Let . In this paper we present an algorithm that computes the cup product structure for the de Rham cohomology ring where U is the complement of an arbitrary Zariski-closed set Y in X. Our method relies on the fact that Tor is a balanced functor, a property which we make algorithmic, as well as a technique to extract explicit representatives of cohomology classes in a restriction or integration complex. We also present an alternative approach to computing V-strict resolutions of complexes that is seemingly much more efficient than the algorithm presented in Walther (J. Symbolic Comput. 29 (2000) 795–839). All presented algorithms are based on Gröbner basis computations in the Weyl algebra.
