The maps $f : C \to D$ and $g : E \to F$ of ZZ/d-graded factorizations induces the map $h = Hom(f,g) : Hom(D,E) \to Hom(C,F)$ defined by $\phi \mapsto g \phi f$.
i1 : S = ZZ/101[a,b];
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i2 : R = S/(a^3+b^3);
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i3 : m = ideal vars R
o3 = ideal (a, b)
o3 : Ideal of R
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i4 : C = tailMF m
2 2 2
o4 = S <-- S <-- S
0 1 0
o4 : ZZdFactorization
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i5 : D = tailMF (m^2)
3 3 3
o5 = S <-- S <-- S
0 1 0
o5 : ZZdFactorization
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i6 : f = randomFactorizationMap(C,D)
2 3
o6 = 0 : S <------------------------------------- S : 0
{2} | 24a-36b -29a+19b -10a-29b |
{3} | -30 19 -8 |
2 3
1 : S <-------------------------------------- S : 1
{4} | -22a-29b -16a+39b 19a-47b |
{4} | -24a-38b 21a+34b -39a-18b |
o6 : ZZdFactorizationMap
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i7 : E = (dual C)
2 2 2
o7 = S <-- S <-- S
0 1 0
o7 : ZZdFactorization
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i8 : F = (dual D)
3 3 3
o8 = S <-- S <-- S
0 1 0
o8 : ZZdFactorization
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i9 : g = randomFactorizationMap(F,E)
3 2
o9 = 0 : S <------------------------ S : 0
{-3} | -13a-43b 2 |
{-3} | -15a-28b 16 |
{-3} | -47a+38b 22 |
3 2
1 : S <----------------------------- S : 1
{-5} | 45a-34b -16a+7b |
{-5} | -48a-47b 15a-23b |
{-5} | 47a+19b 39a+43b |
o9 : ZZdFactorizationMap
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i10 : h = Hom(f,g)
18 8
o10 = 0 : S <------------------------------------------------------------------------------------------------------------------ S : 0
{-6} | 9a2-42ab-33b2 -48a-29b 14a+23b -41 0 0 0 0 |
{-6} | -44a2+31ab+2b2 20a-30b -46a-32b -25 0 0 0 0 |
{-6} | 17a2+22ab-46b2 -23a-16b 4a+29b -47 0 0 0 0 |
{-6} | 27a2+10ab+9b2 -43a-38b 45a+9b -38 0 0 0 0 |
{-6} | -31a2-22ab+27b2 -41a-b -18a+27b -1 0 0 0 0 |
{-6} | -50a2-25ab-15b2 32a-14b -16a-15b -14 0 0 0 0 |
{-6} | -29a2+ab-35b2 20a-43b -3a-41b 16 0 0 0 0 |
{-6} | -49a2-8ab-4b2 -42a-41b -19a-22b 27 0 0 0 0 |
{-6} | 35a2+27ab-9b2 18a+32b 28a+b -26 0 0 0 0 |
{-10} | 0 0 0 0 -20a2-49ab+24b2 -49a2-7ab+b2 -31a2-15ab+21b2 20a2-36ab-37b2 |
{-10} | 0 0 0 0 -46a2-2ab-50b2 27a2+30ab+40b2 -41a2-23ab+32b2 -44a2+18ab+35b2 |
{-10} | 0 0 0 0 24a2-37ab+46b2 50a2-44ab+35b2 17a2+20ab+15b2 27a2-11ab+18b2 |
{-10} | 0 0 0 0 13a2+24ab+13b2 47a2+29ab+30b2 -36a2-8ab+45b2 33a2-7ab-36b2 |
{-10} | 0 0 0 0 40a2+9ab+15b2 38a2-44ab-12b2 -2a2-7ab-18b2 -12a2-27ab-26b2 |
{-10} | 0 0 0 0 45a2-14ab-34b2 18a2-25ab+40b2 23a2+23ab-40b2 -11a2-7ab-48b2 |
{-10} | 0 0 0 0 -47a2+34ab+18b2 a2+24ab+26b2 38a2-11ab-6b2 -18a2-15ab+25b2 |
{-10} | 0 0 0 0 3a2-50ab+13b2 18a2+31ab+30b2 47a2+30ab-38b2 -21a2-21ab-10b2 |
{-10} | 0 0 0 0 16a2+30ab-16b2 -34a2+6ab+b2 15a2-29ab+39b2 6a2-45ab-34b2 |
18 8
1 : S <---------------------------------------------------------------------------------------------------------------- S : 1
{-8} | 31a2+12ab-12b2 -20a2-37ab+50b2 37a-10b 25a+8b 0 0 0 0 |
{-8} | 41a2+6ab+25b2 44a2-19ab-20b2 -26a+4b 46a+17b 0 0 0 0 |
{-8} | -17a2+24ab-23b2 -27a2-32ab+33b2 -4a-36b -42a-23b 0 0 0 0 |
{-8} | -8a2-23ab+40b2 41a2+2ab-32b2 -47a+40b a-32b 0 0 0 0 |
{-8} | 22a2-47ab-16b2 31a2-43ab+33b2 3a-16b 18a+33b 0 0 0 0 |
{-8} | 50a2-39ab+43b2 20a2+ab-9b2 16a+43b -34a-9b 0 0 0 0 |
{-8} | 46a2-45ab+24b2 42a2+10ab+b2 -44a+31b -27a-45b 0 0 0 0 |
{-8} | 25a2-44ab-50b2 49a2+3ab+40b2 20a+28b 19a+18b 0 0 0 0 |
{-8} | -35a2+38ab+46b2 -14a2+46ab+35b2 -28a-50b 9a+41b 0 0 0 0 |
{-8} | 0 0 0 0 17a2-10ab-35b2 44a-43b -9a2-11ab-18b2 48a-25b |
{-8} | 0 0 0 0 -27a2-41ab-4b2 49a-41b 44a2-30ab+47b2 -20a+2b |
{-8} | 0 0 0 0 -24a2-22ab-9b2 -21a+32b -17a2+35ab+30b2 23a+28b |
{-8} | 0 0 0 0 -6a2+21ab-40b2 32a+23b -30a2+32ab+48b2 -42a+33b |
{-8} | 0 0 0 0 -38a2+36ab-19b2 -47a-18b 12a2-13ab+43b2 -33a-39b |
{-8} | 0 0 0 0 -45a2+17ab+33b2 49a-50b -23a2-8ab+21b2 43a-41b |
{-8} | 0 0 0 0 45a2+4ab-b2 -38a-7b -2a2+8ab+34b2 -23a+36b |
{-8} | 0 0 0 0 -18a2+29ab-3b2 -a+45b 21a2-49ab+b2 18a-15b |
{-8} | 0 0 0 0 -16a2-2ab-32b2 -14a+24b -15a2+30ab-23b2 50a-8b |
o10 : ZZdFactorizationMap
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i11 : assert isWellDefined h
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i17 : e = randomFactorizationMap(source h, ZZdfactorization(S^1, 2, Base => 1))
8
o17 = 0 : S <----- 0 : 0
0
8 1
1 : S <--------------------------------------------------------------- S : 1
{-6} | 36a6-3a5b-22a4b2-30a3b3+41a2b4+16ab5-28b6 |
{-6} | -6a6+35a5b-9a4b2-35a3b3+6a2b4+40ab5+3b6 |
{-7} | -31a7+25a6b-2a5b2-41a4b3-49a3b4-13a2b5+4ab6+30b7 |
{-7} | -47a7+27a6b-40a5b2+37a4b3-35a3b4-31a2b5-39ab6-31b7 |
{-6} | -48a6-29a5b-48a4b2+30a3b3-37a2b4+47ab5-49b6 |
{-7} | 28a7-18a6b+46a5b2+a4b3+40a3b4-22a2b5+10ab6+7b7 |
{-6} | 30a6+13a5b-17a4b2-13a3b3+3a2b4-41ab5+8b6 |
{-7} | 8a7-29a6b+30a5b2-46a4b3+49a3b4-18a2b5+42ab6+23b7 |
o17 : ZZdFactorizationMap
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i18 : phi = homomorphism e
2 2
