An application of Kochen’s theorem. (English) Zbl 1057.03027

Introduction: We discuss the impact of Kochen’s Isomorphism Theorem on the definable subsets of the value group of a Hensel field. In Section 1 we characterize the definable subsets of the value group in the language of valued fields with a cross-section. The results in this section are not novel, but are tailored for the application that follows. In Section 2 we apply these results to a specific formula to answer the Invariant Factor Problem for Hecke algebras arising from split reductive group schemes defined over \(\mathbb{Z}\) and Hensel fields whose value group is isomorphic to \((\mathbb{Z},+,\leq)\) posed by M. Kapovich, B. Leeb and J. Millson [The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra. Max Planck Institut Preprint Ser. 133 (2002)]. Originally, the definability of the relevant sets involved the cross-sectional map, but a suggestion of the referee obviated the need for the cross-section. Much of the background material for the application is taken from expositions of Carter, Gross and Waterhouse.


03C60 Model-theoretic algebra
12J10 Valued fields
Full Text: DOI Euclid


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