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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fixed-point subring ： ウィキペディア英語版 Fixed-point subring In algebra, the fixed-point subring $R^f$ of an automorphism ''f'' of a ring ''R'' is the subring of ''R'': :$R^f\; =\; \backslash .$ More generally, if ''G'' is a group acting on ''R'', then the subring of ''R'': :$R^G\; =\; \backslash$ is called the fixed subring or, more traditionally, the ring of invariants. A basic example appears in Galois theory, with ''R'' a field and ''G'' a Galois group; see Fundamental theorem of Galois theory. Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate rings of (affine or projective) GIT quotients and they play fundamental roles in the constructions in geometric invariant theory. Example: Let $R\; =\; k(\backslash dots,\; x\_n)$ be a polynomial ring in ''n'' variables. The symmetric group ''S''''n'' acts on ''R'' by permutating the variables. Then the ring of invariants ''R''''G'' is the ring of symmetric polynomials. If a reductive algebraic group ''G'' acts on ''R'', then the fundamental theorem of invariant theory describes the generators of ''R''''G''. Hilbert's fourteenth problem asks whether the ring of invariants is finitely generated or not (the answer is affirmative if ''G'' is a reductive algebraic group by Nagata's theorem.) The finite generation is easily seen for a finite group ''G'' acting on a finitely generated algebra ''R'': since ''R'' is integral over ''R''''G'',〔Given ''r'' in ''R'', the polynomial $\backslash prod\_\; (t\; -\; g\; \backslash cdot\; r)$ is a monic polynomial over ''R''''G'' and has ''r'' as one of its roots.〕 the Artin–Tate lemma implies ''R''''G'' is a finitely generated algebra. The answer is negative for some unipotent group. Let ''G'' be a finite group. Let ''S'' be the symmetric algebra of a finite-dimensional ''G''-module. Then ''G'' is a reflection group if and only if $S$ is a free module (of finite rank) over ''S''''G'' (Chevalley's theorem). In differential geometry, if ''G'' is a Lie group and $\backslash mathfrak\; =\; \backslash operatorname(G)$ its Lie algebra, then each principal ''G''-bundle on a manifold ''M'' determines a graded algebra homomorphism (called the Chern–Weil homomorphism) :$\backslash mathbb()^G\; \backslash to\; \backslash operatorname^(M;\; \backslash mathbb)$ where $\backslash mathbb()$ is the ring of polynomial functions on $\backslash mathfrak$ and ''G'' acts on $\backslash mathbb()$ by adjoint representation. == Notes == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fixed-point subring」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.