Hom(Complex,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization formed by folding the complex, then taking the ZZ/d-graded Hom factorization

The factorization of homomorphisms is a ZZ/d-graded factorization $D$ whose $i$th component is the direct sum of $Hom(C1_j, C2_{k})$ over all $k-j = i \mod d$. The differential on $Hom(C1_j, C2_{k})$ is the differential $Hom(id_{C1}, dd^{C2}) + t^j Hom(dd^{C1}, id_{C2})$ where $t$ is a primitive $d$th root of unity.

The use of a primitive $d$th root of unity adds some subtlety to this construction, since for $d > 2$ the user may need to adjoin a root of unity using the \texttt{adjoinRoot} command. In general, if $C1$ is a factorizations of $f$ and $C2$ is a factorization of $g$, then the Hom factorization $\operatorname{Hom} (C1,C2)$ is a factorization of the difference $g-f$.

If one of the inputs is a complex, the method will automatically Fold the complex and take the d-periodic Hom.

i1 : S = ZZ/101[a..c];

i2 : f = a^3 + b^3 + c^3

      3    3    3
o2 = a  + b  + c

o2 : S

i3 : C = randomTailMF(f)

      6      6      6
o3 = S  <-- S  <-- S
                    
     0      1      0

o3 : ZZdFactorization

i4 : K = koszulComplex vars S

      1      3      3      1
o4 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o4 : Complex

i5 : D = Hom(K,C)

      48      48      48
o5 = S   <-- S   <-- S
                      
     0       1       0

o5 : ZZdFactorization

i6 : isWellDefined D

o6 = true

i7 : potential C

      3    3    3
o7 = a  + b  + c

o7 : S

i8 : potential D

        3    3    3
o8 = - a  - b  - c

o8 : S