Macaulay2 » Documentation
Packages » MatrixFactorizations :: image(ZZdFactorizationMap)
next | previous | forward | backward | up | index | toc

image(ZZdFactorizationMap) -- make the image of a map of ZZ/d-graded factorizations

Description

If $f : C \to D$ is a map of ZZ/d-graded factorizations of degree $d$, then the image is the ZZ/d-graded factorization $E$ whose $i-th$ is $image(f_{i-d})$, and whose differential is induced from the differential on the target.

In the following example, we first construct a random factorization morphism $f : C \to D$.

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 : f = randomFactorizationMap(D, C, Cycle => true, InternalDegree => 2)

          3                                      2
o6 = 0 : S  <---------------------------------- S  : 0
               {3} | 20a+29b 24a2-17ab+9b2  |
               {3} | 10a-19b 30a2-48ab-7b2  |
               {3} | 32a-36b 19a2-11ab-21b2 |

          3                                2
     1 : S  <---------------------------- S  : 1
               {5} | 20a+21b 24a-36b  |
               {5} | -10a+9b -30a-29b |
               {5} | 32a-7b  19a+19b  |

o6 : ZZdFactorizationMap
 
i7 : isWellDefined image f

o7 = true
 
i8 : potential image f

      3    3
o8 = a  + b

o8 : S
 
i9 : g = inducedMap(D, image f)

          3
o9 = 0 : S  <---------------------------------- image {3} | 20a+29b 24a2-17ab+9b2  | : 0
               {3} | 20a+29b 24a2-17ab+9b2  |         {3} | 10a-19b 30a2-48ab-7b2  |
               {3} | 10a-19b 30a2-48ab-7b2  |         {3} | 32a-36b 19a2-11ab-21b2 |
               {3} | 32a-36b 19a2-11ab-21b2 |

          3
     1 : S  <---------------------------- image {5} | 20a+21b 24a-36b  | : 1
               {5} | 20a+21b 24a-36b  |         {5} | -10a+9b -30a-29b |
               {5} | -10a+9b -30a-29b |         {5} | 32a-7b  19a+19b  |
               {5} | 32a-7b  19a+19b  |

o9 : ZZdFactorizationMap
 
i10 : prune image f

       2      2      2
o10 = S  <-- S  <-- S
                     
      0      1      0

o10 : ZZdFactorization
 
i11 : oo.dd

           2                     2
o11 = 1 : S  <----------------- S  : 0
                {6} | a  b2 |
                {6} | -b a2 |

           2                      2
      0 : S  <------------------ S  : 1
                {4} | a2 -b2 |
                {5} | b  a   |

o11 : ZZdFactorizationMap

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:4318:0.