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

1998/07/31 by Uli Walther, Walther, Uli
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.math/9807176

25 pages, amsart, uses verbatim, amsmath, latexsym, amssymb, xypic, fixed typos

arxiv created 1998/10/19 · arxiv updated 2009/11/30

Abstract

Let X=\Cⁿ. In this paper we present an algorithm that computes the de Rham cohomology groups Hⁱ_dR(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 by T.~Oaku and N.~Takayama (\citeO-T2), who considered the case where Y is a hypersurface, and our methods from \citeW-1 for the computation of local cohomology. We further extend the algorithm to compute de Rham cohomology groups with support Hⁱ_dR,Z(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 the generalization of the restriction process from \citeO-T1 to complexes of modules over the Weyl algebra. All presented algorithms are based on Gröbner basis computations in the Weyl algebra.

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