Definable V-topologies, Henselianity and NIP

2019/01/17 by Halevi, Yatir, Hasson, Assaf, Jahnke, Franziska
#FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1901.05920

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,v₁,v₂) is a bi-valued NIP field with v₁ henselian (resp. t-henselian) then v₁ and v₂ 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.

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