face/degeneracy maps of a simplicial module -- access the face and degeneracy maps of a simplicial module

Usage:

dd^C

ss^C
Inputs:
- C, a Simplicial Module,
Outputs:
- a Map of Simplicial Modules, a map of degree -1 or 1, respectively

A simplicial module is a sequence of modules connected by homomorphisms, called face/degeneracy maps, such that these maps satisfy a set of identities known as the simplicial identities. The face maps are accessed via the symbol dd (the same symbol used for the differentials of a complex), and the degeneracy maps via ss, after the conventional letter $s$ (or $\sigma$) used for degeneracy maps in the literature.

More precisely we should have the following equalities: 1. For face maps: \[ d_j d_i = d_i d_{j-1} \text{ for } 0 \leq i < j \leq n \] 2. For face and degeneracy maps: \[ d_i s_j = s_{j-1} d_i \text{ for } i < j \] \[ d_j s_j = \text{id} \] \[ d_{j+1} s_j = \text{id} \] \[ d_k s_j = s_j d_{k-1} \text{ for } k > j+1 \] 3. For degeneracy maps: \[ s_j s_i = s_i s_{j+1} \text{ for } i \leq j \]

The face/degeneracy maps are considered as double indexed in this package. This is because for a fixed integer $n$, the module $S_n$ comes equipped with $(n+1)$ face and degeneracy maps: $$d_{n,i} : S_n \to S_{n-1}, \quad \text{and} \quad s_{n,i} : S_n \to S_{n+1}.$$ The above maps can be accessed as follows:

i1 : R = QQ[a..d];

i2 : I = ideal(a*d-b*c, b^2-a*c, c^2-b*d);

o2 : Ideal of R

i3 : C = simplicialModule(freeResolution(R^1/I), 4, Degeneracy => true)

      1      4      9      16      25
o3 = R  <-- R  <-- R  <-- R   <-- R  <-- ...
                                   
     0      1      2      3       4

o3 : SimplicialModule

i4 : isWellDefined C

o4 = true

i5 : dd^C

               1                               4
o5 = (0, 0) : R  <--------------------------- R  : (1, 0)
                    | 1 b2-ac bc-ad c2-bd |

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

               4                                               9
     (1, 0) : R  <------------------------------------------- R  : (2, 0)
                    {0} | 1 b2-ac bc-ad 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 -a b  |

               4                                 9
     (1, 1) : R  <----------------------------- R  : (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) : R  <----------------------------- R  : (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 |

               9                                                               16
     (2, 0) : R  <----------------------------------------------------------- R   : (3, 0)
                    {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0 0 0 0  0  0  0  0 0 |
                    {2} | 0 0     0     0     1 0 0 0 0 0 -c d  0  0  0 0 |
                    {2} | 0 0     0     0     0 1 0 0 0 0 b  -c 0  0  0 0 |
                    {2} | 0 0     0     0     0 0 1 0 0 0 -a b  0  0  0 0 |
                    {2} | 0 0     0     0     0 0 0 1 0 0 0  0  -c d  0 0 |
                    {2} | 0 0     0     0     0 0 0 0 1 0 0  0  b  -c 0 0 |
                    {2} | 0 0     0     0     0 0 0 0 0 1 0  0  -a b  0 0 |
                    {3} | 0 0     0     0     0 0 0 0 0 0 0  0  0  0  1 0 |
                    {3} | 0 0     0     0     0 0 0 0 0 0 0  0  0  0  0 1 |

               9                                               16
     (2, 1) : R  <------------------------------------------- R   : (3, 1)
                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |

               9                                               16
     (2, 2) : R  <------------------------------------------- R   : (3, 2)
                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |

               9                                               16
     (2, 3) : R  <------------------------------------------- R   : (3, 3)
                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |

               16                                                                                   25
     (3, 0) : R   <------------------------------------------------------------------------------- R   : (4, 0)
                     {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     1 0 0 0 0 0 0 0 0 -c d  0  0  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 1 0 0 0 0 0 0 0 b  -c 0  0  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 1 0 0 0 0 0 0 -a b  0  0  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 1 0 0 0 0 0 0  0  -c d  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 1 0 0 0 0 0  0  b  -c 0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 0 1 0 0 0 0  0  -a b  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 0 0 1 0 0 0  0  0  0  -c d  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 0 0 0 1 0 0  0  0  0  b  -c 0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 0 0 0 0 1 0  0  0  0  -a b  0 0 0 0 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  1 0 0 0 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 1 0 0 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 1 0 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 1 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0 1 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0 0 1 |

