deRhamAll(RingElement) -- deRham complex for the complement of a hypersurface

Synopsis

Function: deRhamAll
Usage:

deRhamAll f
Inputs:
- f
Optional inputs:
- Strategy => ..., default value Schreyer,
Outputs:
- a hash table, containing explicit cohomology classes in the deRham complex for the complement of the hypersurface {f = 0} and supplementary information

Description

The routine deRhamAll can be used to compute cup product structures as in the paper 'The cup product structure for complements of affine varieties' by Walther(2000).

For a more basic functionality see deRham.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : f = x^2-y^3 

        3    2
o2 = - y  + x

o2 : R

i3 : deRhamAll f

                                        2      1
o3 = HashTable{BFunction => (s - 1)(s - -)(s - -)                                                            }
                                        3      3
                                                    1
               CohomologyGroups => HashTable{0 => QQ }
                                                    1
                                             1 => QQ
                                             2 => 0
               LocalizeMap => | -x_2^3+x_1^2 |
                                               1                         2                         1
               OmegaRes => (QQ[x ..x , D ..D ])  <-- (QQ[x ..x , D ..D ])  <-- (QQ[x ..x , D ..D ])  <-- 0
                                1   2   1   2             1   2   1   2             1   2   1   2         
                                                                                                         3
                           0                         1                         2
               PreCycles => HashTable{0 => | 0 |}
                                           | 1 |
                                      1 => | 0 |
                                           | 1 |
                                           | 0 |
                                      2 => 0
               TransferCycles => HashTable{0 => | 3x_2^3-3x_1^2 |}
                                           1 => | 2x_1   |
                                                | 3x_2^2 |
                                           2 => 0
                                                  1                         3                         2
               VResolution => (QQ[x ..x , D ..D ])  <-- (QQ[x ..x , D ..D ])  <-- (QQ[x ..x , D ..D ])  <-- 0
                                   1   2   1   2             1   2   1   2             1   2   1   2         
                                                                                                            3
                              0                         1                         2

o3 : HashTable

Ways to use this method: