Henselianity in the language of rings, with Franziska Jahnke

I would like to write about a new paper ([AJ15]) which Franziska Jahnke and I have written and put on the arXiv. It’s called Henselianity in the language of rings. In it we investigate the relationship between the following four properties of a field K:

(h) K is henselian, i.e. K admits a nontrivial henselian valuation,

(eh) K is elementarily henselian, i.e. every L\equiv K is henselian,

(def) K admits a definable nontrivial henselian valuation, and

(\emptyset-def) K admits a \emptyset-definable nontrivial henselian valuation.

By `definable’ we mean `definable in the language of rings’. Our paper is the latest work on this topic, which began with the work of Prestel and Ziegler who show, in [PZ78], that there exist t-henselian, nonhenselian fields. Subsequently, several more recent papers explore this topic and related issues of definable valuations; examples include [JK15] and [FJ15]. Our work builds on these three papers, and others. A more complete survey of the literature is available in our paper.

There are some trivial implications between the above properties:

{\bf(\emptyset\textbf{-def})}\Longrightarrow{\bf(\textbf{def})}\Longrightarrow{\bf(h)}\qquad\qquad(a)

and

{\bf(\emptyset\textbf{-def})}\Longrightarrow{\bf(eh)}\Longrightarrow{\bf(h)}.\qquad\qquad(b)

Franziska and I investigate whether any other implications hold between these properties across the class \mathcal{K}_{0} of all fields of characteristic zero, and across the smaller classes

\mathcal{K}_{0,0}:=\{K\in\mathcal{K}_{0}\;|\;\mathrm{char}(Kv_{K})=0\}

and

\mathcal{K}_{0,p}:=\{K\in\mathcal{K}_{0}\;|\;\mathrm{char}(Kv_{K})=p\}, for p a prime;

where v_{K} denotes the canonical henselian valuation on K. Our main theorem gives the `complete picture’ for each of these classes \mathcal{K}_{0,0},\mathcal{K}_{0,p}:

Main Theorem (Theorem 1.1, [AJ15])

  1. In the class {\mathcal{K}_{0,0}} the complete picture is

    {\bf(eh)}  \Longleftrightarrow  {\bf(\emptyset\textbf{-def})}  \Longrightarrow  {\bf(\textbf{def})}  \Longrightarrow  {\bf(h)}.

  2. For each prime {p}, in the class {\mathcal{K}_{0,p}} the complete picture is

    {\bf(\textbf{def})}  \Longleftrightarrow  {\bf(\emptyset\textbf{-def})}  \Longrightarrow  {\bf(eh)}  \Longleftrightarrow  {\bf(h)}.

  3. Consequently, in the class {\mathcal{K}_{0}} the complete picture is simply given by (1) and (2), above.

Given the `trivial’ implications, (a) and (b) above, the proof of Theorem 1.1 boils down to showing the following:

  1. In the class \mathcal{K}_{0,0} the implication {\bf(eh)}\Longrightarrow{\bf(\emptyset\textbf{-def})} holds.
  2. For each prime p, in the class \mathcal{K}_{0,p} the implications {\bf(h)}\Longrightarrow{\bf(eh)} and {\bf(def)}\Longrightarrow{\bf(\emptyset\textbf{-def})} hold.
  3. No implications hold other than the ones listed above and the `trivial’ ones.

The proof of (4) has two stages: first we show that {\bf(eh)}\Longrightarrow{\bf(def)} using Theorem B from [JK15]; then we employ the Omitting Types Theorem to remove the parameter.

The proof of (5) is more tricky. The implication {\bf(h)}\Longrightarrow{\bf(eh)} follows from basic properties of the canonical henselian valuation combined with some model-theoretic trickery. Franziska and I call these arguments `Sirince tricks’ because we wrote them down when we were on a research visit to the wonderful Nesin Mathematics Village, near Sirince, Turkey! The proof of the implication {\bf(def)}\Longrightarrow{\bf(\emptyset\textbf{-def})} depends on some delicate arguments involving uniform definitions of canonical p-henselian valuations.

Finally, the proof of (6) is a set of four counterexamples. The first (given by Prestel-Ziegler, [PZ78]) simply shows the existence (in equicharacteristic zero) of t-henselian nonhenselian fields. The second and third examples were given in [JK15] and are based on the construction in [PZ78]. Finally, the fourth example is based on a more sophisticated construction give in [FJ15] of t-henselian nonhenselian fields with various properties – we extend it to defectless fields of positive characteristic.

There are several key open questions. In my opinion, perhaps the most prominent is: can our results be extended to equicharacteristic p>0?


Bibliography

[AJ15] Sylvy Anscombe and Franziska Jahnke. Henselianity in the language of rings. Manuscript, 2015. arXiv
[FJ15] Arno Fehm and Franziska Jahnke. On the quantifier complexity of definable canonical henselian valuations. Mathematical Logic Quarterly, 61(4-5):347-361, 2015.
[JK15] Franziska Jahnke and Jochen Koenigsmann. Defining coarsenings of valuations. To appear in the Proceedings of the Edinburgh Mathematical Society, 2015. arXiv
[PZ78] Alexander Prestel and Martin Ziegler. Model-theoretic methods in the theory of topological fields. Journal für die reine und angewandte Mathematik, 299(300):318-341, 1978.

Advertisements