5b-20a4b2+15a3b3+9a2b4+16ab5-30b6+5a5c-13a4bc-4a3b2c-2a2b3c+10ab4c-4b5c-29a4c2+12a3bc2+20a2b2c2+31ab3c2-14b4c2+3a3c3-26a2bc3+28ab2c3-33b3c3+33a2c4-6abc4-42b2c4+21ac5-44bc5-5c6 |
o10 : ComplexMap
|
i11 : assert isWellDefined g |
i12 : assert not isCommutative g |
i13 : h = randomComplexMap(D,C, Cycle => true)
1 1
o13 = 0 : R <----------- R : 0
| -17 |
3 3
1 : R <------------------------------------------------------------------------------------------------------------- R : 1
{1} | -17a-29b-44c -49ab-b2+48ac-50bc-40c2 33a2b-9ab2+46b3-43a2c+35abc+27b2c+50ac2+45bc2+49c3 |
{1} | 29a+17b-42c 49a2+ab-17b2+11ac+19bc-45c2 -33a3+9a2b-46ab2-41a2c-29abc-36b2c-47ac2+40bc2-45c3 |
{1} | 44a+42b -48a2+39ab-19b2+40ac+45bc+17c2 43a3+6a2b+2ab2+36b3-50a2c+2abc-40b2c-49ac2+45bc2-17c3 |
3 3
2 : R <---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 2
{2} | 49a3+a2b+35ab2-30b3-a2c+37abc+48b2c+34ac2+16bc2-2c3 -33a4+9a3b-13a2b2-9ab3+46b4+21a3c-40a2bc-ab2c-7b3c-8a2c2-42abc2-44b2c2-49ac3+26bc3+44c4 -33a2b3+9ab4-46b5-16a4c-35a3bc-4a2b2c-7ab3c+2b4c-39a3c2-24a2bc2-23ab2c2-19b3c2+a2c3-27abc3+b2c3-15ac4+15bc4-39c5 |
{2} | -48a3-50a2b+11ab2-41b3+40a2c-34abc+45b2c+17ac2+31bc2+44c3 43a4+45a3b-30a2b2+36ab3+34b4-50a3c-37a2bc-9ab2c-12b3c-49a2c2+49abc2+23b2c2-17ac3+28bc3-44c4 16a4b+35a3b2+47a2b3-28ab4-29b5+39a3bc+24a2b2c-27ab3c-26b4c+32a2bc2+18ab2c2-4b3c2-43a2c3+abc3+11b2c3-3ac4+34bc4+9c5 |
{2} | -12a3+18a2b-42ab2-23b3-22a2c+16abc-45b2c-31ac2-17bc2+42c3 -39a4-11a3b-6a2b2-36ab3-36b4+39a3c+19a2bc+10ab2c+40b3c-4a2c2+26abc2-45b2c2-28ac3+17bc3-42c4 -16a5-35a4b-4a3b2+34a2b3+31ab4+36b5-39a4c-24a3bc-23a2b2c+28ab3c-40b4c-32a3c2-18a2bc2-45ab2c2+45b3c2-7a2c3-13abc3+48b2c3+26ac4-42bc4+11c5 |
1 1
3 : R <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 3
{3} | -16a6-35a5b+12a4b2+7a3b3+46a2b4+8ab5+5b6-39a5c-24a4bc+16a3b2c+13a2b3c-36ab4c-38b5c-32a4c2-18a3bc2-13a2b2c2-34ab3c2+19b4c2-46a3c3-24a2bc3+49ab2c3+5b3c3-48a2c4-5abc4+43b2c4-15ac5+42bc5+42c6 |
o13 : ComplexMap
|
i14 : assert isWellDefined h |
i15 : assert isComplexMorphism h |
This method also checks the following aspects of the data structure: