Let $d$ be the value of the optional argument Degree, or zero, if not given. For each $i$, the terms $D_{i+d}$ and $C_i$ must be subquotients of the same ambient free module. This method returns the factorization map induced by the identity on each of these free modules.
If Verify => true is given, then this method also checks that these identity maps induced well-defined maps. This can be a relatively expensive computation.
i1 : R = ZZ/101[a,b,c]
o1 = R
o1 : PolynomialRing
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i2 : C = koszulMF({a,b,c}, a^3+b^3+c^3)
4 4 4
o2 = R <-- R <-- R
0 1 0
o2 : ZZdFactorization
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i3 : f = randomFactorizationMap(C, C, Cycle => true, InternalDegree => 1)
4 4
o3 = 0 : R <------------------------------------------------------- R : 0
| 24a-36b-30c 0 0 0 |
| -29c 24a-36b-30c 0 0 |
| -29a 0 24a-36b-30c 0 |
| 29b 0 0 24a-36b-30c |
4 4
1 : R <------------------------------------------------------- R : 1
| 24a-36b-30c 0 0 0 |
| -29c 24a-36b-30c -29b -29a |
| 0 0 24a-36b-30c 0 |
| 0 0 0 24a-36b-30c |
o3 : ZZdFactorizationMap
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i4 : isWellDefined coker f
o4 = true
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i5 : (coker f).dd^2 == (a^3 + b^3 + c^3)*id_(coker f)
o5 = true
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i6 : i1 = inducedMap(coker f, C)
4
o6 = 0 : cokernel | 24a-36b-30c 0 0 0 | <--------------- R : 0
| -29c 24a-36b-30c 0 0 | | 1 0 0 0 |
| -29a 0 24a-36b-30c 0 | | 0 1 0 0 |
| 29b 0 0 24a-36b-30c | | 0 0 1 0 |
| 0 0 0 1 |
4
1 : cokernel | 24a-36b-30c 0 0 0 | <--------------- R : 1
| -29c 24a-36b-30c -29b -29a | | 1 0 0 0 |
| 0 0 24a-36b-30c 0 | | 0 1 0 0 |
| 0 0 0 24a-36b-30c | | 0 0 1 0 |
| 0 0 0 1 |
o6 : ZZdFactorizationMap
|
i7 : i2 = inducedMap(C, image f)
4
o7 = 0 : R <------------------------------------------------------- image | 24a-36b-30c 0 0 0 | : 0
| 24a-36b-30c 0 0 0 | | -29c 24a-36b-30c 0 0 |
| -29c 24a-36b-30c 0 0 | | -29a 0 24a-36b-30c 0 |
| -29a 0 24a-36b-30c 0 | | 29b 0 0 24a-36b-30c |
| 29b 0 0 24a-36b-30c |
4
1 : R <------------------------------------------------------- image | 24a-36b-30c 0 0 0 | : 1
| 24a-36b-30c 0 0 0 | | -29c 24a-36b-30c -29b -29a |
| -29c 24a-36b-30c -29b -29a | | 0 0 24a-36b-30c 0 |
| 0 0 24a-36b-30c 0 | | 0 0 0 24a-36b-30c |
| 0 0 0 24a-36b-30c |
o7 : ZZdFactorizationMap
|
i8 : ker i1 == image i2
o8 = true
|