An Algorithm for De Rham Cohomology Groups of the Complement of an Aane Variety via D-module Computation

0 Introduction In this paper, we give an algorithm to compute the following cohomology groups on U = C n n V (f) for any non-zero polynomial f 2 Qx 1 ; : : : ; x n ]: 1. H k (U; C U), where C U is the constant sheaf on U with stalk C. 2. H k (U; V), where V is the locally constant sheaf on U of rank one deened by a multi-valued function f a 1 1 f a d d with polynomials f 1 ; : : : ; f d 2 Qx] such that f = f 1 f d and a 1 ; : : : ; a d 2 Q. We also give partial results on the computation of cohomology groups on U for a locally constant sheaf of general rank as well as on the computation of H k (C n n Z; C), where Z is a general algebraic set of C n. Our algorithm is based on computations of Grr obner bases in the ring of diierential operators with polynomial coeecients, algorithms for functors in the theory of D-modules ((32] and 33]), and Grothendieck-Deligne comparison theorem 15], 10], which relates sheaf cohomology groups and algebraic de Rham cohomology groups. One advantage of the use of the ring of diierential operators in algebraic geometry is that, for example, Qx; 1=x], which is the localized module of Qx] along x, is not nitely generated as a Qx]-module, but it can be regarded as a nitely generated Qhx; @ x i-module with @ x = @=@x. In fact, we 1