Dlocalize -- localization of a D-module

Synopsis

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

Ways to use Dlocalize:

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

For the programmer

The object Dlocalize is a method function with options.

Dlocalize -- localization of a D-module

Synopsis

Description

See also

Ways to use Dlocalize:

For the programmer