Topological aspects of the ideal theory in rings of integer-valued polynomials

Let D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of Int(D) lying over a given maximal ideal m of D. We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the m-adic completion of D. Moreover, these results are naturally extended to overrings of Int(D). As an application we provide explicit constructions of divisorial ideals of Int(D) which are not finitely generated.