localCohom(Ideal,Module) -- local cohomology of a D-module

Synopsis

Function: localCohom
Usage:

H = localCohom(I,M)
Inputs:
- I, an ideal of R = k[x₁,...,x_n]
- M, a holonomic module over Weyl algebra A_n(k)
Optional inputs:
- LocStrategy => ..., default value null, specify localization strategy for local cohomology
- Strategy => ..., default value Walther, specify strategy for local cohomology
Outputs:
- H, 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, W^1 / ideal{dX,dY,dZ})

o3 = HashTable{0 => subquotient (| dZ dY dX |, | dX dY dZ |)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   }
               1 => subquotient (| 0      0        -dY-dZ   0          0           0          0         0             dXdY+dXdZ   0             0             -dYdZ-dZ^2                               0             0                  XdX+2     XdX+4YdZ-4ZdZ-6             0             0             -dX^2dY-dX^2dZ   XdXdZ+2dZ              0                  3dXdYdZ+3dXdZ^2                                            0                  0                  -2XdXY+3XdXZ-4Y+6Z      -XdX^2-3dX     -XdX^2-6dXYdZ+6dXZdZ+9dX                -3dX^2YdZ+3dX^2ZdZ+6dX^2             2dX^2YdZ-2dX^2ZdZ-4dX^2              -2XdXYdZ+3XdXZdZ+2XdX+4YdY-6dYZ+4           2XdXYdZ-3XdXZdZ-6XdX-2YdZ                  -XdX^2dZ-3dXdZ                  -dX^2dYdZ-dX^2dZ^2                                       2XdXYZ-3XdXZ2+8Y2-12YZ+2Z2   2dX^2YdY-3dX^2dYZ+2dX^2YdZ-3dX^2ZdZ            4dX^2Y2-8dX^2YZ+4dX^2Z2 2XdX^2Y-3XdX^2Z+6dXY-9dXZ      4dX^2YdYdZ-6dX^2dYZdZ+4dX^2YdZ^2-6dX^2ZdZ^2-6dX^2dY-6dX^2dZ                                                4dX^2Y2dZ-6dX^2YZdZ+2dX^2Z2dZ-8dX^2Y+4dX^2Z                -2XdX^2YdZ+3XdX^2ZdZ+6XdX^2+3dXYdZ                      2XdX^2YdZ-3XdX^2ZdZ-2XdX^2-6dXYdY+9dXdYZ-6dX           -2XdX^2YZ+3XdX^2Z2-12dXY2+18dXYZ-3dXZ2   2dX^2Y2dY-3dX^2YdYZ+2dX^2YZdZ-3dX^2Z2dZ+4dX^2Y-6dX^2Z      |, | X2Y2-2X2YZ+X2Z2 dY+dZ -YdZ+ZdZ+2 -XdX-2 0      0      0      |)
                                 | -XdX-2 -XdX+YdY -Z2dZ-2Z -dYZdZ-2dY XdXdZ-YdYdZ -XdXdY-2dY XdX^2+3dX 2XdX^2-3dXYdY dXZ2dZ+2dXZ dXdYZdZ+2dXdY dX^2ZdZ+2dX^2 -XdXZdZ+YdYZdZ-dYZ2dZ-Z2dZ^2-2dYZ-4ZdZ-2 XdXdYdZ+2dYdZ -2XdX^2dZ+3dXYdYdZ XdXZ2+2Z2 XdXZ2+4YZ2dZ+4Z3dZ+8YZ+10Z2 dX^2YdY+2dX^2 XdX^2dY+3dXdY -dX^2Z2dZ-2dX^2Z XdXZ2dZ+2XdXZ+2Z2dZ+4Z -dX^2dYZdZ-2dX^2dY 2XdX^2ZdZ-3dXYdYZdZ+3dXdYZ2dZ+3dXZ2dZ^2+6dXdYZ+12dXZdZ+6dX -dX^2YdYdZ-2dX^2dZ -XdX^2dYdZ-3dXdYdZ -2XdXYZ2-XdXZ3-4YZ2-2Z3 -XdX^2Z2-3dXZ2 -XdX^2Z2-6dXYZ2dZ-6dXZ3dZ-12dXYZ-15dXZ2 -3dX^2YZ2dZ-dX^2Z3dZ-6dX^2YZ-2dX^2Z2 2dX^2YZ2dZ+2dX^2Z3dZ+4dX^2YZ+4dX^2Z2 -2XdXYZ2dZ-XdXZ3dZ-4XdXYZ-4XdXZ2+4YdYZ2+4Z2 2XdXYZ2dZ+XdXZ3dZ+4XdXYZ+2XdXZ2-2YZ2dZ-4YZ -XdX^2Z2dZ-2XdX^2Z-3dXZ2dZ-6dXZ dX^2YdYZdZ-dX^2dYZ2dZ-dX^2Z2dZ^2-2dX^2dYZ-2dX^2ZdZ-2dX^2 2XdXYZ3+XdXZ4+8Y2Z2+4YZ3+2Z4 2dX^2YdYZ2+2dX^2YZ2dZ+dX^2Z3dZ+4dX^2YZ+6dX^2Z2 4dX^2Y2Z2               2XdX^2YZ2+XdX^2Z3+6dXYZ2+3dXZ3 4dX^2YdYZ2dZ+3dX^2dYZ3dZ+4dX^2YZ2dZ^2+2dX^2Z3dZ^2+8dX^2YdYZ+6dX^2dYZ2+16dX^2YZdZ+18dX^2Z2dZ+8dX^2Y+24dX^2Z 4dX^2Y2Z2dZ+2dX^2YZ3dZ+2dX^2Z4dZ+8dX^2Y2Z+4dX^2YZ2+4dX^2Z3 -2XdX^2YZ2dZ-XdX^2Z3dZ-4XdX^2YZ-2XdX^2Z2+3dXYZ2dZ+6dXYZ 2XdX^2YZ2dZ+XdX^2Z3dZ+4XdX^2YZ+4XdX^2Z2-6dXYdYZ2-6dXZ2 -2XdX^2YZ3-XdX^2Z4-12dXY2Z2-6dXYZ3-3dXZ4 2dX^2Y2dYZ2+dX^2YdYZ3+2dX^2YZ3dZ+dX^2Z4dZ+8dX^2YZ2+4dX^2Z3 |  | X2Y2Z2          0     0          0      -ZdZ-2 -YdY-2 -XdX-2 |
               2 => cokernel | -X2Y2Z2 X2Y2-2X2YZ+X2Z2 -YdY-ZdZ-6 -XdX-4 YZdZ-Z2dZ+2Y-4Z |

o3 : HashTable

i4 : pruneLocalCohom h

o4 = HashTable{0 => 0                                                                               }
               1 => | dZ dY XdX+3 X3 |
               2 => | dYZ+YdZ+2 YdY+ZdZ+6 Y2-2YZ+Z2 XdX+4 YZdZ-Z2dZ+2Y-4Z 2YZ3-Z4 Z4dZ+2YZ2+4Z3 Z5 |

o4 : HashTable

Caveat

The modules returned are not simplified, use pruneLocalCohom.

Ways to use this method: