Approximation theorems for Krull domains

Abstract

Let R be an integrally closed domain, and denote by I(R) the multiplicative group of all invertible fractional ideals of R. Let {Vi}i∈I be the family of valuation overrings of R, and denote by Gi the corresponding value group of the valuation domain Vi. We show that R = Ti∈I Vi, and there is a map from I(R) into Qi∈I Gi, the cardinal product of the Gi’s. Furthermore, it is well known when R is a Dedekind domain, this map becomes an isomorphism onto `i∈I Gi, the cardinal sum of the Gi’s. In this case, Gi ∼= Z for each i. It is shown, by J. Brewer and L. Klingler, that this map is also an isomorphism onto`i∈I Gi when R is an h-local Prüfer domain. In this thesis, we investigate the existence of such a map, and whether it is injective when R is a Krull domain.