Definable V-topologies, Henselianity and NIP

Abstract

We initiate the study of definable V-topolgies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if (K, v 1 , v 2 ) is a bi-valued NIP field with v 1 henselian (resp. t-henselian) then v 1 and v 2 are comparable (resp. dependent). As a consequence Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.Theorem (Proposition 6.3). Shelah's conjecture for NIP fields implies the henselianity conjecture.The proof of the main result generalizes methods used by Johson [20, Chapter 11], Montenegro [31] and Duret [7] by proving the following strong approximation theorem on curves.Theorem (Proposition 4.2). Let K be a perfect field and τ 1 , τ 2 two distinct V-topologies on K with τ 1 t-henselian. Let C be a geometrically integral separated curve over K, X ∈ τ 1 and Y ∈ τ 2 infinite subsets of C(K). Then X ∩ Y is nonempty.The topological theme is recapitulated in the the final section. We show that a well behaved interaction between definable sets and open sets is equivalent, without further model theoretic assumptions, to t-henselianity. This allows us to deduce the following new (to the best of our knowledge) general characterization of henselianity in terms of t-henselianity.Theorem (Theorem 7.5, Proposition 7.7). Let (K, v) be a valued field. Then (K, v) is henselian if and only if (Kv ∆ ,v) is t-henselian for every coarsening v ∆ of v.Moreover, if K is not real closed then v has a non-trivial henselian coarsening if and only if for every algebraic extension L/K, any extension of v to L is t-henselian.As a final result, we use all of the above to show the following reformulations of Shelah's Conjecture.Theorem (Corollary 7.6). The following are equivalent:(1) Every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation.

Distality in valued fields and related structures

“…Hence if Conjecture 6.9 holds, then every valuation ring on a NIP field is henselian, and its residue field is NIP. Moreover, under Conjecture 6.9, any two (externally) definable valuation rings on a NIP field are comparable [38,Corollary 5.4]. In Theorem 6.12 below we show that Conjecture 6.9 also gives rise to a classification of all fields admitting a distal expansion.…”

“…This conjecture has numerous consequences; for example, by [38,Proposition 6.3], it implies that every NIP valued field is henselian. In [42,Theorem B] it is shown that if K is NIP and O is a henselian valuation ring of K, then the valued field (K, O) is also NIP.…”

“…In Theorem 6.12 below we show that Conjecture 6.9 also gives rise to a classification of all fields admitting a distal expansion. We first note that the non-trivial henselian valuation stipulated in Conjecture 6.9 may be taken to be ∅-definable, by results in [42,43] (see also [38,Corollary 7.6]): Lemma 6.10. Suppose Conjecture 6.9 holds, and suppose K is infinite and NIP; then K is separably closed or real closed, or K has an ∅-definable non-trivial henselian valuation ring.…”

ON NON-COMPACT p-ADIC DEFINABLE GROUPS

“…Let be a collection of definable subsets of . If the sets in are uniformly definable, and is linearly ordered by inclusion, then the sets and are externally definable [6, Kaplan’s Lemma 3.4].…”

$\rm NIP$, and ${\rm NTP}_2$ division rings of prime characteristic

“…NIP Fields. It is believed that an NIP field is either finite, separably closed, real closed or admits a nontrivial henselian valuation (this conjecture is attributed in [22] to S. Shelah). A characterisation of the subclass of dp-minimal fields is given in [28], which also confirms Shelah's conjecture for the particular case of dp-minimal fields.…”