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The arithmetic-arboreal residue structure of a Prüfer domain. I. (English) Zbl 1022.13003
Kuhlmann, Franz-Viktor (ed.) et al., Valuation theory and its applications. Volume I. Proceedings of the international conference and workshop, University of Saskatchewan, Saskatoon, Canada, July 28-August 11, 1999. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 32, 59-79 (2002).
This is the first of a series of papers devoted to a systematic study of the residue structure of a ‘Prüfer domain’. The emphasis is set upon the intimate connection between its arithmetic and ‘arboreal’ properties. In the 1960’s it was proved that the theory of a field of \(p\)-adic numbers ‘admits elimination of quantifiers’, which implies a nice elimination theory for arbitrary sentences of that theory. A central point of the technique was to reduce questions about a \(p\)-adic field to questions about the residue field (the field with \(p\) elements) and the value group (integers). This technique was eventually extended to other henselian valued fields, but only in certain cases.
This is the starting point of the article. The effort is to produce rich residue structures (called mixed) which extend the above technique and produce new elimination results. The author develops the theory of these structures with the intention to apply it to the model theory of rings satisfying a local-global principle and to \(SL_2(K)\), where \(K\) is the field of fractions of a Prüfer domain, in particular of a Dedekind domain.
For the entire collection see [Zbl 0993.00030].
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
03C10 Quantifier elimination, model completeness and related topics
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
12J10 Valued fields
05C05 Trees
03C60 Model-theoretic algebra