               16                                                                 25
     (3, 1) : R   <------------------------------------------------------------- R   : (4, 1)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

               16                                                                 25
     (3, 2) : R   <------------------------------------------------------------- R   : (4, 2)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |

               16                                                                 25
     (3, 3) : R   <------------------------------------------------------------- R   : (4, 3)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |

               16                                                                 25
     (3, 4) : R   <------------------------------------------------------------- R   : (4, 4)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |

o5 : SimplicialModuleMap

i6 : C.dd

               1                               4
o6 = (0, 0) : R  <--------------------------- R  : (1, 0)
                    | 1 b2-ac bc-ad c2-bd |

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

               4                                               9
     (1, 0) : R  <------------------------------------------- R  : (2, 0)
                    {0} | 1 b2-ac bc-ad 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 -a b  |

               4                                 9
     (1, 1) : R  <----------------------------- R  : (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) : R  <----------------------------- R  : (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 |

               9                                                               16
     (2, 0) : R  <----------------------------------------------------------- R   : (3, 0)
                    {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0 0 0 0  0  0  0  0 0 |
                    {2} | 0 0     0     0     1 0 0 0 0 0 -c d  0  0  0 0 |
                    {2} | 0 0     0     0     0 1 0 0 0 0 b  -c 0  0  0 0 |
                    {2} | 0 0     0     0     0 0 1 0 0 0 -a b  0  0  0 0 |
                    {2} | 0 0     0     0     0 0 0 1 0 0 0  0  -c d  0 0 |
                    {2} | 0 0     0     0     0 0 0 0 1 0 0  0  b  -c 0 0 |
                    {2} | 0 0     0     0     0 0 0 0 0 1 0  0  -a b  0 0 |
                    {3} | 0 0     0     0     0 0 0 0 0 0 0  0  0  0  1 0 |
                    {3} | 0 0     0     0     0 0 0 0 0 0 0  0  0  0  0 1 |

               9                                               16
     (2, 1) : R  <------------------------------------------- R   : (3, 1)
                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |

               9                                               16
     (2, 2) : R  <------------------------------------------- R   : (3, 2)
                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |

               9                                               16
     (2, 3) : R  <------------------------------------------- R   : (3, 3)
                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |

               16                                                                                   25
     (3, 0) : R   <------------------------------------------------------------------------------- R   : (4, 0)
                     {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     1 0 0 0 0 0 0 0 0 -c d  0  0  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 1 0 0 0 0 0 0 0 b  -c 0  0  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 1 0 0 0 0 0 0 -a b  0  0  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 1 0 0 0 0 0 0  0  -c d  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 1 0 0 0 0 0  0  b  -c 0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 0 1 0 0 0 0  0  -a b  0  0  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 0 0 1 0 0 0  0  0  0  -c d  0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 0 0 0 1 0 0  0  0  0  b  -c 0 0 0 0 0 0 |
                     {2} | 0 0     0     0     0 0 0 0 0 0 0 0 1 0  0  0  0  -a b  0 0 0 0 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  1 0 0 0 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 1 0 0 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 1 0 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 1 0 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0 1 0 |
                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0 0 1 |

               16                                                                 25
     (3, 1) : R   <------------------------------------------------------------- R   : (4, 1)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

               16                                                                 25
     (3, 2) : R   <------------------------------------------------------------- R   : (4, 2)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |

               16                                                                 25
     (3, 3) : R   <------------------------------------------------------------- R   : (4, 3)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |

               16                                                                 25
     (3, 4) : R   <------------------------------------------------------------- R   : (4, 4)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |

o6 : SimplicialModuleMap

i7 : ss^C

               4                 1
o7 = (1, 0) : R  <------------- R  : (0, 0)
                    {0} | 1 |
                    {2} | 0 |
                    {2} | 0 |
                    {2} | 0 |

