Holomorphically convex compact sets and cohomology (Q1089474)

Let D be a domain of a Stein manifold X of dimension \(n>1\). In the paper are given sufficient conditions for the cohomology groups \(H^ q(D;{\mathcal F})\) and \(H^ q_ k(D;{\mathcal F})\) to be Frechet-Swartz and dual on Frechet-Swartz spaces respectively, for every coherent sheaf \({\mathcal F}\) on X. Moreover it is shown that these conditions are also necessary if X is a two-dimensional manifold. A duality theorem is then given for local cohomology groups with support in a coherent sheaf on X.