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The structure of spaces of \(\mathbb{R}\)-places of rational function fields over real closed fields. (English) Zbl 1380.12009

For a formally real field \(K\), denote by \(\mathcal{X}(K)\) the compact, Hausdorff, totally disconnected (i.e. Boolean) space of orderings of \(K\) endowed with the Harrison topology and let \(M(K)\) be the set of real places \(\xi: K\rightarrow \bar \mathbb{R}=\mathbb{R}\cup \{\infty\}\). There is a natural surjection \(\lambda: \mathcal{X}(K) \rightarrow M(K)\) and one endows \(M(K)\) with the quotient topology (here also called Harrison topology) subordinated to this map. See [E. Becker, Lect. Notes Math. 959, 1–40 (1982; Zbl 0508.14013)].
The author’s main concern is the case of the rational function field \(K=R(y),\) with \(R\) real closed, which we assume throughout. If \(R\) is non-Archimedean, it is found, in her words, that spaces \(M(R(y))\) have an amazingly rich topological structure.
Section 1 introduces the ultrametric on \(R\) defined by \(u(a,b)=v(a-b)\), where \(v\) is the natural valuation coming from the unique ordering of \(R\). For any cut \((S,T)\) in the value group \(vR\), the ultrametric balls of center \(a\) and radius \(T\), \(B_T(a)=\{b\in R: u(a,b)\in T\cup \infty \}\) are defined; they play a pivotal rôle in the study. One important tool is the known Proposition 1.1: The set of cuts \(\mathcal{C}(R)\) endowed with the order topology and the space \(\mathcal{X}(R(y))\) with the Harrison topology are homoeomorphic via a map \(\chi\).
Section 2 finds a small subbasis for the topological space \(M(R(y))\). For \(f\in R(y),\) let \(U(f)=\{\xi\in M(R(y)): \xi(f)\in \mathbb{R}^+\}\). Theorem 2.5: Suppose \(F\) is a field dense in \(R\). Consider the family \(\mathcal{F}=\{a+by, \frac{y-a}{y-b}: a,b \in F\}.\) Then the family \(U(f),\) \(f\in \mathcal{F}\) forms a subbasis for the Harrison topology on \(M(R(y))\).
Two proper subfamilies of \(\mathcal{F}\) are identified that separate points in \(M(R(y))\) in a simple and in a stronger topological sense, respectively. The space \(M(R(y))\) is metrizable iff there is countable real closed field \(F\) which is dense in \(R,\) see F.-V. Kuhlmann et al. [Commun. Algebra 39, No. 9, 3166–3177 (2011; Zbl 1263.12001)]. This metric is now described. Take any bijection \(\sigma: \mathcal{F} \rightarrow \mathbb{N}\). Let \(d_0\) be any fixed metric on \(\bar \mathbb{R}\). Theorem 3.3: The following mapping \(d: M(R(y))\times M(R(y)) \rightarrow [0,\infty[\) is a metric on \(M(R(y))\): \(d(\xi,\eta)=\sup_{f\in \mathcal{F}} \{2^{-\sigma(f)} d_0(\xi(f),\eta(f))\}\).
A theorem in J. Nikiel et al. [Pac. J. Math. 149, 145–155 (1991; Zbl 0687.54023)] says that for the continuous image of a compact ordered space, the covering dimension, the small inductive dimension, and the large inductive dimension are all equal. Since by section 1, \(M(R(y))\) can be seen as image of the compact space \(\mathcal{C}(R)\), some additional argument using connectedness of \(M(R(y)),\) see [J. Harman, Contemp. Math. 8, 141–174 (1982; Zbl 0509.12021)], yields the main result of section 4. Theorem: \(1=\dim M(R(y))=\text{ind} M(R(y))= \text{Ind} M(R(y)).\)
Section 5 studies homeomorphisms on \(M(R(y))\). Using that the automorphisms of \(R(y)\) that leave \(R\) elementwise fixed are the Möbius transformations \(y\mapsto (ay+b)/(cy+d)\) with \(ad-bc\neq 0\) and that these can be written as compositions of translations, multiplication, and inversion, it is first shown in Proposition 5.1 that these actions induce homeomorphisms on \(\mathcal{C}(R)\) which are compatible with equivalence of cuts (defined earlier by means of the ultrametric balls). By the composition \(\lambda \circ\chi \) from \(\mathcal{C}(R)\) to \(M(R(y))\) the author gets results for \(M(R(y))\), in particular cofinal and coinitial chains of subspaces of \(M(R(y))\) which are all mutually homeomorphic.
The field \(R=\mathbb{R}((t^{\mathbb{Q}}))\) is known to be real closed. As the final result of section 5, inspired partially by W. Charatonik and A. Dilks [Topology Appl. 55, No. 3, 215–238 (1994; Zbl 0788.54040)], it is shown in Corollary 5.4 that for this \(R,\) the space \(M(R(y))\) has the remarkable property that every open subset contains a homeomorphic copy of \(M(R(y))\); so \(M(R(y))\) is ‘self-homoeomorphic’.
The final section 6 describes the big picture of what the author calls – apparently with good reasons – the fractal structure of \(M(R(y))\) in case of non-archimedean \(R\). Future papers detailing this are promised.

MSC:

12J15 Ordered fields
12J25 Non-Archimedean valued fields
12J20 General valuation theory for fields
13J30 Real algebra
54E45 Compact (locally compact) metric spaces
54F45 Dimension theory in general topology
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References:

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