Stable Domination and Generic Stability of Linear Algebraic Groups over ℂ[[t]]

Studies in Logic, Vol. 17, No. 6 (2024): 59–75.                PII: 1674-3202(2024)-06-0059-17

Chen Ling, Ningyuan Yao

Abstract. ℂ((t)) is the formal Laurent series over the field ℂ of complex numbers. It is a henselian valued field, and its valuation ring, denoted by ℂ[[t]], is the formal power series over ℂ. Let K be any model of Th(ℂ((t))) with  its valuation ring and k its residue field. Then k is algebraically closed and 𝒪K is elemenatry equivalent to ℂ[[t]].
We first describe the definable subsets of 𝒪K, showing that every definable subset X of 𝒪K is either res-finite or res-cofinite, i.e., the residue res(X) of X, is either finite or cofinite in k. Moreover, X is res-finite iff 𝒪K\X is res-cofinite. Applying this result, we show that GL(n, 𝒪K), the group of invertible n by n matrices over the valuation ring, is stably dominated via the residue map. As a consequence, we conclude that GL(n, 𝒪K) is generically stable, generalizing Y. Halevi’s result, where K is an algebraically closed valued field.