As a first example, consider two matrix factorizations $C$ and $D$. From a random map $f : R^1 \to Hom(C, D)$, we construct the corresponding map of ZZ/d-graded factorizations $g : C \to D$.
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 : g = randomFactorizationMap(D, C, InternalDegree => 2)
3 2
o6 = 0 : S <------------------------------------ S : 0
{3} | 24a-36b -10a2-29ab-8b2 |
{3} | -30a-29b -22a2-29ab-24b2 |
{3} | 19a+19b -38a2-16ab+39b2 |
3 2
1 : S <----------------------------- S : 1
{5} | 21a+34b -13a-43b |
{5} | 19a-47b -15a-28b |
{5} | -39a-18b -47a+38b |
o6 : ZZdFactorizationMap
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i7 : isWellDefined g
o7 = true
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i8 : f = homomorphism' g
12 1
o8 = 0 : S <--------------------------- S : 0
{1} | 24a-36b |
{1} | -30a-29b |
{1} | 19a+19b |
{0} | -10a2-29ab-8b2 |
{0} | -22a2-29ab-24b2 |
{0} | -38a2-16ab+39b2 |
{1} | 21a+34b |
{1} | 19a-47b |
{1} | -39a-18b |
{1} | -13a-43b |
{1} | -15a-28b |
{1} | -47a+38b |
12
1 : S <----- 0 : 1
0
o8 : ZZdFactorizationMap
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i9 : isWellDefined f
o9 = true
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The map $g : C \to D$ corresponding to a random map into $Hom(C,D)$ does not generally commute with the differentials. However, if the element of $Hom(C,D)$ is a cycle, then the corresponding map does commute.
i10 : g = randomFactorizationMap(D, C, Cycle => true, InternalDegree => 3)
3 2
o10 = 0 : S <-------------------------------------------------- S : 0
{3} | 40a2+41ab+48b2 2a3-31a2b-48ab2+26b3 |
{3} | 32a2-24ab-19b2 -45a3+36a2b+49ab2+45b3 |
{3} | -37a2-27ab+22b2 -47a3-9a2b+47ab2+10b3 |
3 2
1 : S <------------------------------------------- S : 1
{5} | 40a2-21ab-10b2 2a2+16ab+22b2 |
{5} | -32a2+8ab+26b2 45a2-34ab-48b2 |
{5} | -37a2-42ab+45b2 -47a2+47ab+19b2 |
o10 : ZZdFactorizationMap
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i11 : isWellDefined g
o11 = true
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i12 : f = homomorphism' g
12 1
o12 = 0 : S <---------------------------------- S : 0
{1} | 40a2+41ab+48b2 |
{1} | 32a2-24ab-19b2 |
{1} | -37a2-27ab+22b2 |
{0} | 2a3-31a2b-48ab2+26b3 |
{0} | -45a3+36a2b+49ab2+45b3 |
{0} | -47a3-9a2b+47ab2+10b3 |
{1} | 40a2-21ab-10b2 |
{1} | -32a2+8ab+26b2 |
{1} | -37a2-42ab+45b2 |
{1} | 2a2+16ab+22b2 |
{1} | 45a2-34ab-48b2 |
{1} | -47a2+47ab+19b2 |
12
1 : S <----- 0 : 1
0
o12 : ZZdFactorizationMap
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i13 : isWellDefined f
o13 = true
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i14 : assert isCommutative g
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i15 : assert(degree f === 0)
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i16 : assert(source f == ZZdfactorization(S^{-3}, 2))
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i17 : assert(target g === D)
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i18 : assert(homomorphism f == g)
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