Algorithms forb-Functions, Restrictions, and Algebraic Local Cohomology Groups ofD-Modules

Abstract

Our purpose is to present algorithms for computing some invariants and functors attached to algebraic D-modules by using Grobner bases for differential operators. Let K be an algebraically closed field of characteristic zero and let X be a Zariski open set of K with a positive integer n. We fix a coordinate system x = (x1, . . . , xn) of X and write ∂ = (∂1, . . . , ∂n) with ∂i := ∂/∂xi. We denote by DX the sheaf of algebraic linear differential operators on X (cf. [3], [6]). Let M be a coherent left DX-module and u a section of M. Suppose that f = f(x) ∈ K[x] is an arbitrary non-constant polynomial of n variables. If M is holonomic, then for each point p of Y := {x ∈ X | f(x) = 0}, there exist a germ P (x, ∂, s) of DX [s] at p and a polynomial b(s) ∈ K[s] of one variable so that P (x, ∂, s)(f u) = b(s)f u (1.1)