Real holomorphy rings and the complete real spectrum
type: Item
instance of: scholarly article
title: Real holomorphy rings and the complete real spectrum
zbMATH Open document ID: 1209.13026
author: Murray Marshall
author: Danielle Gondard
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2010-10-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/115903
review text: Let \(A\) be a commutative ring with \(1\). The prime spectrum \(\mathrm{Spec}\, A\) consisting of all prime ideals of \(A\) is a basic object in algebraic geometry. In real geometry where also inequalities are considered the corresponding notion is the real spectrum \(\mathrm{Sper}\, A\) of \(A\). It consists of all pairs \((\mathfrak{p}, P)\) where \(\mathfrak{p}\) is a prime ideal of \(A\) and \(P\) is an ordering on \(k(\mathfrak{p})\) which is the quotient field of \(A/\mathfrak{p}\). (Recall that an ordering on a field \(K\) is a subset \(P\) of \(K\) such that \(P+P\subset P, PP\subset P, P\cup -P=K\) and \(P\cap -P=\{0\}\).) Both \(\mathrm{Spec}\, A\) and \(\mathrm{Sper}\, A\) carry a natural topology (the Zariski topology in the first case and the Harrison topology in the second case) that make them a spectral space in the sense of [\textit{M. Hochster}, Trans. Am. Math. Soc. 142, 43--60 (1969; Zbl 0184.29401)]. Motivated for example by rigid geometry, the valuation spectrum \(\mathrm{Spv}\, A\) was introduced which consists of all pairs \((\mathfrak{p},v)\) where \(\mathfrak{p}\) is a prime ideal of \(A\) and \(v\) is a valuation on \(k(\mathfrak{p})\) (see for example [\textit{R. Huber} and \textit{M. Knebusch}, Contemp. Math. 155, 167--206 (1994; Zbl 0799.13002)]). It carries also a natural topology making it a spectral space. The authors define in the paper under review the complete real spectrum \(\mathrm{Sper}^c A\) of \(A\). It is related to the valuation spectrum in the same way as the real spectrum is related to the prime spectrum. It consists of all triples \((\mathfrak{p},v,P)\) where \(\mathfrak{p}\) is a prime ideal of \(A\), \(v\) is a valuation on \(k(\mathfrak{p})\) and \(P\) is an ordering on the residue field \(B_v/\mathfrak{m}_v\) of \(v\). There are natural maps \(\mathrm{Sper}^c A\to\mathrm{Spec}\, A, \mathrm{Sper}^c A\to\mathrm{Spv}\, A\) and \(\mathrm{Sper}\, A\to\mathrm{Sper}^c A\). The authors equip \(\mathrm{Sper}^c A\) with a natu…
reviewed by: Tobias Kaiser
Mathematics Subject Classification ID: 13J30
zbMATH DE Number: 5799081
MaRDI profile type: Publication
OpenAlex ID: W2329158408
cites work: Q4889879
cites work: Q4843070
cites work: Sums of powers in rings and the real holomorphy ring.
cites work: Q3770650
cites work: Prime Ideal Structure in Commutative Rings
cites work: Q4274295
cites work: Q4295695
cites work: Manis valuations and Prüfer extensions. I: A new chapter in commutative algebra
cites work: An introduction to real algebra
cites work: Spaces of orderings and abstract real spectra
cites work: A Real Holomorphy Ring without the Schmüdgen Property
cites work: Real reduced multirings and multifields
cites work: Specialization and a local homeomorphism theorem for real Riemann surfaces of rings
cites work: The \(K\)-moment problem for compact semi-algebraic sets
cites work: On a real places of a field and their holomorphy ring
cites work: Iterated rings of bounded elements and generalizations of Schmüdgens Positivstellensatz
DOI: 10.5802/AFST.1275
description: scientific article; zbMATH DE number 5799081