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

Synopsis

Description

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

o1 = W

o1 : PolynomialRing
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})
WARNING! Dlocalization is an obsolete name for Dlocalize
WARNING! Dlocalization is an obsolete name for Dlocalize
WARNING! Dlocalization is an obsolete name for Dlocalize

o3 = HashTable{0 => subquotient (| dZ dY dX |, | dX dY dZ |)          
               1 => subquotient (| 0      0        -dY-dZ   0         
                                 | -XdX-2 -XdX+YdY -Z2dZ-2Z -dYZdZ-2dY
               2 => cokernel | -X2Y2Z2 X2Y2-2X2YZ+X2Z2 YdY+ZdZ+6 XdX+4
     ------------------------------------------------------------------------
                                                   
     0           0          0         0            
     XdXdZ-YdYdZ -XdXdY-2dY XdX^2+3dX 2XdX^2-3dXYdY
     YZdZ-Z2dZ+2Y-4Z dYZ2dZ+Z2dZ^2+4dYZ+8ZdZ+10 |
     ------------------------------------------------------------------------
                                                                             
     -dYdZ-dZ^2                 -XdX-YdZ+ZdZ        dXdY+dXdZ   0            
     -dYZ2dZ-Z2dZ^2-2dYZ-4ZdZ-2 -YZ2dZ+Z3dZ-2YZ+2Z2 dXZ2dZ+2dXZ dXdYZdZ+2dXdY
     ------------------------------------------------------------------------
                                                               
     0             0             0                  4Y2-8YZ+4Z2
     dX^2ZdZ+2dX^2 XdXdYdZ+2dYdZ -2XdX^2dZ+3dXYdYdZ 4Y2Z2      
     ------------------------------------------------------------------------
                                                                             
     -XdX+2YdY-3dYZ-ZdZ+2 XdX+2     XdX+4YdZ-4ZdZ-6             0            
     -XdXZ2+2YdYZ2-Z3dZ   XdXZ2+2Z2 XdXZ2+4YZ2dZ+4Z3dZ+8YZ+10Z2 dX^2YdY+2dX^2
     ------------------------------------------------------------------------
                                                                
     0             2XdX^2+3dXYdZ-3dXZdZ         -dX^2dY-dX^2dZ  
     XdX^2dY+3dXdY 3dXYZ2dZ-3dXZ3dZ+6dXYZ-6dXZ2 -dX^2Z2dZ-2dX^2Z
     ------------------------------------------------------------------------
                                                                 
     XdXdZ+2dZ              0                  0                 
     XdXZ2dZ+2XdXZ+2Z2dZ+4Z -dX^2dYZdZ-2dX^2dY -dX^2YdYdZ-2dX^2dZ
     ------------------------------------------------------------------------
                                                                  
     0                  -4dXY2+8dXYZ-4dXZ2 -2XdXY+3XdXZ-4Y+6Z     
     -XdX^2dYdZ-3dXdYdZ -4dXY2Z2           -2XdXYZ2-XdXZ3-4YZ2-2Z3
     ------------------------------------------------------------------------
                                                           
     -XdX^2-3dX     -XdX^2-6dXYdZ+6dXZdZ+9dX               
     -XdX^2Z2-3dXZ2 -XdX^2Z2-6dXYZ2dZ-6dXZ3dZ-12dXYZ-15dXZ2
     ------------------------------------------------------------------------
                                    
     2XdX^2-6dXYdY+9dXdYZ+3dXZdZ-6dX
     2XdX^2Z2-6dXYdYZ2+3dXZ3dZ      
     ------------------------------------------------------------------------
                                                             
     3dXdYdZ+3dXdZ^2                                         
     -2XdX^2YdY+3dXdYZ2dZ+3dXZ2dZ^2-4XdX^2+6dXdYZ+12dXZdZ+6dX
     ------------------------------------------------------------------------
                                                                  
     -4dX^2YdZ+4dX^2ZdZ+8dX^2 2dX^2YdZ-2dX^2ZdZ-4dX^2             
     -4dX^2YZ2dZ-8dX^2YZ      2dX^2YZ2dZ+2dX^2Z3dZ+4dX^2YZ+4dX^2Z2
     ------------------------------------------------------------------------
                                    
     -XdX^2dZ-3dXdZ                 
     -XdX^2Z2dZ-2XdX^2Z-3dXZ2dZ-6dXZ
     ------------------------------------------------------------------------
                                                             
     -dX^2dYdZ-dX^2dZ^2                                      
     dX^2YdYZdZ-dX^2dYZ2dZ-dX^2Z2dZ^2-2dX^2dYZ-2dX^2ZdZ-2dX^2
     ------------------------------------------------------------------------
                                                                           
     2dX^2YdY-3dX^2dYZ+2dX^2YdZ-3dX^2ZdZ            4dX^2Y2-8dX^2YZ+4dX^2Z2
     2dX^2YdYZ2+2dX^2YZ2dZ+dX^2Z3dZ+4dX^2YZ+6dX^2Z2 4dX^2Y2Z2              
     ------------------------------------------------------------------------
                                                                        
     2XdX^2Y-3XdX^2Z+6dXY-9dXZ      |, | X2Y2-2X2YZ+X2Z2 dY+dZ YdZ-ZdZ-2
     2XdX^2YZ2+XdX^2Z3+6dXYZ2+3dXZ3 |  | X2Y2Z2          0     0        
     ------------------------------------------------------------------------
                               }
     XdX+2 0     0     0     |)
     0     ZdZ+2 YdY+2 XdX+2 |

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.

See also