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

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

• DlocalizeAll -- localization of a D-module (extended version)
• DlocalizeMap -- localization map from a D-module to its localization
• AnnFs -- the annihilating ideal of f^s
• Dintegration -- integration modules of a D-module