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# Dlocalize -- localization of a D-module

- Usage:
    `Dlocalize(M,f), Dlocalize(I,f)`
- Inputs:
  - M, a module, over the Weyl algebra D
  - I, an ideal, which represents the module M = D/I
  - f, a ring element, a polynomial
- Optional inputs:
  - Strategy => ..., default value OTW, strategy for computing a localization of a D-module
- Outputs:
  - a module, 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

## See also

- 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 a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2:47:0.