{"title":"Extensions of Valuation Domains and Going-up","authors":"Shai Sarussi","doi":"10.1007/s40306-021-00461-y","DOIUrl":null,"url":null,"abstract":"<div><p>Suppose <i>F</i> is a field with valuation <i>v</i> and valuation domain <i>O</i><sub><i>v</i></sub>, <i>E</i>/<i>F</i> is a finite-dimensional field extension, and <i>R</i> is an <i>O</i><sub><i>v</i></sub>-subalgebra of <i>E</i> such that <i>F</i> ⋅ <i>R</i> = <i>E</i> and <i>R</i> ∩ <i>F</i> = <i>O</i><sub><i>v</i></sub>. It is known that <i>R</i> satisfies LO, INC, GD and SGB over <i>O</i><sub><i>v</i></sub>; it is also known that under certain conditions <i>R</i> satisfies GU over <i>O</i><sub><i>v</i></sub>. In this paper, we present a necessary and sufficient condition for the existence of such <i>R</i> that does not satisfy GU over <i>O</i><sub><i>v</i></sub>. We also present an explicit example of such <i>R</i> that does not satisfy GU over <i>O</i><sub><i>v</i></sub>.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2021-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-021-00461-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Suppose F is a field with valuation v and valuation domain Ov, E/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.
期刊介绍：
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.
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