Macaulay2 » Documentation
Packages » SimplicialModules :: isHomogeneous(SimplicialModuleMap)
next | previous | forward | backward | up | index | toc

isHomogeneous(SimplicialModuleMap) -- whether a map of simplicial modules is homogeneous

Description

A map of simplicial modules $f \colon C \to D$ 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,d];
 
i2 : I = minors(2, matrix{{a,b,c},{b,c,d}})

               2                        2
o2 = ideal (- b  + a*c, - b*c + a*d, - c  + b*d)

o2 : Ideal of S
 
i3 : C = simplicialModule(freeResolution (S^1/I), Degeneracy => true)

      1      4      9
o3 = S  <-- S  <-- S <-- ...
                    
     0      1      2

o3 : SimplicialModule
 
i4 : assert isHomogeneous dd^C
 
i5 : f = randomSimplicialMap(C, C, Degree => -1)

          1                                                                                                                                                                 3
o5 = 0 : S  <------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 0
               | 24a2-36ab+19b2-30ac+19bc-29c2-29ad-10bd-8cd-22d2 -29a2-24ab+39b2-38ac+21bc+19c2-16ad+34bd-47cd-39d2 -18a2-13ab-28b2-43ac-47bc+2c2-15ad+38bd+16cd+22d2 |

          4                                                                                                                                                                                                      5
     1 : S  <-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 1
               {0} | 24a2-36ab+19b2-30ac+19bc-29c2-29ad-10bd-8cd-22d2 -29a2-24ab+39b2-38ac+21bc+19c2-16ad+34bd-47cd-39d2 -18a2-13ab-28b2-43ac-47bc+2c2-15ad+38bd+16cd+22d2 0               0                |
               {2} | 0                                                0                                                  0                                                 45a-34b-48c-47d -17a-11b+48c+36d |
               {2} | 0                                                0                                                  0                                                 47a+19b-16c+7d  35a+11b-38c+33d  |
               {2} | 0                                                0                                                  0                                                 15a-23b+39c+43d 40a+11b+46c-28d  |

          9                                                                                                                                                                                                                                       7
     2 : S  <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 2
               {0} | 24a2-36ab+19b2-30ac+19bc-29c2-29ad-10bd-8cd-22d2 -29a2-24ab+39b2-38ac+21bc+19c2-16ad+34bd-47cd-39d2 -18a2-13ab-28b2-43ac-47bc+2c2-15ad+38bd+16cd+22d2 0               0                0               0                |
               {2} | 0                                                0                                                  0                                                 45a-34b-48c-47d -17a-11b+48c+36d 0               0                |
               {2} | 0                                                0                                                  0                                                 47a+19b-16c+7d  35a+11b-38c+33d  0               0                |
               {2} | 0                                                0                                                  0                                                 15a-23b+39c+43d 40a+11b+46c-28d  0               0                |
               {2} | 0                                                0                                                  0                                                 0               0                45a-34b-48c-47d -17a-11b+48c+36d |
               {2} | 0                                                0                                                  0                                                 0               0                47a+19b-16c+7d  35a+11b-38c+33d  |
               {2} | 0                                                0                                                  0                                                 0               0                15a-23b+39c+43d 40a+11b+46c-28d  |
               {3} | 0                                                0                                                  0                                                 0               0                0               0                |
               {3} | 0                                                0                                                  0                                                 0               0                0               0                |

o5 : SimplicialModuleMap
 
i6 : assert isHomogeneous f
 
i7 : f = randomSimplicialMap(C, C, InternalDegree => 2)

          1                                                        1
o7 = 0 : S  <---------------------------------------------------- S  : 0
               | a2-3ab-23b2+22ac-7bc+29c2-47ad+2bd-47cd+15d2 |

          4                                                                                                                                                                                                                 4
     1 : S  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 1
               {0} | a2-3ab-23b2+22ac-7bc+29c2-47ad+2bd-47cd+15d2 0                                                 0                                               0                                                  |
               {2} | 0                                            -37a2-13ab-18b2-10ac+39bc-22c2+30ad+27bd+32cd-9d2 13a2-26ab-11b2+22ac-8bc-8c2-49ad+43bd+36cd-3d2  -47a2+27ab-35b2-40ac-31bc-31c2+37ad-39bd-48cd-29d2 |
               {2} | 0                                            -32a2-20ab-48b2+24ac-15bc-30ad+39bd+33cd-49d2     -22a2-30ab-28b2+41ac-6bc-9c2+16ad+35bd-35cd+6d2 -48a2+30ab-49b2-37ac+28bc+46c2+47ad-18bd+cd+40d2   |
               {2} | 0                                            -33a2-19ab+44b2+17ac-39bc+9c2-20ad+36bd-39cd+4d2  40a2+3ab-2b2-31ac-41bc-13c2+25ad-49bd+4cd+30d2  -22a2+10ab+13b2+7ac-17bc+3c2+30ad-13bd-41cd+8d2    |

