# 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:
• , 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