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isHomogeneous(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is homogeneous

Description

A map of factorizations $f \colon C \to D$ is homogeneous (graded) if its underlying ring is graded, and all the component maps $f_i \colon C_i \to D_{d+i}$ are graded of degree zero, where $f$ has degree $d$.

i1 : S = ZZ/101[a,b,c];
 
i2 : R = S/(a^2+b^2+c^2);
 
i3 : m = ideal vars R

o3 = ideal (a, b, c)

o3 : Ideal of R
 
i4 : C = tailMF m

      4      4      4
o4 = S  <-- S  <-- S
                    
     0      1      0

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

      8      8      8
o5 = S  <-- S  <-- S
                    
     0      1      0

o5 : ZZdFactorization
 
i6 : assert isHomogeneous dd^C
 
i7 : f = randomFactorizationMap(C, C, Degree => -1)

          4         4
o7 = 1 : S  <----- S  : 0
               0

          4                                                                   4
     0 : S  <--------------------------------------------------------------- S  : 1
               {3} | 24a-36b-30c  -38a-16b+39c -28a-47b+38c -16a+7b+15c  |
               {3} | -29a+19b+19c 21a+34b+19c  2a+16b+22c   -23a+39b+43c |
               {3} | -10a-29b-8c  -47a-39b-18c 45a-34b-48c  -17a-11b+48c |
               {3} | -22a-29b-24c -13a-43b-15c -47a+47b+19c 36a+35b+11c  |

o7 : ZZdFactorizationMap
 
i8 : f = randomFactorizationMap(C, C, InternalDegree => 2)

          4                                                                                                                                         4
o8 = 0 : S  <------------------------------------------------------------------------------------------------------------------------------------- S  : 0
               {3} | -38a2+33ab+11b2+40ac+46bc-28c2 32a2-9ab-20b2-32ac+24bc-30c2   22a2-49ab-8b2-11ac+43bc-8c2  -49a2-13ab+30b2+4ac-47bc+27c2  |
               {3} | a2-3ab-47b2+22ac-23bc-7c2      -48a2-15ab+39ac+33bc-49c2      36a2-3ab-30b2-22ac+41bc+16c2 -40a2+37ab-31b2-35ac-39bc-31c2 |
               {3} | 2a2+29ab+15b2-47ac-37bc-13c2   -33a2-19ab-20b2+17ac+44bc-39c2 -28a2-6ab-9b2+35ac-35bc+6c2  -48a2-29ab+30b2-48ac-37bc+47c2 |
               {3} | -10a2+30ab+39b2-18ac+27bc-22c2 36a2+9ab+4b2-39ac+13bc-26c2    40a2+3ab+25b2-31ac-2bc-41c2  -49a2+28ab+46b2-18ac+bc+40c2   |

          4                                                                                                                                          4
     1 : S  <-------------------------------------------------------------------------------------------------------------------------------------- S  : 1
               {4} | -22a2+10ab+30b2+7ac+13bc-17c2  -18a2+27ab+23b2-21ac-37bc-23c2 -28a2+42ab+30b2+44ac+4bc+22c2 21a2-30ab-14b2-4ac-33bc-42c2   |
               {4} | -13a2+3ab+8b2-41ac+8bc-29c2    44a2-39ab+19b2+20ac-47c2       5a2-20ab-29b2-13ac+15bc-4c2   -44a2-5ab-35b2-16ac-39bc-4c2   |
               {4} | 30a2-46ab-18b2+49ac+42bc+23c2  -28a2+47ab+6b2-28ac-9bc-33c2   12a2+3ab-2b2+9ac+20bc-26c2    -24a2-32ab-18b2-23ac+27bc-45c2 |
               {4} | -28a2+15ab-16b2+18ac-46bc+12c2 28a2-29ab+5b2+26ac-37bc-33c2   33a2+16ab+31b2+10ac+28bc-6c2  18a2-28ab-11b2+42ac+8bc+42c2   |

o8 : ZZdFactorizationMap

The image of a homogeneous factorization under a nonhomogeneous ring map may fail to be homogeneous.

i9 : use S;
 
i10 : phi = map(S, S, {1,b,c})

o10 = map (S, S, {1, b, c})

o10 : RingMap S <-- S
 
i11 : D = phi C

       4      4      4
o11 = S  <-- S  <-- S
                     
      0      1      0

o11 : ZZdFactorization
 
i12 : dd^D

           4                          4
o12 = 1 : S  <---------------------- S  : 0
                {4} | 0  -b 1 -c |
                {4} | -1 -c 0 b  |
                {4} | -c 1  b 0  |
                {4} | b  0  c 1  |

           4                          4
      0 : S  <---------------------- S  : 1
                {3} | 0  -1 -c b |
                {3} | -b -c 1  0 |
                {3} | 1  0  b  c |
                {3} | -c b  0  1 |

o12 : ZZdFactorizationMap
 
i13 : assert not isHomogeneous dd^D
 
i14 : g = randomFactorizationMap(D, D, InternalDegree => 1)

           4                                                                  4
o14 = 0 : S  <-------------------------------------------------------------- S  : 0
                {3} | 49a+5b-38c   -28a-50b-29c 37a+34b+5c  19a-45b-29c  |
                {3} | -26a+28b-33c -26a-49b+34c 16a-31b-49c -44a-42b-17c |
                {3} | 9a-7b-46c    31a+28b+36c  -a+43b+48c  -23a-4b-2c   |
                {3} | 2a-19b+43c   -40a+45b+42c 8a-40b-47c  34a-8b+27c   |

           4                                                                  4
      1 : S  <-------------------------------------------------------------- S  : 1
                {4} | 7a+38b+37c   4a+5b+36c   23a-47b+10c  -37a+10b+10c |
                {4} | -24a+25b-42c -16a+6b-43c -15a+47b+15c 34a-42b+34c  |
                {4} | 24a-24b-47c  -27a-50b-6c -11a-3b-28c  -44a-39b-40c |
                {4} | 12a-31b+37c  43a+5b-21c  -50a+17b-30c -29a+50b-10c |

o14 : ZZdFactorizationMap

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:2959:0.