          9                                                                                                                                                                                                                                                                                                                                                                                                                                                                         9
     2 : S  <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 2
               {0} | a2-3ab-23b2+22ac-7bc+29c2-47ad+2bd-47cd+15d2 0                                                 0                                               0                                                  0                                                 0                                               0                                                  0                                                 0                                                |
               {2} | 0                                            -37a2-13ab-18b2-10ac+39bc-22c2+30ad+27bd+32cd-9d2 13a2-26ab-11b2+22ac-8bc-8c2-49ad+43bd+36cd-3d2  -47a2+27ab-35b2-40ac-31bc-31c2+37ad-39bd-48cd-29d2 0                                                 0                                               0                                                  0                                                 0                                                |
               {2} | 0                                            -32a2-20ab-48b2+24ac-15bc-30ad+39bd+33cd-49d2     -22a2-30ab-28b2+41ac-6bc-9c2+16ad+35bd-35cd+6d2 -48a2+30ab-49b2-37ac+28bc+46c2+47ad-18bd+cd+40d2   0                                                 0                                               0                                                  0                                                 0                                                |
               {2} | 0                                            -33a2-19ab+44b2+17ac-39bc+9c2-20ad+36bd-39cd+4d2  40a2+3ab-2b2-31ac-41bc-13c2+25ad-49bd+4cd+30d2  -22a2+10ab+13b2+7ac-17bc+3c2+30ad-13bd-41cd+8d2    0                                                 0                                               0                                                  0                                                 0                                                |
               {2} | 0                                            0                                                 0                                               0                                                  -37a2-13ab-18b2-10ac+39bc-22c2+30ad+27bd+32cd-9d2 13a2-26ab-11b2+22ac-8bc-8c2-49ad+43bd+36cd-3d2  -47a2+27ab-35b2-40ac-31bc-31c2+37ad-39bd-48cd-29d2 0                                                 0                                                |
               {2} | 0                                            0                                                 0                                               0                                                  -32a2-20ab-48b2+24ac-15bc-30ad+39bd+33cd-49d2     -22a2-30ab-28b2+41ac-6bc-9c2+16ad+35bd-35cd+6d2 -48a2+30ab-49b2-37ac+28bc+46c2+47ad-18bd+cd+40d2   0                                                 0                                                |
               {2} | 0                                            0                                                 0                                               0                                                  -33a2-19ab+44b2+17ac-39bc+9c2-20ad+36bd-39cd+4d2  40a2+3ab-2b2-31ac-41bc-13c2+25ad-49bd+4cd+30d2  -22a2+10ab+13b2+7ac-17bc+3c2+30ad-13bd-41cd+8d2    0                                                 0                                                |
               {3} | 0                                            0                                                 0                                               0                                                  0                                                 0                                               0                                                  8a2-29ab+49b2+30ac-18bc+23c2-46ad+42bd-28cd+15d2  44a2-39ab+20ac-47bc+47c2+19ad-28bd-28cd+6d2      |
               {3} | 0                                            0                                                 0                                               0                                                  0                                                 0                                               0                                                  18a2-16ab-18b2-46ac+27bc+23c2+12ad-21bd-37cd-23d2 -9a2-33ab+26b2+28ac+5bc-33c2-29ad-37bd-28cd+42d2 |

o7 : SimplicialModuleMap

A map of simplicial modules may be homogeneous even if the source or the target is not homogeneous.

i8 : phi = map(S, S, {1,b,c,d})

o8 = map (S, S, {1, b, c, d})

o8 : RingMap S <-- S
 
i9 : D = phi C

      1      4      9
o9 = S  <-- S  <-- S <-- ...
                    
     0      1      2

o9 : SimplicialModule
 
i10 : dd^D

                1                             4
o10 = (0, 0) : S  <------------------------- S  : (1, 0)
                     | 1 b2-c bc-d c2-bd |

                1                   4
      (0, 1) : S  <--------------- S  : (1, 1)
                     | 1 0 0 0 |

                4                                             9
      (1, 0) : S  <----------------------------------------- S  : (2, 0)
                     {0} | 1 b2-c bc-d c2-bd 0 0 0 0  0  |
                     {2} | 0 0    0    0     1 0 0 -c d  |
                     {2} | 0 0    0    0     0 1 0 b  -c |
                     {2} | 0 0    0    0     0 0 1 -1 b  |

                4                                 9
      (1, 1) : S  <----------------------------- S  : (2, 1)
                     {0} | 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 1 0 0 0 0 |
                     {2} | 0 0 1 0 0 1 0 0 0 |
                     {2} | 0 0 0 1 0 0 1 0 0 |

                4                                 9
      (1, 2) : S  <----------------------------- S  : (2, 2)
                     {0} | 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 |

o10 : SimplicialModuleMap
 
i11 : assert not isHomogeneous dd^D

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:1899:0.