A Localization Algorithm for D-modules

We present a method to compute the holonomic extension of a D-module from a Zariski open set in affine space to the whole space. A particular application is the localization of coherent D-modules which are holonomic on the complement of an affine variety. Throughout this article let R = K[x1, . . . , xn] be the ring of polynomials in n variables over the field K of characteristic zero and D = R〈∂1, . . . , ∂n〉 the n-th Weyl algebra over K. Here, ∂i = ∂ ∂xi . Suppose f ∈ R, let J be a left ideal of D and set M = D/J . Let X = Kn, Y = V(f) ⊆ X, U = X \ Y and j : U →֒ X the natural inclusion. In the landmark paper [K2] M. Kashiwara proved that ∫ j j −1M is holonomic provided that M is holonomic on U , but not necessarily on all of X. In this note we make this important result algorithmic in the sense that we provide an algorithm that computes the module structure of ∫ j j −1M over D. We remark that U is affine and X = Kn, and so ∫ j j −1M = Rf ⊗R M as D-module. An algorithm to compute Rf⊗RM as aD-module has been given in [Oa3] under the assumption that M is holonomic on Kn and f -torsion free, and in [O-T2] under the assumption that M is holonomic on Kn. The advantage of our new algorithm, besides removing the requirement of holonomicity on V(f), is that the natural map M → Rf⊗RM can be traced, which is an important property for other computations as local cohomology ([Oa3], [W1]) and de Rham cohomology ([O-T1], [W2]). 1. Description of the Algorithm We shall start with a motivating construction which will also provide a skeleton of the proof of the correctness of our algorithm. Assume that K is in fact the field C of complex numbers. Let s, v be two new variables and let Dv = D〈v, ∂v〉. Notation 1. Throughout we will denote ∂f/∂xi by fi. If P ∈ Dv is an operator interpreted as a polynomial in ∂1, . . . , ∂n, we will simply write P (∂x). Similarly we will abbreviate P (∂1 − v f1∂v, . . . , ∂n − v fn∂v) by P (∂x − v fx∂v). Consider the ringDv and the C-vectorspace Rf [s]⊗f −s. Here, f−s should be thought of as a symbol that behaves like the complex valued function f−s Date: November 4, 1998. 1 2 TOSHINORI OAKU, NOBUKI TAKAYAMA AND ULI WALTHER under differentiation and the tensor is over C. We shall turn this vectorspace into a module over Dv by means of the action • defined as follows. We require that all xi act as multiplication on the left factor, while ∂i acts by the “product rule” ∂i • ( g(x,s) fm ⊗ f −s) = ∂i( g(x,s) fm )⊗ f −s+ g(x,s) fm+1 (−s)fi⊗ f −s. Thus, we only need to define the action of v and ∂v. We set