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

✍ Scribed by Uli Walther

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 493 KB
• Volume: 29
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.1999.0328

✦ Synopsis

Let X = C n . In this paper we present an algorithm that computes the de Rham cohomology groups H i 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 in Oaku 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 H i 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 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 on V d -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).