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 ...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.展开更多
Let G be type B_2 and denote the two simple roots a and β with α the short one. If B is a Borel subgroup of G, X a character of B, and(X) a induced line bundle on G/B, we denote by H^1 (X)=H^1 (G/B, (X)) the first c...Let G be type B_2 and denote the two simple roots a and β with α the short one. If B is a Borel subgroup of G, X a character of B, and(X) a induced line bundle on G/B, we denote by H^1 (X)=H^1 (G/B, (X)) the first cohomology group of (X). Our main results in this paperare: Theorem Let G be type B_2;x∈X(T) be p-regular and 1≤a<p. Then H^1(X) is a simple G-module iff one of the following conditions Ⅰ)-Ⅳ> holds: Ⅰ) X∈s_α·C_0 Ⅱ) X∈s_β·C_0 Ⅲ) Ⅳ)展开更多
基金supported by The National Social Science Fund of China(Grant No.20CZX050)。
文摘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.
文摘Let G be type B_2 and denote the two simple roots a and β with α the short one. If B is a Borel subgroup of G, X a character of B, and(X) a induced line bundle on G/B, we denote by H^1 (X)=H^1 (G/B, (X)) the first cohomology group of (X). Our main results in this paperare: Theorem Let G be type B_2;x∈X(T) be p-regular and 1≤a<p. Then H^1(X) is a simple G-module iff one of the following conditions Ⅰ)-Ⅳ> holds: Ⅰ) X∈s_α·C_0 Ⅱ) X∈s_β·C_0 Ⅲ) Ⅳ)
