Valued fields, Metastable groups

TL;DR

This paper introduces metastable theories, studies definable groups within them, and characterizes definable fields in algebraically closed valued fields, revealing structural decompositions and classifications.

Contribution

It defines metastability in theories, analyzes groups with stably dominated generics, and classifies all definable fields in ACVF, advancing model theory of valued fields.

Findings

Groups with stably dominated generics have canonical stable quotients.

Abelian groups decompose into parts from the value group and stable limits.

All fields definable in ACVF are classified.

Abstract

We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (over a sort $\Gamma$) if every type over a sufficiently rich base structure can be viewed as part of a $\Gamma$-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from $\Gamma$, and a definable direct limit system of groups with stably dominated generic. In the case of ACVF, among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in ACVF.