An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation

We also give partial results on computation of cohomology groups on U for a locally constant sheaf of general rank and on computation of H(C \ Z,C) where Z is a general algebraic set. Our algorithm is based on computations of Gröbner bases in the ring of differential operators with polynomial coefficients, algorithms for functors in the theory of D-modules ([24] and [25]), and Grothendieck-Deligne comparison theorem [12], [8], which relates sheaf cohomology groups and de Rham cohomology groups. One advantage of the use of the ring of differential operators in algebraic geometry is that, for example, Q[x, 1/x], which is the localized module of Q[x] along x, is not finitely generated as a Q[x]-module, but it can be regarded as a finitely generated Q〈x, ∂x〉-module where ∂x = ∂/∂x. In fact, we have 1/x = (−1)(1/(k − 1)!) (