Extensions of Valuation Domains and Going-up

Abstract

Suppose F is a field with valuation v and valuation domain O_v, E/F is a finite-dimensional field extension, and R is an O_v-subalgebra of E such that F ⋅ R = E and R ∩ F = O_v. It is known that R satisfies LO, INC, GD and SGB over O_v; it is also known that under certain conditions R satisfies GU over O_v. In this paper, we present a necessary and sufficient condition for the existence of such R that does not satisfy GU over O_v. We also present an explicit example of such R that does not satisfy GU over O_v.