# Dlocalize -- localization of a D-module

## Synopsis

• Usage:
Dlocalize(M,f), Dlocalize(I,f)
• Inputs:
• M, , over the Weyl algebra D
• I, an ideal, which represents the module M = D/I
• f, , a polynomial
• Optional inputs:
• Outputs:
• , the localized module Mf = M[f-1] as a D-module

## Description

One of the nice things about D-modules is that if a finitely generated D-module is specializable along f, then it's localization with respect to f is also finitely generated. For instance, this is true for all holonomic D-modules.

There are two different algorithms for localization implemented. The first appears in the paper 'A localization algorithm for D-modules' by Oaku-Takayama-Walther (1999). The second is due to Oaku and appears in the paper 'Algorithmic computation of local cohomology modules and the cohomological dimension of algebraic varieties' by Walther(1999)

 ```i1 : W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}] o1 = W o1 : PolynomialRing``` ```i2 : M = W^1/(ideal(x*Dx+1, Dy)) o2 = cokernel | xDx+1 Dy | 1 o2 : W-module, quotient of W``` ```i3 : f = x^2-y^3 3 2 o3 = - y + x o3 : W``` ```i4 : Mf = Dlocalize(M, f) o4 = cokernel | 3xDx+2yDy+15 y3Dy-x2Dy+6y2 | 1 o4 : W-module, quotient of W```