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Complexes :: HH_ZZ Complex

HH_ZZ Complex -- homology or cohomology module of a complex

Synopsis

Description

The $i$-th homology of a complex $C$ is the quotient (ker dd^C_i/image dd^C_(i+1)).

The first example is the complex associated to a triangulation of the real projective plane, having 6 vertices, 15 edges, and 10 triangles.

i1 : d1 = matrix {
         {1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
         {-1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0},
         {0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1, 0, 0, 0},
         {0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 1, 0},
         {0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 1},
         {0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, -1}};

              6        15
o1 : Matrix ZZ  <--- ZZ
i2 : d2 = matrix {
         {-1, -1, 0, 0, 0, 0, 0, 0, 0, 0},
         {0, 0, -1, -1, 0, 0, 0, 0, 0, 0},
         {1, 0, 1, 0, 0, 0, 0, 0, 0, 0},
         {0, 1, 0, 0, -1, 0, 0, 0, 0, 0},
         {0, 0, 0, 1, 1, 0, 0, 0, 0, 0},
         {0, 0, 0, 0, 0, -1, -1, 0, 0, 0},
         {-1, 0, 0, 0, 0, 0, 0, -1, 0, 0},
         {0, -1, 0, 0, 0, 1, 0, 0, 0, 0},
         {0, 0, 0, 0, 0, 0, 1, 1, 0, 0},
         {0, 0, -1, 0, 0, 0, 0, 0, -1, 0},
         {0, 0, 0, 0, 0, -1, 0, 0, 1, 0},
         {0, 0, 0, -1, 0, 0, -1, 0, 0, 0},
         {0, 0, 0, 0, 0, 0, 0, 0, -1, -1},
         {0, 0, 0, 0, 0, 0, 0, -1, 0, 1},
         {0, 0, 0, 0, -1, 0, 0, 0, 0, -1}};

              15        10
o2 : Matrix ZZ   <--- ZZ
i3 : C = complex {d1,d2}

       6       15       10
o3 = ZZ  <-- ZZ   <-- ZZ
                       
     0       1        2

o3 : Complex
i4 : dd^C

           6                                                         15
o4 = 0 : ZZ  <---------------------------------------------------- ZZ   : 1
                | 1  1  1  1  1  0  0  0  0  0  0  0  0  0  0  |
                | -1 0  0  0  0  1  1  1  1  0  0  0  0  0  0  |
                | 0  -1 0  0  0  -1 0  0  0  1  1  1  0  0  0  |
                | 0  0  -1 0  0  0  -1 0  0  -1 0  0  1  1  0  |
                | 0  0  0  -1 0  0  0  -1 0  0  -1 0  -1 0  1  |
                | 0  0  0  0  -1 0  0  0  -1 0  0  -1 0  -1 -1 |

           15                                          10
     1 : ZZ   <------------------------------------- ZZ   : 2
                 | -1 -1 0  0  0  0  0  0  0  0  |
                 | 0  0  -1 -1 0  0  0  0  0  0  |
                 | 1  0  1  0  0  0  0  0  0  0  |
                 | 0  1  0  0  -1 0  0  0  0  0  |
                 | 0  0  0  1  1  0  0  0  0  0  |
                 | 0  0  0  0  0  -1 -1 0  0  0  |
                 | -1 0  0  0  0  0  0  -1 0  0  |
                 | 0  -1 0  0  0  1  0  0  0  0  |
                 | 0  0  0  0  0  0  1  1  0  0  |
                 | 0  0  -1 0  0  0  0  0  -1 0  |
                 | 0  0  0  0  0  -1 0  0  1  0  |
                 | 0  0  0  -1 0  0  -1 0  0  0  |
                 | 0  0  0  0  0  0  0  0  -1 -1 |
                 | 0  0  0  0  0  0  0  -1 0  1  |
                 | 0  0  0  0  -1 0  0  0  0  -1 |

