  Gr\"obner bases are a fundamental tool when studying ideals in multivariate
polynomial rings. More recently there has been a growing interest in
transferring techniques from the field case to other coefficient rings, most
notably Euclidean domains and principal ideal rings. In this paper we will
consider multivariate polynomial rings over Dedekind domain. By generalizing
methods from the theory of finitely generated projective modules, we show that
it is possible to describe Gr\"obner bases over Dedekind domains in a way
similar to the case of principal ideal domains, both from a theoretical and
algorithmic point of view.
