Generic stability of linear algebraic groups over $\mathbb{C}[[t]]$

Abstract

Let $K$ be a henselian valued field with ${\cal O}_K$ its valuation ring, $\Gamma$ its value group, and $\boldsymbol{k}$ its residue field. We study the definable subsets of ${\cal O}_K$ and algebraic groups definable over ${\cal O}_K$ in the case where $\boldsymbol{k}$ is algebraically closed and $\Gamma$ 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 \ref{def-res-finite-cofinite}). Applying this result, we show that $\mathrm{GL}(n,{\cal O}_K)$ (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 \cite{Y.Halevi}.