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isWellDefined(SimplicialModuleMap) -- whether a map of simplicial modules is well-defined

Description

A map of simplicial modules $f : C \to D$ 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 face/degeneracy maps in the source and target.

This routine checks that $C$ and $D$ are well-defined simplicial modules, 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 SimplicialModules, the basic constructors for simplicial module 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 = simplicialModule(F = freeResolution coker matrix{{a^2-b^2,b^3-c^3,c^4}}, Degeneracy => true)

      1      4      10      20
o2 = R  <-- R  <-- R   <-- R  <-- ...
                            
     0      1      2       3

o2 : SimplicialModule
 
i3 : D = simplicialModule(G = freeResolution coker vars R, Degeneracy => true)

      1      4      10      20
o3 = R  <-- R  <-- R   <-- R  <-- ...
                            
     0      1      2       3

o3 : SimplicialModule
 
i4 : H = hashTable { 0 => map(D_0, C_0, 1),
         1 => map(D_1, C_1, {{1, 0, 0, 0}, {0, a, 0, 0}, {0, -b, b^2, 0}, {0, 0, -c^2, c^3}}),
         2 => map(D_2, C_2, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0},
                              {0, a, 0, 0, 0, 0,0, 0, 0, 0},
                              {0, -b, b^2, 0, 0, 0, 0, 0, 0, 0},
                              {0, 0,-c^2, c^3, 0, 0, 0, 0, 0, 0},
                              {0, 0, 0, 0, a, 0, 0, 0, 0,0},
                              {0, 0, 0, 0, -b, b^2, 0, 0, 0, 0},
                              {0, 0, 0, 0, 0,-c^2, c^3, 0, 0, 0},
                              {0, 0, 0, 0, 0, 0, 0, a*b^2, 0, 0},
                              {0, 0, 0, 0, 0, 0, 0, -a*c^2, a*c^3, 0},
                              {0, 0, 0, 0, 0, 0, 0, b*c^2, -b*c^3, b^2*c^3}}),
         3 => map(D_3, C_3, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0},
                             {0, a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0},
                             {0, -b, b^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0},
                             {0, 0, -c^2, c^3, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
                             {0, 0, 0, 0, a, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
                             {0, 0, 0, 0, -b, b^2, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
                             {0, 0, 0, 0, 0,-c^2, c^3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
                             {0, 0,0, 0, 0, 0, 0, a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
                             {0,0, 0, 0, 0, 0, 0, -b, b^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0},
                             {0, 0, 0, 0, 0, 0, 0, 0, -c^2, c^3, 0, 0, 0, 0, 0, 0,0, 0, 0, 0},
                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a*b^2, 0, 0,0, 0, 0, 0, 0, 0, 0},
                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-a*c^2, a*c^3, 0, 0, 0, 0, 0, 0, 0,0},
                             {0, 0, 0, 0, 0, 0,0, 0, 0, 0, b*c^2, -b*c^3, b^2*c^3, 0, 0, 0, 0, 0, 0, 0},
                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a*b^2, 0, 0, 0, 0,0, 0},
                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -a*c^2,a*c^3, 0, 0, 0, 0, 0},
                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, b*c^2, -b*c^3, b^2*c^3, 0, 0, 0, 0},
                             {0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a*b^2, 0, 0, 0},
                             {0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -a*c^2, a*c^3,0, 0},
                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, b*c^2, -b*c^3, b^2*c^3, 0},
                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, a*b^2*c^3}})
                     }

