Description
A random simplicial module map $f : C \to D$ is obtained from choosing a random map of the underlying normalizations, which uses the randomComplexMap command.
i1 : S = ZZ/101[a..c]
o1 = S
o1 : PolynomialRing
|
i2 : C = simplicialModule(freeResolution coker matrix{{a*b, a*c, b*c}}, 3, Degeneracy => true)
1 4 9 16
o2 = S <-- S <-- S <-- S <-- ...
0 1 2 3
o2 : SimplicialModule
|
i3 : D = simplicialModule(freeResolution coker vars S, Degeneracy => true)
1 4 10 20
o3 = S <-- S <-- S <-- S <-- ...
0 1 2 3
o3 : SimplicialModule
|
i4 : f = randomSimplicialMap(D,C)
1 1
o4 = 0 : S <---------- S : 0
| 24 |
4 4
1 : S <----------------------------------------------------- S : 1
{0} | 24 0 0 0 |
{1} | 0 -36a-30b-29c -29a-24b-38c -39a-18b-13c |
{1} | 0 19a+19b-10c -16a+39b+21c -43a-15b-28c |
{1} | 0 -29a-8b-22c 34a+19b-47c -47a+38b+2c |
10 9
2 : S <---------------------------------------------------------------------------------------------------------------------- S : 2
{0} | 24 0 0 0 0 0 0 0 0 |
{1} | 0 -36a-30b-29c -29a-24b-38c -39a-18b-13c 0 0 0 0 0 |
{1} | 0 19a+19b-10c -16a+39b+21c -43a-15b-28c 0 0 0 0 0 |
{1} | 0 -29a-8b-22c 34a+19b-47c -47a+38b+2c 0 0 0 0 0 |
{1} | 0 0 0 0 -36a-30b-29c -29a-24b-38c -39a-18b-13c 0 0 |
{1} | 0 0 0 0 19a+19b-10c -16a+39b+21c -43a-15b-28c 0 0 |
{1} | 0 0 0 0 -29a-8b-22c 34a+19b-47c -47a+38b+2c 0 0 |
{2} | 0 0 0 0 0 0 0 16a+22b+45c 7a+15b-23c |
{2} | 0 0 0 0 0 0 0 -34a-48b-47c 39a+43b-17c |
{2} | 0 0 0 0 0 0 0 47a+19b-16c -11a+48b+36c |
20 16
3 : S <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S : 3
{0} | 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 -36a-30b-29c -29a-24b-38c -39a-18b-13c 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 19a+19b-10c -16a+39b+21c -43a-15b-28c 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 -29a-8b-22c 34a+19b-47c -47a+38b+2c 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 -36a-30b-29c -29a-24b-38c -39a-18b-13c 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 19a+19b-10c -16a+39b+21c -43a-15b-28c 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 -29a-8b-22c 34a+19b-47c -47a+38b+2c 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 -36a-30b-29c -29a-24b-38c -39a-18b-13c 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 19a+19b-10c -16a+39b+21c -43a-15b-28c 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 -29a-8b-22c 34a+19b-47c -47a+38b+2c 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 16a+22b+45c 7a+15b-23c 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 -34a-48b-47c 39a+43b-17c 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 47a+19b-16c -11a+48b+36c 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 16a+22b+45c 7a+15b-23c 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 -34a-48b-47c 39a+43b-17c 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 47a+19b-16c -11a+48b+36c 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16a+22b+45c 7a+15b-23c |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -34a-48b-47c 39a+43b-17c |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 47a+19b-16c -11a+48b+36c |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
o4 : SimplicialModuleMap
|
i5 : assert isWellDefined f
|
i6 : assert not isCommutative f
|
When the random element is chosen with option Cycle => true, the associated map of simplicial modules commutes with the face/degeneracy map.
