Abstract
We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued field is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn field \(\mathbb {R}((\mathbb {Z}))\) are definable. Similarly, all types over the quotient field of the Witt ring \(W(\mathbb {F}_p^{\text {alg}})\) are definable. This extends a work of Cubides and Delon and of Cubides and Ye.
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Notes
The existence of such compatible angular components follows from the traditional proof of existence of a section of the valuation \({{\,\textrm{RV}\,}}_L \rightarrow \Gamma \): a partial section \(s_A:A\rightarrow {{\,\textrm{RV}\,}}_K\) where A is a pure subgroup of \(\Gamma _K\) can be extended by \(\aleph _1\)-saturation and Zorn Lemma to a full section \(s_K:\Gamma _K \rightarrow {{\,\textrm{RV}\,}}_K\) of the valuation in \({{\,\textrm{RV}\,}}_k\). As \(\Gamma _K\) is a pure subgroup of \(\Gamma _L\), this map can be again extended to a section \(s_L:\Gamma _L\rightarrow {{\,\textrm{RV}\,}}_L\). This section gives then an angular component compatible with the subfield K.
Thanks to the anonymous referee for pointing out this useful remark.
To see this, we must show that the compositions of the interpretation \(\Delta : k^n \rightarrow W_n(k)\) (which identify an element \(a\in W_n\) with its Teichmüller vector \((a_1,\cdots ,a_n)\) ) and \(\chi _{n,1}:W_n(k) \rightarrow k, a\mapsto a_1\) are definable functions. The function \(\chi _{n,1} \circ \Delta \) is the projection to the first coordinate and the graph of the function \(\Delta \circ \chi _{n,1}: W_n(k)^n \rightarrow W_n(k), (a^1, \dots , a^n) \mapsto (\chi _{n,1}(a^1),\dots , \chi _{n,1}(a^n))\) is clearly definable using [23, Fact 1.61].
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Acknowledgements
I deeply thank my PhD advisor Martin Hils for his support and his guidance. Many thanks to Pablo Cubides Kovacsics for his enlightenment of his work in [7]. In particular, my gratitude to both of them for sharing with me their ideas which led to the proof of Proposition 2.4 and motivated the writing of this paper. I thank Artem Chernikov and Martin Ziegler for sharing their results in [2]. Many thanks to Martin Bays and Allen Gehret, whose discussions have been very helpful. Finally, I thank the anonymous referee for their careful proofreading and useful comments, which have greatly improved the quality of this paper.
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The author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through SFB 878 and under Germany ’s Excellence Strategy - EXC 2044 - 390685587, Mathematics Münster: Dynamics - Geometry - Structure. This research has been also supported by the DAAD through the ‘Kurzstipendien für Doktoranden 2020/21’ programme, by Mathematisches Forschungsinstitut Oberwolfach in 2020 through the “Oberwolfach Leibniz Fellows” program - Project-ID2043r, and by the University of Campania ‘Luigi Vanvitelli’ in the framework of V:ALERE 2019 (GoAL project).
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Touchard, P. Stably embedded submodels of Henselian valued fields. Arch. Math. Logic 63, 279–315 (2024). https://doi.org/10.1007/s00153-023-00894-2
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DOI: https://doi.org/10.1007/s00153-023-00894-2