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Dmodules :: Dlocalize

Dlocalize -- localization of a D-module

Synopsis

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 variables
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

Ways to use Dlocalize :

For the programmer

The object Dlocalize is a method function with options.