o18 = 1 : S <--------------------------------------------------------------------------------------------------------- S : 0
{-4} | 36a6-3a5b-22a4b2-30a3b3+41a2b4+16ab5-28b6 -31a7+25a6b-2a5b2-41a4b3-49a3b4-13a2b5+4ab6+30b7 |
{-4} | -6a6+35a5b-9a4b2-35a3b3+6a2b4+40ab5+3b6 -47a7+27a6b-40a5b2+37a4b3-35a3b4-31a2b5-39ab6-31b7 |
2 2
0 : S <------------------------------------------------------------------------------------------------------------ S : 1
{-2} | -48a6-29a5b-48a4b2+30a3b3-37a2b4+47ab5-49b6 30a6+13a5b-17a4b2-13a3b3+3a2b4-41ab5+8b6 |
{-3} | 28a7-18a6b+46a5b2+a4b3+40a3b4-22a2b5+10ab6+7b7 8a7-29a6b+30a5b2-46a4b3+49a3b4-18a2b5+42ab6+23b7 |
o18 : ZZdFactorizationMap
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i19 : assert(degree phi == 1)
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i20 : psi = homomorphism'(g * phi * f)
18 1
o20 = 1 : S <---------------------------------------------------------------------- S : 0
{-8} | -25a8+19a7b+37a6b2-26a5b3-19a4b4+46a3b5+20a2b6-31ab7-39b8 |
{-8} | 43a8-2a7b-32a6b2+12a5b3+33a4b4-13a3b5+20a2b6+44ab7-45b8 |
{-8} | -32a8-14a7b+20a6b2-21a5b3-21a4b4+31a3b5-42a2b6+2ab7+28b8 |
{-8} | 33a8-38a7b+50a6b2-5a5b3+4a4b4-39a3b5-35a2b6+23ab7+34b8 |
{-8} | 30a8+5a6b2-33a5b3-7a4b4+14a3b5+9a2b6-28ab7+47b8 |
{-8} | 46a8-35a7b-8a6b2+a5b3-43a4b4-5a3b5+14a2b6-46ab7-35b8 |
{-8} | 3a8+13a7b+26a6b2+50a5b3-39a4b4-28a3b5+37a2b6+21ab7-40b8 |
{-8} | -2a8-44a7b-16a6b2-42a5b3-36a4b4-45a3b5-15a2b6-48ab7+16b8 |
{-8} | 24a8+26a7b+43a6b2-5a5b3-38a4b4-17a3b5+46a2b6-9ab7+15b8 |
{-8} | -25a8+8a7b-32a6b2-11a5b3-29a4b4+48a3b5-28a2b6-33ab7+12b8 |
{-8} | 10a8+21a7b-31a6b2+44a5b3-26a4b4-9a3b5+35a2b6-19ab7-28b8 |
{-8} | -36a8+22a7b+20a6b2-28a5b3+12a4b4-3a3b5+44a2b6+33ab7-34b8 |
{-8} | -49a8+27a7b+22a6b2-10a5b3+15a4b4-22a3b5-21a2b6+10ab7-32b8 |
{-8} | 2a8-26a7b-3a6b2-2a5b3-40a4b4+24a3b5-15a2b6-44ab7-50b8 |
{-8} | 46a8-32a7b+33a6b2-4a5b3-11a4b4+37a2b6-14ab7+15b8 |
{-8} | 34a8+42a7b-18a6b2+36a5b3+31a4b4+8a3b5+22a2b6+17ab7+11b8 |
{-8} | 6a8+17a7b-29a6b2-24a5b3-a4b4+45a3b5-17a2b6-50ab7-24b8 |
{-8} | -23a8+32a7b-44a6b2+5a5b3+37a4b4-49a3b5-11a2b6-26ab7+46b8 |
18
0 : S <----- 0 : 1
0
o20 : ZZdFactorizationMap
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i21 : i = map(ZZdfactorization(S^1, 2), source e, id_(source e), Degree => 1)
o21 = 1 : 0 <----- 0 : 0
0
1 1
0 : S <--------- S : 1
| 1 |
o21 : ZZdFactorizationMap
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i22 : assert((degree h, degree e, degree psi, degree i) === (0, 0, 1, 1))
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