Infinitesimally stable theories of henselian rings
type: Item
instance of: scholarly article
title: Infinitesimally stable theories of henselian rings
zbMATH Open document ID: 0792.03023
DOI: 10.1016/0022-4049(93)90054-W
author: Luc Bélair
published in: Journal of Pure and Applied Algebra
publication date: 1993-12-20T00:00:00Z
review text: This note considers first-order theories of henselian rings which are stable for the étale factorization system and gives continuum many examples as stabilizations of model-complete complete theories of fields. The framework of the paper is \textit{E. Robinson's} work ``Stable theories of local rings'' [Var. Publ. Ser., Aarhus Univ. 35, 257-280 (1983; Zbl 0531.03044)]. Going from a suitable theory of fields to its ``infinitesimal'' stabilization can be regarded as a logical process from the point of view of categorical logic. From that angle, the examples of henselian rings given arise from the first-order theory of their residue field. The fields considered are certain fields \(K\) of algebraic numbers such that every absolutely irreducible affine curve over \(K\) has a \(K\)- rational point (i.e. regularly closed or p.a.c.). The result follows by exhibiting an adjusted axiomatization of these fields. It consists roughly of expressing the relevant known properties as sequents \(\varphi(x)\to \psi(x)\), where \(\varphi(x)\), \(\psi(x)\) define open sets in the étale topology. In the end, the examples consist of henselian local rings with a residue field elementarily equivalent to one of the above fields.
reviewed by: Luc Bélair
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 13L05
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 13J15
zbMATH DE Number: 445048
zbMATH Keywords: fields of algebraic numbers
zbMATH Keywords: first-order theories of henselian rings
zbMATH Keywords: étale factorization system
zbMATH Keywords: stabilizations of model-complete complete theories of fields
zbMATH Keywords: absolutely irreducible affine curve
zbMATH Keywords: étale topology
zbMATH Keywords: henselian local rings
MaRDI profile type: Publication
cites work: Indécidabilité de la théorie des anneaux de séries formelles à plusieurs indéterminées
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full work available at URL: 0022-4049(93)90054-w
OpenAlex ID: W2012938087
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description: scientific article; zbMATH DE number 445048