o4 = HashTable{0 => | 1 |                                                                                     }
               1 => {0} | 1 0  0   0  |
                    {1} | 0 a  0   0  |
                    {1} | 0 -b b2  0  |
                    {1} | 0 0  -c2 c3 |
               2 => {0} | 1 0  0   0  0  0   0  0    0    0    |
                    {1} | 0 a  0   0  0  0   0  0    0    0    |
                    {1} | 0 -b b2  0  0  0   0  0    0    0    |
                    {1} | 0 0  -c2 c3 0  0   0  0    0    0    |
                    {1} | 0 0  0   0  a  0   0  0    0    0    |
                    {1} | 0 0  0   0  -b b2  0  0    0    0    |
                    {1} | 0 0  0   0  0  -c2 c3 0    0    0    |
                    {2} | 0 0  0   0  0  0   0  ab2  0    0    |
                    {2} | 0 0  0   0  0  0   0  -ac2 ac3  0    |
                    {2} | 0 0  0   0  0  0   0  bc2  -bc3 b2c3 |
               3 => {0} | 1 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 a  0   0  0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 -b b2  0  0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 0  -c2 c3 0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 0  0   0  a  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 0  0   0  -b b2  0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 0  0   0  0  -c2 c3 0  0   0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 0  0   0  0  0   0  a  0   0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 0  0   0  0  0   0  -b b2  0  0    0    0    0    0    0    0    0    0    0     |
                    {1} | 0 0  0   0  0  0   0  0  -c2 c3 0    0    0    0    0    0    0    0    0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  ab2  0    0    0    0    0    0    0    0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  -ac2 ac3  0    0    0    0    0    0    0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  bc2  -bc3 b2c3 0    0    0    0    0    0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    ab2  0    0    0    0    0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    -ac2 ac3  0    0    0    0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    bc2  -bc3 b2c3 0    0    0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    ab2  0    0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    -ac2 ac3  0    0     |
                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    bc2  -bc3 b2c3 0     |
                    {3} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    ab2c3 |

o4 : HashTable
 
i5 : f = map(D, C, H)

          1             1
o5 = 0 : R  <--------- R  : 0
               | 1 |

          4                           4
     1 : R  <----------------------- R  : 1
               {0} | 1 0  0   0  |
               {1} | 0 a  0   0  |
               {1} | 0 -b b2  0  |
               {1} | 0 0  -c2 c3 |

          10                                                    10
     2 : R   <------------------------------------------------ R   : 2
                {0} | 1 0  0   0  0  0   0  0    0    0    |
                {1} | 0 a  0   0  0  0   0  0    0    0    |
                {1} | 0 -b b2  0  0  0   0  0    0    0    |
                {1} | 0 0  -c2 c3 0  0   0  0    0    0    |
                {1} | 0 0  0   0  a  0   0  0    0    0    |
                {1} | 0 0  0   0  -b b2  0  0    0    0    |
                {1} | 0 0  0   0  0  -c2 c3 0    0    0    |
                {2} | 0 0  0   0  0  0   0  ab2  0    0    |
                {2} | 0 0  0   0  0  0   0  -ac2 ac3  0    |
                {2} | 0 0  0   0  0  0   0  bc2  -bc3 b2c3 |

          20                                                                                                  20
     3 : R   <---------------------------------------------------------------------------------------------- R   : 3
                {0} | 1 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 a  0   0  0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 -b b2  0  0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 0  -c2 c3 0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 0  0   0  a  0   0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 0  0   0  -b b2  0  0  0   0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 0  0   0  0  -c2 c3 0  0   0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 0  0   0  0  0   0  a  0   0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 0  0   0  0  0   0  -b b2  0  0    0    0    0    0    0    0    0    0    0     |
                {1} | 0 0  0   0  0  0   0  0  -c2 c3 0    0    0    0    0    0    0    0    0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  ab2  0    0    0    0    0    0    0    0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  -ac2 ac3  0    0    0    0    0    0    0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  bc2  -bc3 b2c3 0    0    0    0    0    0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    ab2  0    0    0    0    0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    -ac2 ac3  0    0    0    0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    bc2  -bc3 b2c3 0    0    0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    ab2  0    0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    -ac2 ac3  0    0     |
                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    bc2  -bc3 b2c3 0     |
                {3} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0    0    0    0    0    ab2c3 |

o5 : SimplicialModuleMap
 
i6 : assert isWellDefined f
 
i7 : assert isHomogeneous f
 
i8 : assert(degree f == 0)
 
i9 : assert isSimplicialMorphism f

We construct two random maps of simplicial modules, and check to see that, as should be the case, both are well defined.

i10 : g = randomSimplicialMap(D,C); --outputs are large
 
i11 : normalize g

           1              1
o11 = 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 |

o11 : ComplexMap
 
i12 : assert isWellDefined g
 
i13 : assert not isCommutative g
 
i14 : h = randomSimplicialMap(D,C, Cycle => true);
 
i15 : normalize h

           1               1
o15 = 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 |

o15 : ComplexMap
 
i16 : assert isWellDefined h
 
i17 : assert isSimplicialMorphism 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 simplicial modules
  • The degree is an integer
  • All keys in the map field are integers or sequences, in the range of the top degree 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 face/degeneracy maps is checked

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:1691:0.