Algorithmic Computation of de Rham Cohomology of Complements of Complex Affine Varieties

AbstractLet X=Cn. In this paper we present an algorithm that computes the de Rham cohomology groups HdRi(U,C ) where U is the complement of an arbitrary Zariski-closed set Y in X. Our algorithm is a merger of the algorithm given inOaku and Takayama (1999), who considered the case where Y is a hypersurface, and our methods from Walther (1999) for the computation of local cohomology. We further extend the algorithm to compute de Rham cohomology groups with supports HdR, Zi(U,C ) where again U is an arbitrary Zariski-open subset of X and Z is an arbitrary Zariski-closed subset of U. Our main tool is a generalization of the restriction process from Oaku and Takayama (in press) to complexes of modules over the Weyl algebra. The restriction rests on an existence theorem onVd -strict resolutions of complexes that we prove by means of an explicit construction via Cartan–Eilenberg resolutions. All presented algorithms are based on Gröbner basis computations in the Weyl algebra and the examples are carried out using the computer system Kan by Takayama (1999)