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

C[[t]]上的线性代数群的稳定支配与泛稳定性

作　　者：Chen Ling Ningyuan Yao 凌晨;姚宁远(复旦大学哲学学院)

机构地区：[1]School of Philosophy,Fudan University

出　　处：《逻辑学研究》2024年第6期59-75,共17页Studies in Logic

基　　金：supported by The National Social Science Fund of China(Grant No.20CZX050)。

摘　　要：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 OK 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.C((t))是复数域C上的形式洛朗级数,它是一种亨泽尔赋值域,其赋值环为C上的形式幂级数,记作C[[t]]。令K为Th(C((t)))的一个模型,OK表示K的赋值环,k表示K的剩余域,则OK和k也分别与C[[t]]和C初等等价。本文首先刻画了OK的可定义子集,证明了OK的可定义子集X或者是剩余-有限的,或者是剩余-余有限的,即X在剩余映射res下的像总是k的有限子集或者余有限子集。此外,剩余-有限集在OK中的补集恰好是剩余-余有限集。基于这个性质,我们证明了OK上的n阶可逆矩阵群GL(n,OK)被其剩余映射稳定支配。作为一个推论,我们证明了GL(n,OK)具有泛稳定性,该结果推广了Y.Halevi在代数闭赋值域中证明的一个定理。

关 键 词：FIELD FINITE RESIDUE 

分 类 号：O151.2[理学—数学] B812[理学—基础数学]