The profinite hull of special groups and local-global principles
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 that if \(f\star \varphi\) is a subform of \(f\star\psi\) over \(H\), then \(\varphi\) is a subform of \(\psi\) over \(G\). The main theorem now states that the special group embedding \(\eta_G\) is subform reflecting. Several consequences of this theorem are also given. For example, it is shown that to any reduced special group \(G\) there exists a Pythagorean field \(F\) and a special group morphism \(j:G\to {\mathcal G}(F)\) that reflects subforms. As another corollary, one obtains the analogue of Pfister's subform criterion for reduced special groups \(G\), stating that a form \(\varphi\) over \(G\) is a subform of a form \(\psi\) over \(G\) if and only if for each special group \(H\) and each special group morphism \(f:G\to H\) one has \(D_H(f\star\varphi)\subseteq D_H(f\star\psi)\) (where \(D_H\) denotes the value set of a form over \(H\)).
scientific article; zbMATH DE number 5865394
scientific article