Quotients of index two and general quotients in a space of orderings. (English) Zbl 1379.11033
In this paper, the authors study spaces of orderings \((X,G)\) in the sense of M. Marshall [Spaces of orderings and abstract real spectra. Berlin: Springer (1996; Zbl 0866.12001)] where \(X\) is a nonempty set, \(G\) is a subgroup of \(\{ 1,-1\}^X\) that contains the constant function \(-1\), separates points in \(X\) and satisfies some further axioms. If \(k\) is a formally real field and if \(X_k\) is the usual space of orderings and \(G_k=k^*/(\sum k^2)\), then \((X_k,G_k)\) is a space of orderings in the above sense.
A quotient structure \((X_0,G_0)\) of a space of orderings \((X,G)\) is a subgroup \(G_0\) of \(G\) containing \(-1\) together with \(X_0=X|_{G_0}\). If \((X_0,G_0)\) is itself a space of orderings, it is called a quotient space of \((X,G)\). It turns out that quotient spaces are indeed quotient objects in the category of spaces of orderings. An important open problem is to find general necessary and sufficient criteria for a quotient structure to be a quotient space. In the present paper, the authors provide such criteria in the case where \(G_0\) is of index \(2\) and the so-called stability index of \((X,G)\) is \(1\) (stability index \(\leq 1\) is equivalent to \((X,G)\) having the strong approximation property SAP), and a refinement yields such criteria also for stability index \(2\) under some mild extra condition on \((X,G)\). For higher stability indices one can still obtain necessary conditions by further refinements. They also obtain various results in the case where the index of \(G_0\) in \(G\) is greater than \(2\), possibly infinite.
The authors study in some detail the case where \((X_{\mathbb{Q}(x)}, G_{\mathbb{Q}(x)})\) is the space of orderings of the rational function field \(\mathbb{Q}(x)\). In this situation, the stability index is \(2\) and the above results can be applied. They give a new proof of the fact that this space is profinite (cf. the first author and B. Jacob [J.Pure Appl.Algebra 216, No.12, 2608–2613 (2012; Zbl 1270.11036)]) and they provide necessary and sufficient criteria for certain quotient structures to be quotient spaces. In particular, they show that a quotient structure of \((X_{\mathbb{Q}(x)}, G_{\mathbb{Q}(x)})\) of finite index is a quotient space if and only if it is profinite. Various examples illustrate their results.
A quotient structure \((X_0,G_0)\) of a space of orderings \((X,G)\) is a subgroup \(G_0\) of \(G\) containing \(-1\) together with \(X_0=X|_{G_0}\). If \((X_0,G_0)\) is itself a space of orderings, it is called a quotient space of \((X,G)\). It turns out that quotient spaces are indeed quotient objects in the category of spaces of orderings. An important open problem is to find general necessary and sufficient criteria for a quotient structure to be a quotient space. In the present paper, the authors provide such criteria in the case where \(G_0\) is of index \(2\) and the so-called stability index of \((X,G)\) is \(1\) (stability index \(\leq 1\) is equivalent to \((X,G)\) having the strong approximation property SAP), and a refinement yields such criteria also for stability index \(2\) under some mild extra condition on \((X,G)\). For higher stability indices one can still obtain necessary conditions by further refinements. They also obtain various results in the case where the index of \(G_0\) in \(G\) is greater than \(2\), possibly infinite.
The authors study in some detail the case where \((X_{\mathbb{Q}(x)}, G_{\mathbb{Q}(x)})\) is the space of orderings of the rational function field \(\mathbb{Q}(x)\). In this situation, the stability index is \(2\) and the above results can be applied. They give a new proof of the fact that this space is profinite (cf. the first author and B. Jacob [J.Pure Appl.Algebra 216, No.12, 2608–2613 (2012; Zbl 1270.11036)]) and they provide necessary and sufficient criteria for certain quotient structures to be quotient spaces. In particular, they show that a quotient structure of \((X_{\mathbb{Q}(x)}, G_{\mathbb{Q}(x)})\) of finite index is a quotient space if and only if it is profinite. Various examples illustrate their results.
Reviewer: Detlev Hoffmann (Dortmund)
MSC:
| 11E10 | Forms over real fields |
| 12D15 | Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) |
Keywords:
spaces of orderings; quotient structures; quotient spaces; stability index; axiomatic theory of quadratic forms; Witt ringReferences:
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