Description
For a simplicial module map $f : C \to D$, this method checks whether, for all $i$, we have $dd^D_{i} * f_i = f_{i-1} * dd^C_i$ and, if the source at target are equipped with degeneracy maps, it also checks the equality $ss^D_{i} * f_i = f_{i+1} * ss^C_i$.
We first construct a random simplicial module map which commutes with the face/degeneracy map.
i1 : S = ZZ/101[a,b,c];
|
i2 : C = simplicialModule(freeResolution coker vars S, Degeneracy => true)
1 4 10 20
o2 = S <-- S <-- S <-- S <-- ...
0 1 2 3
o2 : SimplicialModule
|
i3 : D = C ** C
1 16 100 400
o3 = S <-- S <-- S <-- S <-- ...
0 1 2 3
o3 : SimplicialModule
|
i4 : isWellDefined D
o4 = true
|
i5 : f1 = prune randomSimplicialMap(D, C, Cycle => true, InternalDegree => 1);
|
i6 : prune normalize f1
1 1
o6 = 0 : S <------------------- S : 0
| -14a-6b+26c |
15 3
1 : S <-------------------------------------------------- S : 1
{1} | 4a-b-32c 4a+25b+10c -44a+41b+26c |
{1} | -20a+28b-27c -23a-14b+13c -22a+28b+34c |
{1} | -34a+25b-43c -2a-9b+27c 14a-12b-28c |
{1} | -40a+43b+17c -31a+48b-12c -35a+16b+29c |
{1} | -27a-13b+8c 30a+17b-8c -49a+29b+37c |
{1} | -39a-5b-23c -38a-27b-12c -11a+12b+7c |
{2} | 22 27 -22 |
{2} | 15 34 20 |
{2} | 7 -11 -50 |
{2} | -16 -27 -6 |
{2} | -15 -9 44 |
{2} | 7 10 -2 |
{2} | 6 -48 -22 |
{2} | -8 47 26 |
{2} | -35 -15 47 |
69 3
2 : S <-------------------------------------------------- S : 2
{2} | -17a-15b-41c 21a-9b+6c 11a-35b-17c |
{2} | -28a-19b+5c 11a+4b-22c -2a+4b+31c |
{2} | 28a+46b+40c 16a-9b-35c -38a-47b+10c |
{2} | -31a-31b+25c -35a-47b-17c 9b-48c |
{2} | 35a-28b-38c 50a+13b+40c -29a-5c |
{2} | -45a+41b+2c 20a+47b+29c -10a-27b+2c |
{2} | 27a+40b-48c -22a-14b+13c -28b+5c |
{2} | -46a+12b-24c -41a+6b-47c -22a-30b-25c |
{2} | 19a-9b+3c 42a+15b+2c 8a-31b-22c |
{2} | 13a+27b-33c -36a-16b-32c -15a+37b-18c |
{2} | 2a+13b+19c 32a+31c 31a+29b+45c |
{2} | 48a-31b-19c 18a+39b+13c 17a+23b+21c |
{2} | -9a-47b+12c -32a-43b+25c 28a+32b-5c |
{2} | 27a+15b-31c -20a+26c 12a+44b-25c |
{2} | -28a+42b+27c 28a+36b-12c -2a+31b-14c |
{2} | -30a-38c -46a-12b+25c -12a-25b+41c |
{2} | -31a+9b-22c -28a+44b+26c -2a-36b-25c |
{2} | 24a-37b -25a-42b+23c 7a+46b+12c |
{2} | -30a-13b-3c -48a-38b-36c -40a-2b+24c |
{2} | 5a-10b-22c 42a-47b-42c -14a+27b-38c |
{2} | -15a+30b -33a-46b+35c -44a+42b+15c |
{2} | 12a+20b+49c 28a-47b+50c 45a+48b+48c |
{2} | -17a-47b+11c -14a-45b+5c -23a-15b-5c |
{2} | 6a-45b+22c 26a-46b-14c -15a+14b-32c |
{3} | -17 -34 39 |
{3} | 0 -36 0 |
{3} | -37 23 49 |
{3} | 34 -33 26 |
{3} | 4 -48 -44 |
{3} | 41 -17 -12 |
{3} | -43 -7 5 |
{3} | 36 1 15 |
{3} | 4 24 -35 |
{3} | -21 -40 22 |
{3} | -30 9 -8 |
{3} | 26 36 -8 |
{3} | 38 -43 38 |
{3} | 14 20 -9 |
{3} | -49 -36 -33 |
{3} | -9 -49 -3 |
{3} | -3 -30 -24 |
{3} | -22 -50 9 |
{3} | 5 -41 -30 |
{3} | -5 -41 37 |
{3} | 10 14 18 |
{3} | -18 26 -28 |
