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

## Synopsis

• Usage:
H = localCohom I
• Function: localCohom
• Inputs:
• I, an ideal, an ideal of R = k[x1,...,xn]
• Optional inputs:
• 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 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 |, | dX dY dZ |) } 1 => subquotient (| -dY-dZ -Y+Z 0 0 0 -dXdY-dXdZ dXY-dXZ XdX+1 0 0 |, | XY-XZ dY+dZ XdX+YdZ-ZdZ -YdZ+ZdZ+1 0 0 0 |) | -ZdZ-1 -YZ -YdY-ZdZ-2 -XdX-1 -3dXZdZ-3dX -dXZdZ-dX dXYZ XdXZ+Z dXYdY+dXZdZ+2dX XdXdY+dY | | 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.