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Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes

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 $\phi \mapsto g \phi f$. When the period of a factorization is greater than $2$, one should adjoin a root of unity in order for the Hom factorization to be defined.

i1 : S = ZZ/101[a,b];
 
i2 : C = linearMF(a^3+b^3, t)

     /   S[t]   \3     /   S[t]   \3     /   S[t]   \3     /   S[t]   \3
o2 = |----------|  <-- |----------|  <-- |----------|  <-- |----------|
     | 2        |      | 2        |      | 2        |      | 2        |
     \t  + t + 1/      \t  + t + 1/      \t  + t + 1/      \t  + t + 1/
                                                            
     0                 1                 2                 0

o2 : ZZdFactorization
 
i3 : C.cache.?rootOfUnity

o3 = true
 
i4 : H = Hom(C,C) --no need to adjoin root of unity

     /   S[t]   \27     /   S[t]   \27     /   S[t]   \27     /   S[t]   \27
o4 = |----------|   <-- |----------|   <-- |----------|   <-- |----------|
     | 2        |       | 2        |       | 2        |       | 2        |
     \t  + t + 1/       \t  + t + 1/       \t  + t + 1/       \t  + t + 1/
                                                               
     0                  1                  2                  0

o4 : ZZdFactorization
 
i5 : A = ZZdfactorization {a,a,a}

      1      1      1      1
o5 = S  <-- S  <-- S  <-- S
                           
     0      1      2      0

o5 : ZZdFactorization
 
i6 : A.cache.?rootOfUnity

o6 = false
 
i7 : H' = Hom(A,A,t) --need to adjoin root of unity

     /   S[t]   \3     /   S[t]   \3     /   S[t]   \3     /   S[t]   \3
o7 = |----------|  <-- |----------|  <-- |----------|  <-- |----------|
     | 2        |      | 2        |      | 2        |      | 2        |
     \t  + t + 1/      \t  + t + 1/      \t  + t + 1/      \t  + t + 1/
                                                            
     0                 1                 2                 0

o7 : ZZdFactorization

If one instead wants to define a new factorization over the same ambient ring, use the tensor product instead.

i8 : St = ring C;
 
i9 : At = A**St

       1       1       1       1
o9 = St  <-- St  <-- St  <-- St
                              
     0       1       2       0

o9 : ZZdFactorization
 
i10 : Ht = Hom(At,At)

        3       3       3       3
o10 = St  <-- St  <-- St  <-- St
                               
      0       1       2       0

o10 : ZZdFactorization
 
i11 : Ht**H

        243       243       243       243
o11 = St    <-- St    <-- St    <-- St
                                     
      0         1         2         0

o11 : ZZdFactorization
 
i12 : isZZdComplex Ht

o12 = true

The homomorphism and homomorphism' commands are also implemented for factorizations of longer length, as well as randomFactorizationMap.

i13 : e = randomFactorizationMap(C,C, Cycle => true)

            3                                          3
o13 = 0 : St  <------------------------------------- St  : 0
                 {0, 2} | 19     -41t-12 -12t+24 |
                 {0, 2} | 36t+12 19      12t-29  |
                 {0, 2} | 29t+41 -24t-36 19      |

            3                                          3
      1 : St  <------------------------------------- St  : 1
                 {0, 2} | 19      29t+41 -24t-36 |
                 {0, 2} | -12t+24 19     -41t-12 |
                 {0, 2} | 12t-29  36t+12 19      |

            3                                          3
      2 : St  <------------------------------------- St  : 2
                 {0, 2} | 19      12t-29  36t+12 |
                 {0, 2} | -24t-36 19      29t+41 |
                 {0, 2} | -41t-12 -12t+24 19     |

o13 : ZZdFactorizationMap
 
i14 : phi = homomorphism' e

            27                    1
o14 = 0 : St   <--------------- St  : 0
                  | 19      |
                  | 36t+12  |
                  | 29t+41  |
                  | -41t-12 |
                  | 19      |
                  | -24t-36 |
                  | -12t+24 |
                  | 12t-29  |
                  | 19      |
                  | 19      |
                  | -12t+24 |
                  | 12t-29  |
                  | 29t+41  |
                  | 19      |
                  | 36t+12  |
                  | -24t-36 |
                  | -41t-12 |
                  | 19      |
                  | 19      |
                  | -24t-36 |
                  | -41t-12 |
                  | 12t-29  |
                  | 19      |
                  | -12t+24 |
                  | 36t+12  |
                  | 29t+41  |
                  | 19      |

            27
      1 : St   <----- 0 : 1
                  0

            27
      2 : St   <----- 0 : 2
                  0

o14 : ZZdFactorizationMap

If either of the arguments is a ZZdFactorization, that argument is understood to be the identity map on the factorization.

i15 : assert(Hom(e, C) == Hom(e, id_C))
 
i16 : assert(Hom(C, e) == Hom(id_C, e))

See also

Ways to use this method:

  • Hom(ZZdFactorization,Matrix,RingElement)
  • Hom(ZZdFactorization,Matrix,Symbol)
  • Hom(ZZdFactorization,ZZdFactorizationMap,RingElement)
  • Hom(ZZdFactorization,ZZdFactorizationMap,Symbol)
  • Hom(ZZdFactorizationMap,Matrix,RingElement)
  • Hom(ZZdFactorizationMap,Matrix,Symbol)
  • Hom(ZZdFactorizationMap,Module,RingElement)
  • Hom(ZZdFactorizationMap,Module,Symbol)
  • Hom(ZZdFactorizationMap,Ring,RingElement)
  • Hom(ZZdFactorizationMap,Ring,Symbol)
  • Hom(ZZdFactorizationMap,ZZdFactorization,RingElement)
  • Hom(ZZdFactorizationMap,ZZdFactorization,Symbol)
  • Hom(ZZdFactorizationMap,ZZdFactorizationMap,RingElement)
  • Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3447:0.