{3} | 27 6 -50 |
{3} | 24 -50 11 |
{3} | 9 49 35 |
{3} | 8 10 37 |
{3} | -9 41 -5 |
{3} | -18 26 40 |
{3} | -50 33 17 |
{3} | 3 9 -33 |
{3} | -25 -40 9 |
{3} | 41 2 34 |
{3} | -41 35 45 |
{3} | -2 -11 -23 |
{3} | 18 47 -1 |
{3} | 27 -5 -4 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
147 1
3 : S <------------------------ S : 3
{3} | 24a-36b-30c |
{3} | 39a+8b+15c |
{3} | 23b-47c |
{3} | 37a-39b+5c |
{3} | -49a+33b-23c |
{3} | -19a-33b-29c |
{3} | -16a+17b+4c |
{3} | -39a+44b-20c |
{3} | -39a+9b+36c |
{3} | 47a+13b+4c |
{3} | -29a+19b+19c |
{3} | -10a-29b-8c |
{3} | -22a-29b-24c |
{3} | 22a-26b-18c |
{3} | -8a-11b-49c |
{3} | -35a-31b-48c |
{3} | -38a-16b+39c |
{3} | 21a+34b+19c |
{3} | -47a-39b-18c |
{3} | 3a+43b-22c |
{3} | 20a+33b-29c |
{3} | 10a+36b-8c |
{3} | -13a-43b-15c |
{3} | -28a-47b+38c |
{3} | 2a+16b+22c |
{3} | -33a-45b-26c |
{3} | -5a-11b-3c |
{3} | -35a+9b-22c |
{3} | 45a-34b-48c |
{3} | -47a+47b+19c |
{3} | -16a+7b+15c |
{3} | -30a-28b-49c |
{3} | 5a+41b-40c |
{3} | -13a+15b-48c |
{3} | -28a+16b+46c |
{3} | 9a-6b+10c |
{3} | -26a+16b+30c |
{3} | -23a+39b+43c |
{3} | -17a-11b+48c |
{3} | 36a+35b+11c |
{3} | 35a-49b+8c |
{3} | 21a-4b-44c |
{3} | -4a-32b-9c |
{3} | 40a+6b-35c |
{3} | -42a+18b+28c |
{3} | -16a+8b+3c |
{3} | -38a+33b+40c |
{3} | 11a+46b-28c |
{3} | a-3b+22c |
{3} | 25a-31b-45c |
{3} | -2a-2b+24c |
{3} | 7a-18b-41c |
{3} | 4a-13b-49c |
{3} | -28a-47b+30c |
{3} | 49a-7b+27c |
{3} | -47a-23b-7c |
{4} | -38 |
{4} | -27 |
{4} | -42 |
{4} | 12 |
{4} | 20 |
{4} | -23 |
{4} | 27 |
{4} | -48 |
{4} | 17 |
{4} | -26 |
{4} | -25 |
{4} | -16 |
{4} | 44 |
{4} | -47 |
{4} | 42 |
{4} | -37 |
{4} | -33 |
{4} | -37 |
{4} | 27 |
{4} | -33 |
{4} | 40 |
{4} | 35 |
{4} | -29 |
{4} | 5 |
{4} | 15 |
{4} | -34 |
{4} | 35 |
{4} | 30 |
{4} | -19 |
{4} | 27 |
{4} | 29 |
{4} | -9 |
{4} | 5 |
{4} | -7 |
{4} | -43 |
{4} | 41 |
{4} | 32 |
{4} | 31 |
{4} | -29 |
{4} | 42 |
{4} | 37 |
{4} | 34 |
{4} | 0 |
{4} | 0 |
{4} | 0 |
{4} | 16 |
{4} | -1 |
{4} | 49 |
{4} | -42 |
{4} | -17 |
{4} | -23 |
{4} | 0 |
{4} | 0 |
{4} | 0 |
{4} | 30 |
{4} | -20 |
{4} | -17 |
{4} | -24 |
{4} | -47 |
{4} | 12 |
{4} | 0 |
{4} | 0 |
{4} | 0 |
{4} | -31 |
{4} | -33 |
{4} | 41 |
{4} | -16 |
{4} | 6 |
{4} | -43 |
{4} | -27 |
{4} | -50 |
{4} | -6 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{5} | 0 |
{6} | 0 |
o6 : ComplexMap
|
i7 : isCommutative oo
o7 = true
|
i8 : isCommutative f1
o8 = true
|
i9 : assert(degree f1 == 0)
|
We next generate a simplicial module map that is commutative and (likely) induces a nontrivial map on homology.
i10 : f2 = randomSimplicialMap(D, C, Cycle => true);
|
i11 : isCommutative f2
o11 = true
|
i12 : assert(degree f2 == 0)
|
i13 : assert isSimplicialMorphism f2
|
If the debugLevel is greater than zero, then the location of the first failure of commutativity is displayed.