The profinite hull of special groups and local-global principles
type: Item
instance of: scholarly article
title: The profinite hull of special groups and local-global principles
zbMATH Open document ID: 1229.11065
author: Hugo L. Mariano
author: Francisco Miraglia
published in: Studia Logica
publication date: 2011-03-14T00:00:00Z
review text: The theory of special groups is a first-order axiomatization of the algebraic theory of quadratic forms. It was developed by \textit{M. A. Dickmann} and the second author [``Special groups. Boolean-theoretic methods in the theory of quadratic forms'', Mem. Am. Math. Soc. 689 (2000; Zbl 1052.11027)] where analogues of many classical results on quadratic forms over fields have been proved in the context of special groups. The category of reduced special groups is dual to Marshall's category of abstract order spaces. There is a covariant functor \({\mathcal G}\) from the category of fields into the category of special groups by which the theory of quadratic forms over a field is encoded by the theory of special groups. This functor maps Pythagorean fields into the category of reduced special groups. The Boolean hull of a reduced special group has been introduced in [loc. cit.] and was used to prove Marshall's signature conjecture in the general context of reduced special groups. The main theme of the present paper is the construction of the profinite hull functor for special groups which constitutes a refinement of the Boolean hull construction and which uses earlier work by the authors in [The many sides of logic. Studies in Logic (London) 21, 201--224 (2009; Zbl 1245.03049)]. Let \({\mathcal P}(G)\) denote the profinite hull of a reduced special group \(G\) as constructed in the present paper. It is shown that the map \(\eta_G: G\to {\mathcal P}(G)\) is a special group embedding with dense image which satisfies the following universal property: if \(f: G\to P\) is any special group morphism with \(P\) a profinite reduced special group, then there exists a unique continuous special group morphism \(\tilde{f}: {\mathcal P}(G)\to P\) such that \(\tilde{f}\circ\eta_G=f\). To explain the authors' main result on profinite hulls, recall that a special group morphism \(f: G\to H\) is said to be subform reflecting if for all forms \(\varphi\) and \(\psi\) over \(G\) we have t…
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 03C65
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 5865394
zbMATH Keywords: quadratic form
zbMATH Keywords: special group
zbMATH Keywords: reduced special group
zbMATH Keywords: Boolean hull
zbMATH Keywords: profinite hull
zbMATH Keywords: subform reflection property
zbMATH Keywords: local-global principle
zbMATH Keywords: pure embedding
zbMATH Keywords: Pythagorean field
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: s11225-010-9301-3
OpenAlex ID: W1998819016
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DOI: 10.1007/S11225-010-9301-3
description: scientific article; zbMATH DE number 5865394