The Profinite Hull of Special Groups and Local-Global Principles

Abstract

We introduce the Profinite Hull functor of special groups, showing that it gives rise to a new (and strong) local-global principle, the subform reflection property. We also indicate applications of this principle to the abstract algebraic theory of quadratic forms.The theory of special groups (SGs) is a first-order axiomatization of the algebraic theory of quadratic forms, introduced by M. Dickmann and F. Miraglia (see [9]). The local-global principles in the algebraic theory of quadratic forms were developed, initially, in the context of fields. Of fundamental importance is Pfister's local-global principle for isometry in the reduced theory of quadratic forms, a vast generalization of Sylvester's inertia law for forms with unit real coefficients. This was in turn generalized to reduced special groups context in [9]. There are other interesting abstract formulations of the reduced theory of quadratic forms, as the theory of abstract order spaces introduced M. Marshall (see [26]), that is shown in [9] to be essentially equivalent to the theory of reduced special groups (RSGs).The Boolean hull of a RSG is introduced and studied in [9], being an essential ingredient in the solution of Marshall's signature conjecture in [10]. The profinite hull functor introduced here, defined on the category of RSGs, is, in some sense, a refinement of the Boolean hull construction, yielding a new (and stronger) local-global principle, the subform reflection property.The paper is divided into three sections. In section 1 we recall basic results on special groups, needed in the sequel. We also prove two new results: the first gives a very useful sufficient condition for the reflection of Special Issue: The Legacy of Newton da Costa Edited by Daniele Mundici and Itala M. Loffredo D'Ottaviano

Anel de Witt para semigrupos reais, envoltória von Neumann e B-pares

“…Na Teoria dos Grupos Especiais Reduzidos (RSG), existe a envoltória Booleana introduzida e estudada em [DM1] que foi um dos ingredientes essenciais na solução da conjectura de Marshall ([DM2]). Outra envoltória dos RSGs é a profinita, introduzida e analisada em [MM1]. Num sentido preciso, ela é um refinamento da envoltória Booleana que prova um novo princípio local-global: a reflexão de subformas.…”