Separably real closed local rings
type: Item
instance of: scholarly article
title: Separably real closed local rings
zbMATH Open document ID: 0604.18003
DOI: 10.1016/0022-4049(86)90068-X
author: André Joyal
author: G. E. Reyes
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: In topos theory several different constructions of ''variable reals'' (Dedekind reals using cuts, Cauchy reals using sequences) can be performed, giving in some well chosen toposes various sheaves of real functions (continuous, \(C^{\infty}\) analytic, etc...). These sheaves of functions are not real closed fields (in fact they are not fields). This fact is discussed in the introduction of the paper. A. Kock introduced the definition of separably real closed local ring (i.e. local Henselian with real closed residue field) and conjectured that the Dedekind reals in any elementary topos is always a separably real local ring. The paper is devoted to the proof of that conjecture, which was known in particular cases, and of other related results. The proof relies on a coherent axiomatization of separably real closed local rings, which uses the finiteness theorem of semi-algebraic geometry (a topological refinement of Tarski-Seidenberg elimination of quantifiers for real closed fields). The paper is clearly written; I regret that it does not include any precise references to the construction of ''variable reals'' in elementary toposes.
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 13J15
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 03C10
zbMATH DE Number: 3975172
zbMATH Keywords: topos
zbMATH Keywords: variable reals
zbMATH Keywords: separably real closed local ring
zbMATH Keywords: Dedekind reals
zbMATH Keywords: coherent axiomatization
zbMATH Keywords: semi-algebraic geometry
zbMATH Keywords: elimination of quantifiers
reviewed by: Marie-Françoise Roy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3975172