Macaulay2 » Documentation
Packages » BernsteinSato :: logCohomology(RingElement)
next | previous | forward | backward | up | index | toc

# logCohomology(RingElement) -- logarithmic cohomology groups in two variables

- Function: logCohomology
- Usage:
    `logCohomology f`
- Inputs:
  - f, a ring element, polynomial in two variables
- Outputs:
  - a hash table, with entries {VResolution, Input, TransferCycles, CohomologyGroups, PreCycles, OmegaRes, LocalizeMap, BFunction}

## Description

For a polynomial f in two variables executes the algorithm described in Castro-Jimenez and Takayama "The Computation of the Logarithmic Cohomology for Plane Curves" (arXiv:0712.0001).

    i1 : S=QQ[x,y];

    i2 : f=x*y*(x-y);

    i3 : logCohomology(f)

    o3 = HashTable{BFunction => (s - 1)                                                                    }
                                                        1
                   CohomologyGroups => HashTable{0 => QQ }
                                                        3
                                                 1 => QQ
                                                        2
                                                 2 => QQ
                   Input => cokernel | xdx+ydy+3 xydy-y2dy+x-2y |
                   LocalizeMap => LocMap
                                                 1                       2                       1
                   OmegaRes => (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- 0

                               0                       1                       2                       3
                   PreCycles => HashTable{0 => | x |       }
                                               | 1 |
                                          1 => | -x -y 0  |
                                               | 0  0  -1 |
                                               | 0  0  0  |
                                          2 => | -y -x |
                   TransferCycles => HashTable{0 => | -x2y+xy2 |    }
                                               1 => | xy y2 -xy+y2 |
                                                    | x2 xy 0      |
                                               2 => | -y -x |
                                                    1                       3                       2
                   VResolution => (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- 0

                                  0                       1                       2                       3

    o3 : HashTable

## See also

- deRham -- deRham cohomology groups for the complement of a hypersurface

## Ways to use this method:

- logCohomology(RingElement) -- logarithmic cohomology groups in two variables

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BernsteinSato/DOC/other.m2:107:0.