isWellDefined(ComplexMap) -- whether a map of chain complexes is well-defined

Synopsis

Function: isWellDefined
Usage:

isWellDefined f
Inputs:
- f, a map of complexes,
Outputs:
- a Boolean value, that is true when f determines a well defined complex map

Description

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:

The underlying hash table has exactly the expected keys, namely, source, target, degree, map, cache
The ring of the source and target are the same
The source and target are well defined complexes
The degree is an integer
All keys in the map field are integers, in the range of the concentration of the source
The source and target of each $f_i$ is as expected
If the isCommutative key is present in the cache table, then commutativity of the map with the differentials is checked

Ways to use this method: