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Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map

Description

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];
 
i2 : R = S/(a^3+b^3);
 
i3 : m = ideal vars R

o3 = ideal (a, b)

o3 : Ideal of R
 
i4 : C = tailMF m

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

o4 : ZZdFactorization
 
i5 : D = tailMF (m^2)

      3      3      3
o5 = S  <-- S  <-- S
                    
     0      1      0

o5 : ZZdFactorization
 
i6 : K = koszulComplex vars S

      1      2      1
o6 = S  <-- S  <-- S
                    
     0      1      2

o6 : Complex
 
i7 : H = Hom(K, C)

      8      8      8
o7 = S  <-- S  <-- S
                    
     0      1      0

o7 : ZZdFactorization
 
i8 : potential H

        3    3
o8 = - a  - b

o8 : S
 
i9 : H' = Hom(D, K)

      12      12      12
o9 = S   <-- S   <-- S
                      
     0       1       0

o9 : ZZdFactorization
 
i10 : potential H'

       3    3
o10 = a  + b

o10 : S
 
i11 : isZZdComplex (H**H')

o11 = true

See also

Ways to use this method:

  • Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
  • Hom(ComplexMap,ZZdFactorizationMap,RingElement)
  • Hom(ComplexMap,ZZdFactorizationMap,Symbol)
  • Hom(ZZdFactorizationMap,ComplexMap)
  • Hom(ZZdFactorizationMap,ComplexMap,RingElement)
  • Hom(ZZdFactorizationMap,ComplexMap,Symbol)

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3377:0.