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isShortExactSequence(ZZdFactorizationMap,ZZdFactorizationMap) -- whether a pair of ZZ/d-graded factorization maps forms a short exact sequence

Description

A short exact sequence of ZZ/d-graded factorizations \[ 0 \to B \xrightarrow{f} C \xrightarrow{g} D \to 0\] consists of two morphisms of factorizations $f \colon B \to C$ and $g \colon C \to D$ such that $g f = 0$, $\operatorname{image} f = \operatorname{ker} g$, $\operatorname{ker} f = 0$, and $\operatorname{coker} g = 0$.

From a factorization morphism $h \colon B \to C$, one obtains a short exact sequence \[ 0 \to \operatorname{image} h \to C \to \operatorname{coker} h \to 0. \]

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 : h = randomFactorizationMap(C, D, Cycle => true)

          2                                         3
o6 = 0 : S  <------------------------------------- S  : 0
               {2} | 47a+17b -36a-48b -29a+24b |
               {3} | 19      -30      19       |

          2                                        3
     1 : S  <------------------------------------ S  : 1
               {4} | 47a-19b 36a+19b -29a-30b |
               {4} | 19a+24b 30a-17b 19a+48b  |

o6 : ZZdFactorizationMap
 
i7 : f = inducedMap(C, image h)

          2
o7 = 0 : S  <------------------------------------- image {2} | 47a+17b -36a-48b -29a+24b | : 0
               {2} | 47a+17b -36a-48b -29a+24b |         {3} | 19      -30      19       |
               {3} | 19      -30      19       |

          2
     1 : S  <------------------------------------ image {4} | 47a-19b 36a+19b -29a-30b | : 1
               {4} | 47a-19b 36a+19b -29a-30b |         {4} | 19a+24b 30a-17b 19a+48b  |
               {4} | 19a+24b 30a-17b 19a+48b  |

o7 : ZZdFactorizationMap
 
i8 : g = inducedMap(coker h, C)

                                                                      2
o8 = 0 : cokernel {2} | 47a+17b -36a-48b -29a+24b | <--------------- S  : 0
                  {3} | 19      -30      19       |    {2} | 1 0 |
                                                       {3} | 0 0 |

                                                                     2
     1 : cokernel {4} | 47a-19b 36a+19b -29a-30b | <--------------- S  : 1
                  {4} | 19a+24b 30a-17b 19a+48b  |    {4} | 1 0 |
                                                      {4} | 0 1 |

o8 : ZZdFactorizationMap
 
i9 : assert isShortExactSequence(g,f)

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:4885:0.