o4 : ComplexMap
i5 : HH C

o5 = cokernel | 1  1  1  1  1  0  0  0  0  0  0  0  0  0  0  | <-- subquotient (| 0  1  0  0  0  0  0  0  0  0  |, | -1 -1 0  0  0  0  0  0  0  0  |) <-- image 0
              | -1 0  0  0  0  1  1  1  1  0  0  0  0  0  0  |                  | 1  0  0  0  0  0  0  0  0  0  |  | 0  0  -1 -1 0  0  0  0  0  0  |       
              | 0  -1 0  0  0  -1 0  0  0  1  1  1  0  0  0  |                  | 0  -1 1  0  -1 0  1  0  1  0  |  | 1  0  1  0  0  0  0  0  0  0  |      2
              | 0  0  -1 0  0  0  -1 0  0  -1 0  0  1  1  0  |                  | 0  0  0  0  0  1  0  0  0  0  |  | 0  1  0  0  -1 0  0  0  0  0  |
              | 0  0  0  -1 0  0  0  -1 0  0  -1 0  -1 0  1  |                  | -1 0  -1 0  1  -1 -1 0  -1 0  |  | 0  0  0  1  1  0  0  0  0  0  |
              | 0  0  0  0  -1 0  0  0  -1 0  0  -1 0  -1 -1 |                  | 0  0  0  0  1  0  0  0  0  0  |  | 0  0  0  0  0  -1 -1 0  0  0  |
                                                                                | 0  0  0  1  0  0  0  0  1  1  |  | -1 0  0  0  0  0  0  -1 0  0  |
     0                                                                          | 0  1  -1 0  0  0  0  0  -1 0  |  | 0  -1 0  0  0  1  0  0  0  0  |
                                                                                | 0  0  1  -1 -1 0  0  0  0  -1 |  | 0  0  0  0  0  0  1  1  0  0  |
                                                                                | 0  0  0  0  0  0  0  1  0  0  |  | 0  0  -1 0  0  0  0  0  -1 0  |
                                                                                | 0  0  0  0  0  0  0  0  0  1  |  | 0  0  0  0  0  -1 0  0  1  0  |
                                                                                | 1  0  0  0  1  0  0  -1 0  -1 |  | 0  0  0  -1 0  0  -1 0  0  0  |
                                                                                | 0  -1 1  0  -1 0  1  1  2  1  |  | 0  0  0  0  0  0  0  0  -1 -1 |
                                                                                | 0  0  0  1  0  0  0  0  0  0  |  | 0  0  0  0  0  0  0  -1 0  1  |
                                                                                | 0  0  0  0  -1 1  1  1  1  2  |  | 0  0  0  0  -1 0  0  0  0  -1 |
                                                                    
                                                                   1

o5 : Complex
i6 : prune HH_0 C

       1
o6 = ZZ

o6 : ZZ-module, free
i7 : prune HH_1 C

o7 = cokernel | 2 |

                              1
o7 : ZZ-module, quotient of ZZ
i8 : prune HH_2 C

o8 = 0

o8 : ZZ-module

The $i$-th cohomology of a complex $C$ is the $(-i)$-th homology of $C$.

i9 : S = ZZ/101[a..d, DegreeRank=>4];
i10 : I = intersect(ideal(a,b),ideal(c,d))

o10 = ideal (b*d, a*d, b*c, a*c)

o10 : Ideal of S
i11 : C = dual freeResolution (S^1/I)

       1      4      4      1
o11 = S  <-- S  <-- S  <-- S
                            
      -3     -2     -1     0

o11 : Complex
i12 : prune HH^1 C

o12 = 0

o12 : S-module
i13 : prune HH^2 C

o13 = cokernel {-1, -1, 0, 0} | b a 0 0 |
               {0, 0, -1, -1} | 0 0 d c |

                             2
o13 : S-module, quotient of S
i14 : prune HH^3 C

o14 = cokernel {-1, -1, -1, -1} | d c b a |

                             1
o14 : S-module, quotient of S

See also