               9                       4
     (2, 0) : R  <------------------- R  : (1, 0)
                    {0} | 1 0 0 0 |
                    {2} | 0 0 0 0 |
                    {2} | 0 0 0 0 |
                    {2} | 0 0 0 0 |
                    {2} | 0 1 0 0 |
                    {2} | 0 0 1 0 |
                    {2} | 0 0 0 1 |
                    {3} | 0 0 0 0 |
                    {3} | 0 0 0 0 |

               9                       4
     (2, 1) : R  <------------------- R  : (1, 1)
                    {0} | 1 0 0 0 |
                    {2} | 0 1 0 0 |
                    {2} | 0 0 1 0 |
                    {2} | 0 0 0 1 |
                    {2} | 0 0 0 0 |
                    {2} | 0 0 0 0 |
                    {2} | 0 0 0 0 |
                    {3} | 0 0 0 0 |
                    {3} | 0 0 0 0 |

               16                                 9
     (3, 0) : R   <----------------------------- R  : (2, 0)
                     {0} | 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 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 |
                     {2} | 0 0 0 0 1 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 1 |

               16                                 9
     (3, 1) : R   <----------------------------- R  : (2, 1)
                     {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 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 1 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |

               16                                 9
     (3, 2) : R   <----------------------------- R  : (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 |
                     {2} | 0 0 0 0 1 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 1 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |

               25                                               16
     (4, 0) : R   <------------------------------------------- R   : (3, 0)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

               25                                               16
     (4, 1) : R   <------------------------------------------- R   : (3, 1)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

               25                                               16
     (4, 2) : R   <------------------------------------------- R   : (3, 2)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

               25                                               16
     (4, 3) : R   <------------------------------------------- R   : (3, 3)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

o7 : SimplicialModuleMap

i8 : C.ss

               4                 1
o8 = (1, 0) : R  <------------- R  : (0, 0)
                    {0} | 1 |
                    {2} | 0 |
                    {2} | 0 |
                    {2} | 0 |

               9                       4
     (2, 0) : R  <------------------- R  : (1, 0)
                    {0} | 1 0 0 0 |
                    {2} | 0 0 0 0 |
                    {2} | 0 0 0 0 |
                    {2} | 0 0 0 0 |
                    {2} | 0 1 0 0 |
                    {2} | 0 0 1 0 |
                    {2} | 0 0 0 1 |
                    {3} | 0 0 0 0 |
                    {3} | 0 0 0 0 |

               9                       4
     (2, 1) : R  <------------------- R  : (1, 1)
                    {0} | 1 0 0 0 |
                    {2} | 0 1 0 0 |
                    {2} | 0 0 1 0 |
                    {2} | 0 0 0 1 |
                    {2} | 0 0 0 0 |
                    {2} | 0 0 0 0 |
                    {2} | 0 0 0 0 |
                    {3} | 0 0 0 0 |
                    {3} | 0 0 0 0 |

               16                                 9
     (3, 0) : R   <----------------------------- R  : (2, 0)
                     {0} | 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 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 |
                     {2} | 0 0 0 0 1 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 1 |

               16                                 9
     (3, 1) : R   <----------------------------- R  : (2, 1)
                     {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 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 1 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |

               16                                 9
     (3, 2) : R   <----------------------------- R  : (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 |
                     {2} | 0 0 0 0 1 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 1 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 |

               25                                               16
     (4, 0) : R   <------------------------------------------- R   : (3, 0)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

               25                                               16
     (4, 1) : R   <------------------------------------------- R   : (3, 1)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

               25                                               16
     (4, 2) : R   <------------------------------------------- R   : (3, 2)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

               25                                               16
     (4, 3) : R   <------------------------------------------- R   : (3, 3)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

o8 : SimplicialModuleMap

i9 : assert(dd^C === C.dd)

i10 : assert(source dd^C === C)

i11 : assert(target dd^C === C)

i12 : assert(degree dd^C === -1)

i13 : assert(source ss^C === C)

i14 : assert(target ss^C === C)

i15 : assert(degree ss^C === 1)

The individual maps between terms are indexed by their source.

i16 : dd^C_(2,0)

o16 = {0} | 1 b2-ac bc-ad 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 -a b  |

              4      9
o16 : Matrix R  <-- R

i17 : assert(source dd^C_2 === C_2)

i18 : assert(target dd^C_2 === C_1)

face/degeneracy maps of a simplicial module -- access the face and degeneracy maps of a simplicial module

See also