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# 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:
• Strategy => ..., default value OTW, strategy for computing a localization of a D-module
• Outputs:
• , the localized module M_f = 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, 2 differential variable(s) 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

• DlocalizeAll -- localization of a D-module (extended version)
• DlocalizeMap -- localization map from a D-module to its localization
• AnnFs -- differential annihilator of a polynomial in a Weyl algebra
• Dintegration -- integration modules of a D-module

## Ways to use Dlocalize :

• Dlocalize(Ideal,RingElement)
• Dlocalize(Module,RingElement)

## For the programmer

The object Dlocalize is .