i7 : g = randomSimplicialMap(D,C, Cycle => true)
1 1
o7 = 0 : S <----------- S : 0
| -50 |
4 4
1 : S <----------------------------------------- S : 1
{0} | -50 0 0 0 |
{1} | 0 40b-35c -46b+18c 28b+3c |
{1} | 0 11a-11c 46a+22c -28a-37c |
{1} | 0 35a+11b 33a-22b -3a-13b |
10 9
2 : S <--------------------------------------------------------------------------------------------- S : 2
{0} | -50 0 0 0 0 0 0 0 0 |
{1} | 0 40b-35c -46b+18c 28b+3c 0 0 0 0 0 |
{1} | 0 11a-11c 46a+22c -28a-37c 0 0 0 0 0 |
{1} | 0 35a+11b 33a-22b -3a-13b 0 0 0 0 0 |
{1} | 0 0 0 0 40b-35c -46b+18c 28b+3c 0 0 |
{1} | 0 0 0 0 11a-11c 46a+22c -28a-37c 0 0 |
{1} | 0 0 0 0 35a+11b 33a-22b -3a-13b 0 0 |
{2} | 0 0 0 0 0 0 0 46b-49c -28a-46b+17c |
{2} | 0 0 0 0 0 0 0 -30b-35c -3a+b |
{2} | 0 0 0 0 0 0 0 -38a-22b-11c -47a+22b |
20 16
3 : S <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S : 3
{0} | -50 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 40b-35c -46b+18c 28b+3c 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 11a-11c 46a+22c -28a-37c 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 35a+11b 33a-22b -3a-13b 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 40b-35c -46b+18c 28b+3c 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 11a-11c 46a+22c -28a-37c 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 35a+11b 33a-22b -3a-13b 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 40b-35c -46b+18c 28b+3c 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 11a-11c 46a+22c -28a-37c 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 35a+11b 33a-22b -3a-13b 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 46b-49c -28a-46b+17c 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 -30b-35c -3a+b 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 -38a-22b-11c -47a+22b 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 46b-49c -28a-46b+17c 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 -30b-35c -3a+b 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 -38a-22b-11c -47a+22b 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 46b-49c -28a-46b+17c |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -30b-35c -3a+b |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -38a-22b-11c -47a+22b |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
o7 : SimplicialModuleMap
|
i8 : assert isWellDefined g
|
i9 : assert isCommutative g
|
i10 : assert isSimplicialMorphism g
|
When the random element is chosen with option Boundary => true, the associated map of simplicial modules is a simplicial null homotopy.
i11 : h = randomSimplicialMap(D,C, Boundary => true)
1 1
o11 = 0 : S <----- S : 0
0
4 4
1 : S <--------------------------------------- S : 1
{0} | 0 0 0 0 |
{1} | 0 23b+7c -29b+47c 37b+13c |
{1} | 0 -23a-2c 29a-15c -37a+10c |
{1} | 0 -7a+2b -47a+15b -13a-10b |
10 9
2 : S <----------------------------------------------------------------------------------------- S : 2
{0} | 0 0 0 0 0 0 0 0 0 |
{1} | 0 23b+7c -29b+47c 37b+13c 0 0 0 0 0 |
{1} | 0 -23a-2c 29a-15c -37a+10c 0 0 0 0 0 |
{1} | 0 -7a+2b -47a+15b -13a-10b 0 0 0 0 0 |
{1} | 0 0 0 0 23b+7c -29b+47c 37b+13c 0 0 |
{1} | 0 0 0 0 -23a-2c 29a-15c -37a+10c 0 0 |
{1} | 0 0 0 0 -7a+2b -47a+15b -13a-10b 0 0 |
{2} | 0 0 0 0 0 0 0 29b-48c -37a-29b-18c |
{2} | 0 0 0 0 0 0 0 24b+7c -13a-36b |
{2} | 0 0 0 0 0 0 0 30a+15b-2c -28a-15b |
20 16
3 : S <------------------------------------------------------------------------------------------------------------------------------------------------------------------- S : 3
{0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 23b+7c -29b+47c 37b+13c 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 -23a-2c 29a-15c -37a+10c 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 -7a+2b -47a+15b -13a-10b 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 23b+7c -29b+47c 37b+13c 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 -23a-2c 29a-15c -37a+10c 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 -7a+2b -47a+15b -13a-10b 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 23b+7c -29b+47c 37b+13c 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 -23a-2c 29a-15c -37a+10c 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 -7a+2b -47a+15b -13a-10b 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 29b-48c -37a-29b-18c 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 24b+7c -13a-36b 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 30a+15b-2c -28a-15b 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 29b-48c -37a-29b-18c 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 24b+7c -13a-36b 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 30a+15b-2c -28a-15b 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 29b-48c -37a-29b-18c |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24b+7c -13a-36b |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30a+15b-2c -28a-15b |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
o11 : SimplicialModuleMap
|
i12 : assert isWellDefined h
|
i13 : assert isCommutative h
|
i14 : assert isSimplicialMorphism h
|
i15 : assert isNullHomotopic normalize h
|
i16 : nullHomotopy normalize h
3 1
o16 = 1 : S <----- S : 0
0
3 3
2 : S <----------------------- S : 1
{2} | -23 29 -37 |
{2} | -7 -47 -13 |
{2} | 2 15 -10 |
1 2
3 : S <------------------ S : 2
{3} | 30 -18 |
o16 : ComplexMap
|
When the degree of the random element is chosen with a specific degree, the associated map of simplicial modules will be a well-defined degree 0 simplicial morphism mapping to the Dold-Kan image of the shift of the normalization. Thus, even when specifying nonzero degree this constructor will still yield a simplicial morphism.
