Infinitesimally stable theories of henselian rings
This note considers first-order theories of henselian rings which are stable for the étale factorization system and gives continuum many examples as stabilizations of model-complete complete theories of fields. The framework of the paper is \textit{E. Robinson's} work ``Stable theories of local rings'' [Var. Publ. Ser., Aarhus Univ. 35, 257-280 (1983; Zbl 0531.03044)]. Going from a suitable theory of fields to its ``infinitesimal'' stabilization can be regarded as a logical process from the point of view of categorical logic. From that angle, the examples of henselian rings given arise from the first-order theory of their residue field. The fields considered are certain fields \(K\) of algebraic numbers such that every absolutely irreducible affine curve over \(K\) has a \(K\)- rational point (i.e. regularly closed or p.a.c.). The result follows by exhibiting an adjusted axiomatization of these fields. It consists roughly of expressing the relevant known properties as sequents \(\varphi(x)\to \psi(x)\), where \(\varphi(x)\), \(\psi(x)\) define open sets in the étale topology. In the end, the examples consist of henselian local rings with a residue field elementarily equivalent to one of the above fields.
stabilizations of model-complete complete theories of fields
scientific article; zbMATH DE number 445048
scientific article