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

# localCohom(Ideal) -- local cohomology of a polynomial ring

- Function: localCohom
- Usage:
    `H = localCohom I`
- Inputs:
  - I, an ideal, an ideal of R = k[x1,...,xn]
- Optional inputs:
  - LocStrategy => ..., default value null, specify localization strategy for local cohomology
  - Strategy => ..., default value Walther, specify strategy for local cohomology
- Outputs:
  - H, a hash table, each entry of H has an integer key and contains the cohomology module in the corresponding degree.

## Description

    i1 : W = QQ[X, dX, Y, dY, Z, dZ, WeylAlgebra=>{X=>dX, Y=>dY, Z=>dZ}]

    o1 = W

    o1 : PolynomialRing, 3 differential variable(s)

    i2 : I = ideal (X*(Y-Z), X*Y*Z)

    o2 = ideal (X*Y - X*Z, X*Y*Z)

    o2 : Ideal of W

    i3 : h = localCohom I

    o3 = HashTable{0 => subquotient (| dZ dY dX |, | dZ dY dX |)                                                                                                                                                }
                   1 => subquotient (| 0          0      0           -dXdY-dXdZ dXY-dXZ XdX+1  0               0        XdXdZ-dY   -XdXZ-Y      |, | XY-XZ dY+dZ XdX+YdZ-ZdZ -YdZ+ZdZ+1 0       0       0     |)
                                     | -YdY-ZdZ-2 -XdX-1 -3dXZdZ-3dX -dXZdZ-dX  dXYZ    XdXZ+Z dXYdY+dXZdZ+2dX XdXdY+dY XdXZdZ+XdX -XdXZ2-YZ-Z2 |  | XYZ   0     0           0          YdY-ZdZ XdX-ZdZ ZdZ+1 |
                   2 => cokernel | -XYZ XY-XZ 3XdX-2YdY-2ZdZ YdY+ZdZ+3 Y2dY-2YdYZ-2YZdZ+Z2dZ |

    o3 : HashTable

    i4 : pruneLocalCohom h

    o4 = HashTable{0 => 0                         }
                   1 => | dZ dY X |
                   2 => | Y-Z Z2 dYZ+ZdZ+2 XdX+2 |

    o4 : HashTable

## Caveat

The modules returned are not simplified, use pruneLocalCohom.

## See also

- pruneLocalCohom -- prunes local cohomology modules

## Ways to use this method:

- localCohom(Ideal) -- local cohomology of a polynomial ring

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BernsteinSato/DOC/localCohom.m2:110:0.