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nullHomotopy(ZZdFactorizationMap) -- a map which is a candidate for being a null homotopy

Description

A map of ZZ/d-graded factorizations $f \colon C \to D$ is null-homotopic if there exists a map of factorizations $h : C \to D$ of degree $\deg(f)+1$, such that we have the equality \[ f = \operatorname{dd}^D h + (-1)^{\deg(f)} h \operatorname{dd}^C. \]

As a first example, we verify that the identity map on the "trivial" factorization is nullhomotopic.

i1 : S = ZZ/101[a,b];
 
i2 : C = trivialMF(S^2, a^3+b^3)

      2      2      2
o2 = S  <-- S  <-- S
                    
     0      1      0

o2 : ZZdFactorization
 
i3 : h = map(C, C, i -> if i == 1 then id_(S^2), Degree => 1)

          2         2
o3 = 1 : S  <----- S  : 0
               0

          2               2
     0 : S  <----------- S  : 1
               | 1 0 |
               | 0 1 |

o3 : ZZdFactorizationMap
 
i4 : isWellDefined h

o4 = true
 
i5 : assert isNullHomotopyOf(h, id_C)
 
i6 : assert isNullHomotopic id_C

A random map of factorizations, arising as a boundary in the associated Hom complex, is automatically null homotopic. We use the method nullHomotopy to construct a witness and verify it is a null homotopy.

i7 : D = randomTailMF(a^3+b^3, 1,2,4)

      2      2      2
o7 = S  <-- S  <-- S
                    
     0      1      0

o7 : ZZdFactorization
 
i8 : D.dd

          2                                 2
o8 = 1 : S  <----------------------------- S  : 0
               {1} | -26b2 a2-3ab+32b2 |
               {2} | a+3b  3a+8b       |

          2                              2
     0 : S  <-------------------------- S  : 1
               | -3a-8b a2-3ab+32b2 |
               | a+3b   26b2        |

o8 : ZZdFactorizationMap
 
i9 : f = randomFactorizationMap(D, D[1], Boundary => true)

          2                                   2
o9 = 0 : S  <------------------------------- S  : 0
               | -19a-38b -7a2+33ab-37b2 |
               | 19a-20b  -5a2+35ab-30b2 |

          2                                            2
     1 : S  <---------------------------------------- S  : 1
               {1} | 5a2-35ab-48b2 -4a2+24ab-42b2 |
               {2} | 22a-11b       28a-39b        |

o9 : ZZdFactorizationMap
 
i10 : assert isNullHomotopic f
 
i11 : h = nullHomotopy f

           2                           2
o11 = 1 : S  <----------------------- S  : 0
                {1} | -10 -29a-8b |
                {2} | 0   -22     |

           2                   2
      0 : S  <--------------- S  : 1
                | -29 -38 |
                | -24 0   |

o11 : ZZdFactorizationMap
 
i12 : assert isNullHomotopyOf(h, f)

When a map of factorizations is not null-homotopic, this method nevertheless returns a map $h$ of factorizations, having the correct source, target and degree, but cannot be a null homotopy.

i13 : g = randomFactorizationMap(D, D[1])

           2                                    2
o13 = 0 : S  <-------------------------------- S  : 0
                | -16a+39b 19a2-47ab-39b2  |
                | 21a+34b  -18a2-13ab-43b2 |

           2         2
      1 : S  <----- S  : 1
                0

o13 : ZZdFactorizationMap
 
i14 : assert not isNullHomotopic g
 
i15 : h' = nullHomotopy g

           2                    2
o15 = 1 : S  <---------------- S  : 0
                {1} | 0 0  |
                {2} | 0 19 |

           2         2
      0 : S  <----- S  : 1
                0

o15 : ZZdFactorizationMap
 
i16 : assert isWellDefined h'
 
i17 : assert(degree h' === degree g + 1)
 
i18 : assert not isNullHomotopyOf(h', g)

Caveat

The output is only a null homotopy when one exists.

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:5116:0.