filteredComplex(SpectralSequence) -- obtain the filtered complex associated to the spectral sequence

Synopsis

Function: filteredComplex
Usage:

K = filteredComplex E
Inputs:
- E, a spectral sequence,
Optional inputs:
- ReducedHomology => a Boolean value, default value true,
- Shift => an integer, default value 0,
Outputs:
- K, a filtered chain complex,

Description

Produces the filtered complex which determined the spectral sequence. Consider the spectral sequence $E$ which arises from a nested list of simplicial complexes.

i1 : A = QQ[a,b,c,d];

i2 : D = simplicialComplex {a*d*c, a*b, a*c, b*c};

i3 : F2D = D;

i4 : F1D = simplicialComplex {a*c, d};

i5 : F0D = simplicialComplex {a,d};

i6 : K = filteredComplex {F2D, F1D, F0D};

i7 : E = spectralSequence(K) ;

The underlying filtered chain complex can be recovered from the spectral sequence by:

i8 : C = filteredComplex E

o8 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0
                                               
          -1          0           1           2

     0 : image | 1 | <-- image | 1 0 | <-- image 0 <-- image 0
                               | 0 0 |                  
         -1                    | 0 0 |     1           2
                               | 0 1 |
                          
                         0

     1 : image | 1 | <-- image | 1 0 0 | <-- image | 0 | <-- image 0
                               | 0 0 0 |           | 1 |      
         -1                    | 0 1 0 |           | 0 |     2
                               | 0 0 1 |           | 0 |
                                                   | 0 |
                         0                    
                                             1

           1       4       5       1
     2 : QQ  <-- QQ  <-- QQ  <-- QQ
                                  
         -1      0       1       2

o8 : FilteredComplex

Ways to use this method: