Model theory and spectra
type: Item
instance of: scholarly article
title: Model theory and spectra
zbMATH Open document ID: 0698.03025
DOI: 10.1016/0022-4049(90)90071-O
author: Robert O. Robson
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: The aim of the present paper is to consider some spectrum functors which have significance in algebraic geometry. The author sets up a spectrum functor \({\mathcal T}\)-spec from the category of commutative rings with unit into the category of topological spaces, where \({\mathcal T}\) is a theory of commutative rings with unit admitting elimination of quantifiers, and \({\mathcal T}\)-spec A consists of equivalence classes of homomorphisms. Closed formulae are defined as those equivalent to positive quantifier- free formulae and are used to define a topology on \({\mathcal T}\)-spec A making it a spectral space. Criteria for \({\mathcal T}\)-closed formulae and spectrally closed constructible subsets, respectively, are given. Some examples illustrate additional properties such as the existence of open projections and enough prime models for a theory \({\mathcal T}\). Using results of \textit{G. Cherlin} and \textit{M. Dickmann} [Ann. Pure Appl. Logic 25, 213-231 (1983; Zbl 0538.03028)], the author discusses the theory RCVR of real closed valuation rings. The constructed spectrum functor RCVR- spec is new, therefore more details are described, especially for real Prüfer rings. An affine space of dimension n over a fixed model R of a theory \({\mathcal T}\) with enough prime models is defined as a certain n-fold fibred product. Such affine spaces enjoy an ``Artin-Lang'' property. The projection from the affine space \({\mathcal A}^ m\) to the affine space \({\mathcal A}^ n\) is open if \({\mathcal T}\) has open projections (both spaces are endowed with the same type of topology). Sheaves of definable sections and sheaves of closed and bounded definable sections (denoted by \({\mathcal D}\) and \({\mathcal S}\), respectively), are considered, where a section over \(X\subset {\mathcal T}\)-spec A is a function s: \(X\to {\mathcal T}\)-spec A[T] such that \(pr(s(\alpha))=\alpha\). Closed and bounded sections are continuous and a closed section is bounded if and only if …
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 4143961
zbMATH Keywords: continuous section
zbMATH Keywords: spectrum functors
zbMATH Keywords: algebraic geometry
zbMATH Keywords: commutative rings
zbMATH Keywords: topological spaces
zbMATH Keywords: spectral space
zbMATH Keywords: open projections
zbMATH Keywords: prime models
zbMATH Keywords: real closed valuation rings
zbMATH Keywords: affine space
zbMATH Keywords: fibred product
zbMATH Keywords: sheaves
reviewed by: Hans-Jürgen Vogel
MaRDI profile type: Publication
cites work: On the real spectrum of a ring and its application to semialgebraic geometry
cites work: Q4722187
cites work: Model theory
cites work: Real closed rings. II. Model theory
cites work: The homotopy axiom in semialgebraic cohomology.
cites work: Semialgebraic topology over a real closed field. II: Basic theory of semialgebraic spaces
cites work: Prime Ideal Structure in Commutative Rings
cites work: Local semianalytic geometry
cites work: Q3961580
cites work: Q4724752
cites work: Compactification of varieties
cites work: Q3324077
description: scientific article; zbMATH DE number 4143961