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Acta Mathematica Vietnamica

Extensions of Valuation Domains and Going-up

icon-email Shai Sarussi

Abstract

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