Algorithms for D-modules — restriction, tensor product, localization, and local cohomology groups

We consider an algebraic D-module M on the affine space, i.e. a system of linear partial differential equations with polynomial coefficients. We give an algorithm for computing the cohomology groups of the restriction of M to a linear subvariety by using a free resolution of M adapted to the V-filtration. Our algorithm works, at least, if M is holonomic. As applications, we obtain algorithms for computing tensor product, localization, and algebraic local cohomology groups of holonomic systems.