  We give algorithms for the computation of the algebraic de Rham cohomology of
open and closed algebraic sets inside projective space or other smooth complex
toric varieties. The methods, which are based on Gr\"obner basis computations
in rings of differential operators, can also be used to compute the cohomology
of intersections of smooth closed and open subsets, and in certain situations
the cup-product structure.
  We give some examples which were carried out with the help of \Mtwo.
