Valued fields, metastable groups

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 Γ) if every type over a sufficiently rich base structure can be viewed as part of a Γ-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 Γ, 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.

Definability in the group of infinitesimals of a compact Lie group

“…The techniques we applied here will not readily adapt to handle imaginaries. In the case of algebraically closed valued fields, a result of [HR18] is that the only interpretable fields, up to definable isomorphism, are the valued field and its residue field. It would be natural to expect that, correspondingly, the only interpretable fields in RCVF up to definable isomorphism are the valued field, its residue field, and their algebraic closures.…”

“…ACF is self-recollecting by [Poi88], RCF is self-recollecting by [NP91], and Th(Qp) is self-recollecting for definitions by [Pil89]. It follows directly from the characterisation of interpretable fields in [HR18] that the theory of non-trivially valued algebraically closed fields ACVF is self-recollecting.…”

On Stably Pointed Varieties and Generically Stable Groups in ACVF

“…Proof. The proof is a slight generalization, due to descent, but mostly identical to the an argument given in [6,Proposition 6.11]. If F is trivially valued there is nothing to prove.…”

“…We will first need to recall the notion of a strongly stably dominated type. It was first defined in [6,Section 2.3] and later shown in [4, Proposition 8.1.2] to be equivalent to the following when the type is concentrated on a variety. Let q be a definable type on a variety V over a field.…”

“…In [6,Theorem 6.11], Hrusovski-Rideau show that given a pair (G, p), where G is an affine algebraic group and p is a generically stable generic type of a definable subgroup H, H is definably isomorphic to the O-valued points of some group scheme defined over the valuation ring O. In this paper we expand the ideas from [6] to give a geometric interpretation of the category of varieties with generically stable Zariski dense types concentrated on them. We expand and elaborate:…”