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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 Lang's theorem ： ウィキペディア英語版 Lang's theorem In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if ''G'' is a connected smooth algebraic group over a finite field $\backslash mathbf\_q$, then, writing $\backslash sigma:\; G\; \backslash to\; G,\; \backslash ,\; x\; \backslash mapsto\; x^q$ for the Frobenius, the morphism of varieties :$G\; \backslash to\; G,\; \backslash ,\; x\; \backslash mapsto\; x^\; \backslash sigma(x)$  is surjective. Note that the kernel of this map (i.e., $G\; =\; G(\backslash overline)\; \backslash to\; G(\backslash overline)$) is precisely $G(\backslash mathbf\_q)$. The theorem implies that $H^1(\backslash mathbf\_q,\; G)\; =\; H\_\}^1(\backslash operatorname\backslash mathbf\_q,\; G)$   vanishes,〔This is "unwinding definition". Here, $H^1(\backslash mathbf\_q,\; G)\; =\; H^1(\backslash operatorname(\backslash overline/\backslash mathbf\_q),\; G(\backslash overline))$ is Galois cohomology; cf. Milne, Class field theory.〕 and, consequently, any ''G''-bundle on $\backslash operatorname\; \backslash mathbf\_q$ is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of finite groups of Lie type. It is not necessary that ''G'' is affine. Thus, the theorem also applies to abelian varieties (e.g., elliptic curves.) In fact, this application was Lang's initial motivation. If ''G'' is affine, the Frobenius $\backslash sigma$ may be replaced by any surjective map with finitely many fixed points (see below for the precise statement.) The proof (given below) actually goes through for any $\backslash sigma$ that induces a nilpotent operator on the Lie algebra of ''G''. == The Lang–Steinberg theorem == gave a useful improvement to the theorem. Suppose that ''F'' is an endomorphism of an algebraic group ''G''. The Lang map is the map from ''G'' to ''G'' taking ''g'' to ''g''−1''F''(''g''). The Lang–Steinberg theorem states that if ''F'' is surjective and has a finite number of fixed points, and ''G'' is a connected affine algebraic group over an algebraically closed field, then the Lang map is surjective. 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「Lang's theorem」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.