Generic stability of linear algebraic groups over ℂ[[t]]

2023/07/09 by Ling Chen, Ling, Chen, Ningyuan Yao +1
Mathematics · #20G30 03C98 12L12 #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2307.05546

openalex publication_date 2023/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a henselian valued field with \cal O_K its valuation ring, Γ its value group, and \boldsymbolk its residue field. We study the definable subsets of \cal O_K and algebraic groups definable over \cal O_K in the case where \boldsymbolk is algebraically closed and Γ is a \mathbb Z-group. We first describe the definable subsets of \cal O_K, showing that every definable subset of \cal O_K is either res-finite or res-cofinite (see Definition \refdef-res-finite-cofinite). Applying this result, we show that GL(n,\cal O_K) (the invertible n by n matrices over \cal O_K) are generically stable for each n, generalizing Y. Halevi's result, where K is an algebraically closed valued field \citeY.Halevi.

Related

• On residual domination and types orthogonal to the value group
• Valued fields, Metastable groups
• Models of Abelian varieties over valued fields, using model theory
• A short note on groups in separably closed valued Fields
• Residually Dominated Groups in Henselian Valued Fields of Equicharacteristic Zero