i17 : p = randomSimplicialMap(D, C, Cycle => true, Degree => -1)
1 3
o17 = 0 : S <----------------------------------------------------------------------------------------------- S : 0
| 19a2-30ab+36b2+40ac+4bc-32c2 36a2+42ab-33b2+13ac-30bc+9c2 20a2+12ab-44b2-13ac-16bc+26c2 |
4 5
1 : S <--------------------------------------------------------------------------------------------------------------------------------------------------------------- S : 1
{0} | 19a2-30ab+36b2+40ac+4bc-32c2 36a2+42ab-33b2+13ac-30bc+9c2 20a2+12ab-44b2-13ac-16bc+26c2 0 0 |
{1} | 0 0 0 -10ab-31b2-20ac+31bc-39c2 -20a2+24ab-48b2-30ac-15bc+39c2 |
{1} | 0 0 0 -26a2-11ab+33b2+34bc+4c2 33ab-33b2-49ac-19bc+17c2 |
{1} | 0 0 0 39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc |
10 7
2 : S <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S : 2
{0} | 19a2-30ab+36b2+40ac+4bc-32c2 36a2+42ab-33b2+13ac-30bc+9c2 20a2+12ab-44b2-13ac-16bc+26c2 0 0 0 0 |
{1} | 0 0 0 -10ab-31b2-20ac+31bc-39c2 -20a2+24ab-48b2-30ac-15bc+39c2 0 0 |
{1} | 0 0 0 -26a2-11ab+33b2+34bc+4c2 33ab-33b2-49ac-19bc+17c2 0 0 |
{1} | 0 0 0 39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc 0 0 |
{1} | 0 0 0 0 0 -10ab-31b2-20ac+31bc-39c2 -20a2+24ab-48b2-30ac-15bc+39c2 |
{1} | 0 0 0 0 0 -26a2-11ab+33b2+34bc+4c2 33ab-33b2-49ac-19bc+17c2 |
{1} | 0 0 0 0 0 39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc |
{2} | 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 |
20 9
3 : S <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S : 3
{0} | 19a2-30ab+36b2+40ac+4bc-32c2 36a2+42ab-33b2+13ac-30bc+9c2 20a2+12ab-44b2-13ac-16bc+26c2 0 0 0 0 0 0 |
{1} | 0 0 0 -10ab-31b2-20ac+31bc-39c2 -20a2+24ab-48b2-30ac-15bc+39c2 0 0 0 0 |
{1} | 0 0 0 -26a2-11ab+33b2+34bc+4c2 33ab-33b2-49ac-19bc+17c2 0 0 0 0 |
{1} | 0 0 0 39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc 0 0 0 0 |
{1} | 0 0 0 0 0 -10ab-31b2-20ac+31bc-39c2 -20a2+24ab-48b2-30ac-15bc+39c2 0 0 |
{1} | 0 0 0 0 0 -26a2-11ab+33b2+34bc+4c2 33ab-33b2-49ac-19bc+17c2 0 0 |
{1} | 0 0 0 0 0 39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc 0 0 |
{1} | 0 0 0 0 0 0 0 -10ab-31b2-20ac+31bc-39c2 -20a2+24ab-48b2-30ac-15bc+39c2 |
{1} | 0 0 0 0 0 0 0 -26a2-11ab+33b2+34bc+4c2 33ab-33b2-49ac-19bc+17c2 |
{1} | 0 0 0 0 0 0 0 39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
o17 : SimplicialModuleMap
|
i18 : assert isWellDefined p
|
i19 : assert isCommutative p
|
i20 : assert isSimplicialMorphism p
|
Given an internal degree, the random element is constructed as maps of modules with this degree.
i21 : q = randomSimplicialMap(D, C, Boundary => true, InternalDegree => 2);
|
i22 : assert isCommutative q
|
i23 : assert isSimplicialMorphism q
|
i24 : source q === C
o24 = false
|
i25 : target q === D
o25 = false
|
i26 : assert isNullHomotopic normalize q
|