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Composition Ax-Kochen/Ershov principles and tame fields of mixed characteristic

Margarete Ketelsen, Philip Dittmann

TL;DR

The paper investigates when a composition AKE principle (CAKE^≡) holds for valued fields by decomposing valuations into coarsenings and induced valuations. It proves CAKE^≡ for tame fields of equal characteristic via the resplendent AKE^≡ framework and the relative embedding property, but provides counterexamples in mixed characteristic showing CAKE^≡ can fail even for tame fields with the same underlying field. The results highlight that, in mixed characteristic, the theory of a valued field cannot in general be determined by the theory of its underlying field, residue field, and value group, and they emphasize the role of pointed value groups in negating AKE^≡. The methods combine resplendent model theory, ultrapower/back-and-forth constructions, and explicit valuation-theoretic counterexamples.

Abstract

We study in which settings we have a composition AKE principle, i.e. when the theory of the coarsening $(K,w)$ and the theory of the induced valuation $(Kw,\overline{v})$ determine the theory of the composition $(K,v)$. We show that this is the case when $(K,w)$ is tame of equal characteristic, and provide counterexamples in mixed characteristic. We further show that, for a tame field of mixed characteristic, the theory of the valued field cannot, in general, be determined solely by the theories of its underlying field, its residue field, and its value group.

Composition Ax-Kochen/Ershov principles and tame fields of mixed characteristic

TL;DR

The paper investigates when a composition AKE principle (CAKE^≡) holds for valued fields by decomposing valuations into coarsenings and induced valuations. It proves CAKE^≡ for tame fields of equal characteristic via the resplendent AKE^≡ framework and the relative embedding property, but provides counterexamples in mixed characteristic showing CAKE^≡ can fail even for tame fields with the same underlying field. The results highlight that, in mixed characteristic, the theory of a valued field cannot in general be determined by the theory of its underlying field, residue field, and value group, and they emphasize the role of pointed value groups in negating AKE^≡. The methods combine resplendent model theory, ultrapower/back-and-forth constructions, and explicit valuation-theoretic counterexamples.

Abstract

We study in which settings we have a composition AKE principle, i.e. when the theory of the coarsening and the theory of the induced valuation determine the theory of the composition . We show that this is the case when is tame of equal characteristic, and provide counterexamples in mixed characteristic. We further show that, for a tame field of mixed characteristic, the theory of the valued field cannot, in general, be determined solely by the theories of its underlying field, its residue field, and its value group.
Paper Structure (7 sections, 7 theorems, 13 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Resplendent AKE and CAKE
  3. Counterexamples in mixed characteristic
  4. Lifting automorphisms
  5. Counterexamples for CAKE${}^\equiv$
  6. Tame counterexamples to AKE${}^\equiv$ with the same underlying field
  7. Counterexamples to AKE principles for elementary equivalence in mixed characteristic ( by Philip Dittmann)

Key Result

Theorem 2.2

The class of equicharacteristic zero henselian valued fields admits resplendent AKE${}^\equiv$.

Figures (1)

  • Figure 1: Back and forth construction for the proof of \ref{['lem:resplendent-relative-subcompleteness']}

Theorems & Definitions (24)

  • Definition 2.1: Resplendent AKE${}^\equiv$
  • Theorem 2.2: vdD2014lectures
  • Proposition 2.3: CAKE${}^\equiv$ from resplendent AKE${}^\equiv$
  • proof
  • Remark 2.4
  • Definition 2.5: kuhlmann2016algebra
  • Example 2.6: kuhlmann2016algebra
  • Lemma 2.7: Resplendent relative subcompleteness from the relative embedding property
  • proof
  • Corollary 2.8
  • ...and 14 more